SCIENTIFIC REVIEW AND CRITIQUE OF

                             R. L. PATTERSON'S:

           PRODUCTION, MORTALITY, AND POWER-PLANT ENTRAPMENT OF

                 LARVAL YELLOW PERCH IN WESTERN LAKE ERIE*
                       D. S.  Vaughan and S.  M.  Adams

                      Environmental Sciences Division
                       Oak  Ridge National  Laboratory
                            Oak Ridge,  Tennessee
                                     for
                    U.S.  Environmental  Protection Agency

                                  Region V
                                February 1981
*Research  sponsored by Region V of the  Environmental Protection Agency,
 under  Interagency Agreement 40-650-77  (Task Order No. ORNL/EPA-V5)
 with the  U.S. Department  of Energy under contract W-7405-eng-26 with
 Union  Carbide Corporation.
                         By acceptance of this artic.lc, the publisher or
                         recipient acknawlodjjcs the U S. Govt rnment's
                         right to retain a non - ex-.iuiiiye, royalty - free
                         license in and to any copyright covering the
                         article.

-------
                      SCIENTIFIC REVIEW AND CRITIQUE OF
                              R.  L.  PATTERSON'S;
            PRODUCTION. MORTALITY, AND POWER-PLANT ENTRAPMENT OF
                   LARVAL  YELLOW  PERCH  IN WESTERN LAKE ERIE

                        D. S. Vaughan and S.  M.  Adams

                                   SUMMARY

      This   report  presents   a   review  and  critique  conducted  for  EPA,
 Region  V,   of   the   report,   Production,  Mortality,   and   Power-Plant
 Entrapment  of   Larval   Yellow   Perch   in    Western   Lake   Erie.   by
 R.  L. Patterson  (1979).   Patterson  enumerates  six  objectives  for  his
 study,  including:
 1.  estimate  the  production  of  larval  yellow  perch  in  1975-76  in
    Michigan and Ohio waters of the western basin of Lake Erie,
 2.  estimate  natural  mortality  among  larval   yellow  perch  for  the
    25  days following hatching,
 3.  estimate the number  of larval  yellow perch entrained and killed in
    1975-76 in cooling water used by the Monroe  Power Plant,
 4.  estimate  the  percentage  of  total   larval  production  in  Michigan
    waters lost to the cooling waters used by the Monroe  Power Plant,
 5.  estimate the percent  loss  in young-of-year  recruitment attributable
    to  entrainment mortality at the Monroe Power Plant,
 6.  estimate the loss to  the  yellow perch fishery in western Lake  Erie
    attributable to entrainment and impingement mortality at  the Monroe
    Power Plant.
 The  first  objective  is  met  through  Patterson's  "materials  balance"
model   (discussed  in  Sect. 4  of  this  report).  Estimates  of  natural
mortality for larval  yellow  perch  (objective 2)  are not obtained  from
 the "material balance" model,  but  are stated without adequate  justifi-
cation  (Patterson 1979, p. 114).   Ranges of estimates of larval yellow

-------
perch entrained in the cooling water used by the Monroe  Power  Plant  are
provided  and  combined  with subjective  weightings for which no  justifi-
cation is given; and little explanation is given for the  source of data
used  in  Appendix  E  of  Patterson  for estimating  the  through  plant
entrainment  mortality   (objective  3).   Objective  4  is  obtained from
objectives  1  and  3;   and  objective 5  is  obtained  from  objective   4,
assuming  no  density-dependent  mechanism occurs  during  the larval life
stage.  Objective 5 was attempted via the  Patterson models 1 and 2.
     This report provides  a detailed discussion and  evaluation of  the
data  used in the  Patterson "materials balance  model"  and the  use   of
this  model   to  obtain  estimates  of  larval   natural   survival   and
abundance.  A principal  deficiency of  the  Patterson  study  is the lack
of attention  to detail  and  clarity in  presenting the data used in this
model  to  obtain estimates  of larval  abundance.  An  additional  defi-
ciency relates to  objective 2, in that estimates  of the daily instan-
taneous natural mortality  parameter  p  are  not estimated, but are given
without adequate explanation or justification.
     The  two models (models 1  and 2) proposed by  Patterson to  estimate
long-term  impacts  on  the  yellow  perch  fishery  are  also  critically
evaluated.  A problem of significant importance relates to a duality  in
meaning  of  the  life-stage  survivals,  which  should  represent   the
survival  of an individual  having  an  average age within its life  stage.
The  survival  must  also  represent  the probability  of  surviving   the
entire life stage because of impacts to later life stages.  A secondary
problem relates to the  lack of stability  in model 1  for the  ranges   of
life history parameters.  Furthermore,  the abundance estimates obtained
from the materials  balance  model  are  not used  in  either model 1 or 2.

-------
     The  weaknesses  and  strengths  of  the  Patterson  approach,   an
introduction to  an age  structure  approach  to assessing  the  long-term
impacts  resulting  from  operating  the Monroe  Power  Plant,  and  future
data  needs  to  improve  further assessments  are also  present  in our
report.  The age structure model described in detail  in Appendix  A will
appear in Quantitative Population Dynamics,  edited by D. G.  Chapman and
V. F. Callucci   (International  Cooperative Publishing House,  Fairland,
fid.).  Ranges of losses  to the yellow perch fishery have been  estimated
based  on this  age  structure  model  and  entrainment  and   impingement
estimates from Patterson and  C.  D.  Goodyear  (1978).   A higher range  of
impact estimates were  obtained based on  C.  D.  Goodyear's estimates  of
entrainment and  impingement  due to  much  higher  estimates  of  impinged
yellow  perch.   However,  lack of  adjustment  in  Appendix  A of  larval
concentrations in western Lake  Erie  for  gear  bias is believed to  cause
a  considerable  underestimate  of larval  abundance  and  results  in   an
overestimate of conditional  mortality rates and projected losses  to the
fishery.  On the  other  hand,  the  assumption  of  an  equilibrium  popula-
tion, when  the  population is  declining,  would lead  to an overestimate
of age class 0 survival  and abundance of  later life stages.   Additional
assumptions and sources of error are discussed in Appendix A (Sect. 4),
including  three sources of  error:   (1) natural  variability  in the
environment, (2) errors  in the  estimation of  both model  parameters and
inputs, and (3) errors in model structure.

-------
                                 CONTENTS


                                                                    Page

SUMMARY	   ill

1 .  INTRODUCTION	     1

2.  SITE AND PLANT DESCRIPTION	     3

3.  YELLOW PERCH FISHERY IN WESTERN LAKE ERIE  	     6

4.  LARVAL YELLOW PERCH ABUNDANCE - EVALUATION
    OF METHODOLOGIES 	     9

    4.1  Status of Larval Concentration Data 	     9

    4.2  Materials Balance Model  of Larval Production  	    11

    4.3  Data Presentation and Reduction	    16

5.  LONG-TERM IMPACTS ON YELLOW PERCH FISHERY -
    EVALUATION OF MODELS 	    18

    5.1  Model 1	    19

    5.2  Model 2	    23

6.  DISCUSSION	    26

    6.1  Weaknesses and Strengths of Patterson's Approach  ....    26

    6.2  Alternative Age-Structure Approach  	    27

    6.3  Future Data Needs	    29

REFERENCES	    32

Appendix A.  AN AGE STRUCTURE MODEL OF YELLOW PERCH
             IN WESTERN LAKE ERIE	   A-l

Appendix B.  MISCELLANEOUS EDITORIAL COMMENTS  	   B-l

-------
                            1.   INTRODUCTION

     Oak  Ridge  National  Laboratory  under  contract  with  the   U.S.
Environmental  Protection  Agency,  Region  V  (EPA  V), has  undertaken  a
scientific  review  and  critique  of  the  report  by  R.   L.   Patterson
entitled  Production,  Mortality,  and  Power-Pi ant  Entrainment  of  Larval
Yellow  Perch  in  Western  Lake   Erie  (Patterson 1979).   The   following
principal questions  (as called  for  in  the  EPA task description)  were
considered as part of the review and critique:
1.  Are the models appropriately applied?
2.  Are the assumptions reasonable?
3.  What are the weaknesses and strengths of  the total approach?
4.  What could be done  or  ought  to be done to  overcome or more
    fully support weaknesses or problem  areas?
5.  What  strengths  need to  be  pointed  out or expressed more
    fully?
6.  Are the estimates of impact about  right?
7.  What other or better approaches  (models  or other) might be
    pursued for comparative purposes?
8.  If additional data  are collected, where  should more effort
    be applied?
9.  If the report is  unclear  or  cannot  be understood as it now
    stands,  what information  is  needed  to make it clearer and
    more susceptible to independent evaluation?
Considerable overlap exists among  the first  seven  questions,  and these
will be addressed as similar problems  in  the  following critique.
     Patterson's report follows  a  logical progression  in  the develop-
ment of his modeling approach.   Data for concentration of larval yellow
perch in  the  U.S.  waters  of  the western  basin of Lake  Erie  are  pre-
sented  in  Sect. 3.   These  data  are  used  as  input  to  a  "materials

-------
balance model"  to obtain  estimates  of  larval  yellow perch  mortality and
production  (abundance)  (Sect.  4).   Two models are  then  described  which
assess the  loss  to  the  yellow  perch fishery  attributable to  entrainment
and  impingement mortality at the Monroe  Power Plant (Sect.  5).   Long-
term  losses to  the yellow  perch  fishery are  estimated  (model  1)  by
using an extention of Horst's (1975) "equivalent adult"  model.
      In this  review and critique,  the  western basin of Lake Erie  and
the  Monroe  Power   Plant  are  described  along  with  the  yellow  perch
fishery of  Lake  Erie.   Next, we examine the methodology for  estimating
larval abundance used  by Patterson and  examine models  1  and 2,  which
consider the  "long-term" effects  (i.e.,  at  equilibrium following  the
onset  of  power-plant  impact)  of  entrainment  and  impingement  by  the
Monroe Power  Plant  on  the yellow  perch fishery in western  Lake  Erie.
Finally,  we summarize  the   weaknesses  and  strengths  of  the modeling
approaches   used by  Patterson,  compare  an  alternative age  structure
approach   (see  Appendix   A)  with  Patterson's  approach,  and  briefly
discuss future data  needs.

-------
                     2.  SITE AND PLANT DESCRIPTION
     The Monroe  Power  Plant is located  on  the western  shores  of Lake
Erie near  the city  of  Monroe,  Mich.  (Fig.  2.1).   Lake  Erie  is  the
                                                                     2
fourth  largest  of  the  Great  Lakes  by   surface  area  (25,690  km  )
(Hartman 1973, Nepszy 1977) and  receives about  95% of its total inflow
(Hartman 1973)   (annual  average  of  5465 nT/s) from  the three upper
Great  Lakes   (Lakes  Huron,  Michigan,  and  Superior)  via  the   Detroit
River.   The   annual   average  outflow  to  Lake  Ontario   is  about
5720 m /s,   implying  a  theoretical   flushing  time  of  approximately
920 days (Hartman).
     Lake  Erie  is naturally divided  into  three basins  (Fig  2.1),  the
western  basin being the smallest  and  shallowest.   This basin,  from
which the Monroe  Power  Plant  receives its cooling  water, has about 13%
of the total  surface area of Lake Erie, but  only 5% of the total volume
(Hartman 1973,  Nepszy  1977).  The  average  depth of  the western basin
(7.4 m) is only 40% of  the  average depth of all of Lake Erie (18.5 m).
The western basin  has  a flushing time of about 60 days (Hartman),  and
is  considered to  have the  most important  fish spawning and   nursery
areas in Lake Erie (Hartman 1973,  Nepszy 1977).
     The Monroe  Power  Plant has  four coal-fired units  with  a   maximum
expected  net  capacity  of  3150 MW  (Detroit  Edison   Company   1976a).
Unit 1   came  on-line  in  May 1971,   and   the   remaining  units began
operation  in  annual  increments:   June 1972,  March  1973, and  May 1974
respectively.   Cooling water is obtained from the  Raisin River  through
an  intake  canal,  which  is  screened  by   3-in.-mesh  trash   racks  and
traveling screens with  3/8-in.-sq. openings.   The  maximum flow  through

-------
                                ONTARIO
MICHIGAN


    LAKE ST. CLAIR


       DETROIT •
 MONROE
 POWER
 PLANT
 TOLEDO
                  OHIO
                                                              ORNL-DWG 80-16976 ESD


                                                                  /
                                                                    BUFFALO
                                                             NEW YORK
                                  CLEVELAND
                                                  PENNSYLVANIA
0  20 40 60 80    (km)
 •—. '  . • .  •.	•   DISTANCE
0 10 20 30 40 50    (mi)
 Fig.  2.1.   Drawing  of   Lake   Erie,  showing  detail   of  the   western,
            central, and  eastern basins and location  of the Monroe  Power
            Plant.

-------
the Monroe  Power  Plant is  about  92 m /s,  and up  to 20% of  the water
is recirculated in  the winter  to  prevent icing.   Chlorination, lasting
about 30 min,  occurs  twice daily in  the summer  and once daily  in the
winter.   From April  1975  through   May  1976  the  Monroe   Power  Plant
operated at about 56%  capacity  and  at about  64%  maximum cooling water
flow  (59 m /s),  resulting  in AT's  across  the  condensers   of  2  to
16°C (mean 9°C).

-------
              3.  YELLOW PERCH FISHERY IN WESTERN  LAKE ERIE

     Western Lake Erie is in  the  center  of the geographic range of the
yellow perch (Perca flavescens).  Spawning usually occurs in late April
and  early  May  as the  water temperature  reaches  7 to 10°C  (Van Meter
1960, Wolfert  et  al.  1975).  Spawning takes  place at night  and early
morning over submerged vegetation and brush,  which anchors  the charac-
teristic ribbon-like egg mass  (Muncy  1962,  Scott and  Grossman 1973).
Hatching time  varies with  temperature,  ranging from 6 days at 20°C to
50 days at 5 to 6eC (Ney 1978).   Patterson  (1979)  reports a  range of C
to 12 days  from spawning until  hatching.  Yellow  perch  hatch  at about
5 mm in  total  length,  and  the  yolk  sac  is  absorbed at  about 7 mm in
total length (Ney  1978,  Ney and  Smith 1975).   Schneider (1973), while
studying  size-selective  feeding  of  yellow  perch  on   zooplankton  in
ponds, found the onset of  feeding to be a critical  phase of  life.  He
further found  that the survival  of larval yellow  perch  and  the forma-
tion of year-class strength depend  on the density of older  perch.  The
swimming ability  of larval  yellow  perch is  insufficient  to sustain
directed  movement  against  wind-driven  surface  currents (Ney  1978).
Larval  yellow perch are  initially photopositive and pelagic,  but become
demersal  at total  lengths of about 25 to  40 mm (Ney 1978, Ney and Smith
1975).  Larvae  are more  abundant offshore in  western  Lake  Erie, where
they concentrate  near  the  bottom,  while  juveniles  are   common  in the
shallow  inshore waters   (Detroit  Edison  Company  1976a).  Noble (1975)
found that yellow  perch  mortality rates  in Oneida Lake  were   inversely
proportional  to the biomass of young  fish  and that growth  rates were

-------
not strongly  density  dependent.   Adult yellow perch are  active during
the day, with  peak movement  shortly  after  sunrise  (offshore migration)
and before  sunset (onshore  migration)  (Ney  1978).  Growth,  fecundity
and survival data for yellow  perch  are  given in Sect. 2.1 of Appendix A.
     Yellow perch is  one  of  the two most  exploited  percid populations
in North  America (the other  being walleye) (Ney 1978).   Marked varia-
tions in year-class strength  are characteristic  of  yellow perch popula-
tions (Hartman 1973,  Nepszy  1977).   Annual  landings  for  Lake  Erie from
1923 to 1972  averaged  2600 metric  tons (Hartman  1973).   The commercial
harvest of  yellow perch  rose slowly from  1915  to 1927,  averaged 5700
metric  tons  between  1927 and  1936  (this upsurge  attributed to  the
decline in lake herring), and increased greatly  in  the  1950s (primarily
in Canadian waters) (Hartman  1973, Nepszy 1977).   Catch  per unit effort
was high, with low effort from 1950 to  1970 in  the western basin; how-
ever, since 1970, effort has increased  in  the west and  central basins,
while  catch  per unit  effort   has decreased  in  all  Canadian  waters
(Nepszy 1977).
     Attempts  to  relate  yellow  perch  and walleye year-class  strength
fluctuations  to  environmental   and  biological   variables  have  been
largely unsuccessful  (Ney 1978),  although  Smith  (1977)  relates  year-
class strength of yellow perch  and walleye to climatic  factors.  Strong
year-classes of yellow perch  have been  found in  1952, 1954, 1956, 1959,
1962, and  1965,  with relatively strong year  classes in 1970  and 1975
(Hartman  1973,  Nepszy  1977).    Coincidental  strong  year-classes  of
walleye have  been found  in 1959,  1962,  1965, and 1970  (Hartman  1973,
Nepszy 1977).  These coincidental  strong year-classes may  imply either

-------
a  similar environmental  mechanism controlling  the  success  of hatch
(such as  rate and regularity  of water  warming  during spawning)  or a
common controlling predator  (rainbow  smelt  has  been suggested)  (Nepszy
1977).  Wells  (1977)  has suggested that the sudden  decline in yellow
perch in Lake Michigan during the early and mid-1960s was brought on by
the  interference of  nonnative  alewife with  yellow perch reproduction.
This decline was preceded by a period of conspicuously high commercial
production, although  the sport  fishery declined a few years earlier.

-------
      4.   LARVAL  YELLOW  PERCH ABUNDANCE - EVALUATION OF METHODOLOGIES

      The purpose  of  this section  is  to  describe  briefly  the  data
 available to Patterson  (1979)  at  the  time of his analysis  (and  listed
 in  his references), to  discuss  Patterson's  "materials  balance  model,"
 which is used to obtain estimates of  larval abundance, and to  discuss
 the  data included  in  Patterson's (1979,  Sect.  3) report  in  terms  of
 clarity  of  presentation  and accessibility  of data  in  a  form susceptible
 to an independent evaluation of his methodology.

                 4.1  Status of Larval  Concentration Data
      In  1975 and 1976,  under the sponsorship of the  U.S.  Environmental
 Protection  Agency,  Region  V,  a  sampling  program  for the western  basin
 of Lake  Erie  (U.S. waters) was conducted jointly by the  Center for Lake
 Erie   Area   Research   (CLEAR),   the   Michigan  Department  of   Natural
 Resources  (Michigan DNR),  and  Michigan State University  (MSU).   Inves-
 tigations  by  CLEAR included  the  Ohio  waters  of  the  western  basin
 (Herdendorf  et al. 1976,  1977),  while the  Michigan DNR  surveyed  the
 Michigan  waters  of  Lake Erie  (Hemmick  et  al. 1975,  Michigan DNR  1975,
 1976).  The area near the Monroe Power Plant  (Cole 1977) was sampled  by
 MSU.
      The Michigan DNR and  CLEAR  sampled with  nets  having different mesh
 sizes  (471  and  760  y  respectively),  while MSU  used  plankton nets  with
 571-y mesh  (C.  D.  Goodyear 1978).  The comparability of the data  col-
lected by these agencies is subject to question because  gear efficiency
most  likely differs between  the  different   sampling  gear.   Patterson

-------
                                   10
ignores  the  question of  gear efficiency,  both  in  terms  of combining
data from  disparate  sampling  gear and the resultant underestimation of
larval   perch  concentrations.   Numerous  papers  discuss  the  problem of
gear   efficiency   (Schnack 1974,   Bjorke   et al.  1974,  Murphy   and
Clutter 1972,   Leithiser  et al.  1979, Tranter  1968).   Inaccuracies in
larval   abundance estimates  can be introduced by  changes in filtration
performance (interaction between  net  and  water),  avoidance of  sampling
gear (especially by older  larvae),  and  the loss  of organisms through
the meshes  (extrusion).   Filtration  performance (including clogging of
nets),  avoidance of  gear, and  extrusion  through  the mesh will  all  lead
to  an   underestimate of  the  larval  concentrations  and  hence  their
abundance.
     Patterson (1979; Fig. 2)  presents the location of the western  Lake
Erie larval sampling stations  in  1975 and 1976.  Samples were obtained
from five  depth  zones  (6-ft  increments  from the  surface)  in Michigan
waters   during  the  periods  June  4  through  September  5,  1975,  and
April  13 through August 3,  1976.   Samples were  obtained  from six depth
zones  (2-m increments from the surface)  and  five  sectors (A-E) in  Ohio
waters   during  the  periods   May  12  through  September 8, 1975,  and
April  12 through July 7,  1976.  Sampling, both  day and night, occurred
irregularly over time  in both Michigan  and Ohio  waters,  ranging  from
weekly   to  biweekly  intervals.   Although  Patterson  (1979,  Appendix A)
finds  differences between day and night samples, he does not attempt to
adjust  upward  the  concentration  estimates to account  for  greater  gear
avoidance in day samples  compared with  night samples.   Patterson notes
the lack  of larval yellow  perch  concentration estimates  for Michigan

-------
                                    11

 waters  during  May  1975.   Larval  abundance  estimates  for  Michigan  waters
 in  1975 are suspect because  data for the start  (and  probably  the peak)
 of  the  spawning season is lacking.
      Entrapment  and impingement data  for the  Monroe  Power Plant  are
 reported  by Cole  (1977),  Detroit  Edison  Company  (1976a,  1976b),  and
 Hubbell  and Herdendorf  (1977).   An  excellent evaluation  of  the  entrain-
 ment  and impingement data  is provided  by  C.  D. Goodyear (1978).   This
 study found that  impinged  yellow  perch  were   seriously  underestimated
 [122,000 according  to Detroit Edison Company (1976a)  vs  her  estimate of
 626,000].   The possibility  also exists that  entrapment estimates in
 the   1976a    report   may   also   be    significantly    underestimated
 (C. D.  Goodyear 1978).   This  underestimate could have occurred because
 entrainment  sampling was conducted  only at 1-  and  3-m  depths, and  the
 intake  canal  is  5  to 7m  deep.   It is  suspected  that  larvae  would be
 more  abundant  near the  bottom of  the  Monroe  Power  Plant intake canal
 (C. D.  Goodyear, 1978).

            4.2  Materials Balance Model  of Larval  Production

      A  brief description of  Patterson's "materials  balance model" is
 given in  the Appendix to Appendix  A.   In  this  model the abundance of
 larval yellow  perch is  viewed as an integration of  several processes,
 including (1)  production of  larvae,  h(t);  (2)  natural mortality  during
 the larval  stage  (25  days),  m(t);  (3)  recruitment to the young-of-year
 life  stage,  r(t);  (4)  net emigration from U.S. waters  of  the western
basin,  v(t);  (5)  losses due  to  entrainment of larvae   in  the cooling
water  of  the  Monroe  Power  Plant,  Eft);  and  (6)  losses   due  to

-------
                                   12
entrainment of  larvae  by  other water users,  L(t).   The rate of  change
                       •
of  larval  abundance,  N(t),  is  then given  by a  linear combination  of
these processes [Patterson 1979, Eq. (3)]:

            N(t) = h(t) - v(t) - r(t) -  m(t) - L(t)  - E(t)   .     (4.2.1)
     Total producton is defined as  all  yellow perch larvae entering  or
hatched in the  U.S.  waters of  western  Lake Erie, including Maumee Bay
(Patterson 1979,  p.  4).   The binomial  distribution  (with  parameters m
and  q  [Eq.  (A-l ]}  is  used  to distribute  total  production  over the
spawning  period  in  Michigan  and   Ohio  waters of  western  Lake  Erie.
Losses  due  to natural mortality  are assumed  to  follow an exponential
decay  with  a  daily  instantaneous   natural  mortality  rate,  £,   while
recruits to  the young-of-year life  stage  (25  days  after hatching) are
given by  Eq.  (A-2).  Since  Patterson assumes that v(t), E(t), and  L(t)
are equal to  zero for  all  values of t,  then net  emigration and  losses
to  all  cooling  water users (including  the  Monroe  Power  Plant) are
incorporated among the natural mortality and production  expressions.
     This modeling  approach  assumes  that  the   binomial  distribution
reasonably reflects  the  input  of yellow perch larvae  to  Michigan and
Ohio waters during the spawning season.   The parameters  m  and q of the
binomial distribution in Eq.  (A-l) are obtained by inspection; that is,
Patterson compares the shapes of curves  generated by different combina-
tions of  these  parameters with  the field  survey data  (larval  concen-
tration as a  function of the day of year).    The parameter d, the  period
of time over  which spawning  is  assumed constant, is also  obtained  by
inspection.   The parameters m, q,  and d  are  given in Table 4.1 for Ohio

-------
                                   13
Table 4.1   Parameter values for Patterson's  binomial
           distribution of spawning by state waters  and year
                               Ohio  waters            Michigan waters

     Parameter               1975       1976          1975        1976


B

m
q
, d (days) ,
(100-m3 units x 107)
(1!) for x:
5
0.
7
9.


10
393

5
0.
7
9.


10
393

5
0.15
14
0.4976

5
0.
14
0.


15
4976

          0                 0.5905      0.5905        0.4437      0.4437
          1                  0.3280      0.3280        0.3915      0.3915
          2                 0.0729      0.0729        0.1382      0.1382
          3                 0.0081      0.0081        0.0244      0.0244
          4                 0.0004      0.0004        0.0022      0.0022
          5                 0.00001     0.00001       0.0001      0.0001

-------
                                   14
and Michigan  waters in 1975  and  1976.   A comparison of  the binomial
coefficients,  (^),  in  Table  4.1  with  those   given   by  Patterson
(1979, pp.  82-83)  will  show that  the  coefficient  given  by 0.00001  was
left  out  of  Patterson's  analysis  for  Ohio  waters  in  1975  and  1976.
Because of  the small  value  of this coefficient, the  effect of leaving
it out would  be  expected  to be small.   However,  the coefficient given
by 0.0022 was  left out of Patterson's  analysis for  Michigan  waters in
1975  [but  not 1976),  and  the effect  of leaving   out  this coefficient
could have  a more  significant  effect  on  predicted  larval abundances.
Since the value of  0.0001  was used in  place  of 0.0022 for the binomial
coefficient for  x  = 4, and 0 was  used  in place of  0.0001 for x  = 5,
then  the  relationship  between h and p  will  be biased in  some unknown
direction.  These  coefficients  are  also missing from the  solutions to
Eq. (A-3) as  given by  Patterson (1979, Appendix F).   These errors will
have  no effect on  Patterson's long-term impacts, since he  does not use
estimates of abundance in models 1 or  2 which assess long-term impacts
on the yellow perch fishery  (Sect.  5).
      In the model,  natural  mortality  is assumed  to occur  uniformly
during the  larval   life stage throughout  the  period of  occurrence of
larvae in the western basin.   The  probability of a  larva  dying on day 1
of the larval stage is the same as the  probability of a  larva dying on
day 25  of  the  larval  stage.  Furthermore,  this probability  remains
constant over  the  entire  period of occurrence  of  larvae.   Patterson's
assumption of 25 days  for  larval  development from  hatching to recruit-
ment  to  the  young-of-year life  stage  will  affect  both  the  natural
mortality and  recruitment terms in Eq.  (A-3);  that  is,  increasing the

-------
                                    15
 life-stage  duration decreases the  dally  Instantaneous mortality  rate,
 p.   A decrease (or  increase)  in  p will affect  the  calibration of  the
 model.  The expression  for  recruitment  [Eq.  (A-2)] also assumes  that no
 density-dependence mechanism occurs during the larval  life stage.
      Patterson  (1979,  p.  114) does not estimate natural  survival  from
 his  materials  balance  model, but  "judges"  that the  "overall   25-day
 survival  is  ...  in  the  2 to 10  percent range."  The  corresponding
 values  of  p  range  from 0.156  to  0.092  day   .   Patterson  (personal
 communication)  argues  that  an inspection of  Figs.  35, 37,  40,  and  42
 (Patterson  1979)  justifies  his selection of  his range for  p (0.09  to
 0.16).  However,  we believe that  no  unique  (p,h) pair  is superior  to
 any  other  "best"  point estimate.   The  direct relation  between p and  h
 allows one  to estimate p or h from  the materials balance model,  given
 an independently  obtained  estimate of  the  other parameter.   Patterson
 (1979, p.  114)  points out  that  this  range  of estimates  of p reflects
 net emigration and water intake losses  other  than these from  the Monroe
 Power  Plant.    If the  natural  mortality  is fixed  so  as  to  already
 reflect these additional  losses,  then  the  addition  of  further  terms
having nonzero  values  to  represent  these  losses  will  underestimate  the
actual larval  production.
     The larval concentration  data  that are  compared  to  the model  for
various combinations  of  p and  h  are  assumed  to   provide unbiased
estimates  of  these  concentrations  during the larval  production season
for  larvae  in  western Lake Erie.   Since  these  data  have  not  been
adjusted  for   gear  inefficiencies,  the   resultant   estimates   of   h
 (productivity)  are   likely   to  be  underestimated.    In   addition,   the

-------
                                    16
 estimates  based  on  Michigan waters  in  1975  are  suspect because  sampling
 was  not  conducted during the earlier portions of the production  season.

                   4.3  Data Presentation and Reduction

      Patterson does not clearly delineate  the four sets of larval  con-
 centration data  that  he uses in his materials balance model for obtain-
 ing  estimates  of  larval  abundance  in Michigan   and  Ohio  waters  of
 western  Lake  Erie in  1975  and  1976.   In order to permit an independent
 evaluation  of Patterson's methodology,  the report  should  contain  the
 following information:
 1.   the  raw  larval  concentration data  for  Michigan and  Ohio  waters  in
     western  Lake Erie  in  1975 and 1976  (should   be  presented  as  an
     appendix with uniform format for the two state  waters and two years
     of data),
 2.   a description in  the main  text of  how  the  data  were  reduced to  the
     form directly usable in estimating larval  production and abundance
     (including comparability of Michigan DNR  and MSU data,  presence  of
     dead larvae  in  Lake Erie,  gear efficiencies,   and pooling  of  data
     over depth and day vs night),
 3.   a  summarization of  these   reduced  data in  tabular  form by  state
    water and year.
 Most of  the  raw  larval  concentration  data  appear in Patterson's report
 in Tables 1 (Michigan waters in 1975), 9 (Michigan  waters in 1976),  12A
 (Ohio waters in  1975),  and  12D (Ohio  waters in 1976).   Additional  lar-
 val concentration data appear  in Tables 2-7 (MSU night samples in 1975)
 and  Table  A.I  (Michigan waters in 1975).   It  is  not clear which  data
were used in Patterson's analysis.
     No tabular summary of larval concentrations in Michigan waters  for
1975 is  given  for  the raw data presented  in Tables 1-7 and  A.I;  only
Fig.   36  gives  any  indication  of  the  magnitudes  of  the  reduced

-------
                                   17
concentration  data.   Tables  9,   12A,   and   12D  are  summarized  in
Tables 10, 12C,  and  12F  respectively.  In comparing the  raw data pre-
sented in Table  9 with the reduced  data presented in Table 10, the raw
data in Table 9  for 6/14-29 were apparently not  used in the summary in
Table 10  (Patterson 1979).  Further confusion  ensues as a result of two
dating systems  used interchangeably  among  Tables  1-12  and Appendixes
A-D  (Patterson   1979).   Only  the   "Note" below  Table  10  presents  a
correspondence between  these  two   dating systems;  however,   there  is
little  agreement between  the  month/day  system  in Table  9  and  the
day-of-the-year  system  in  Table 10.

-------
                                    18
   5.  LONG-TERM IMPACTS  ON  YELLOW  PERCH FISHERY - EVALUATION OF MODELS

      In  this  section  we  discuss  two models  which  Patterson  (1979,
 Sect.  5)  employs  to  assess  long-term   Impacts  by  entrainment   and
 Impingement  at  the  Monroe   Power  Plant  on  the yellow  perch  fishery.
 Model  1 considers only the entrained and  Impinged portion of the yellow
 perch  population  and  is  an  extension   of  the   "equivalent adults"
 approach  introduced  by  Horst (1975).   Model  2  describes  the entire
 population of  yellow perch   and  includes  a density-dependent  component
 based  on  the   logistic  function.   There is  no  indication  that   the
 estimates  of  larval  abundance  are  incorporated  in  either  of  these
models.
     An  underestimate  of the  number  of  entrained  or  impinged yellow
 perch  will cause an  underestimate  in  predicted losses  by both models.
 Gear  inefficiencies  in entrainment samples (Sect. 4.1)  will  lead to an
 underestimate  of the  number  of  larvae  and  young-of-year  entrained.
 C. D.  Goodyear  (1978)  discussed  the  question  of  inaccuracies  in
 impingement estimates and presented estimates  of  impinged yellow perch
based  on  actual  counts   considerably   higher   than  Detroit  Edison
Company's  estimates   (513%).   Errors  in  estimates  of  survival   of
entrained and  impinged yellow perch  will also  cause errors  in  model
predictions.   Patterson (1979, Appendix E) discussed estimation methods
and an example to obtain an  entrainment survival of  about 0.7, but  the
source  of  the  data  was   not   given.    Patterson   did  not  discuss
impingement survival.

-------
                                   19

                              5.1  Model 1
     Model  1  follows  only  the  subpopulation  of   the  yellow  perch
entrained and impinged at the Monroe  Power  Plant.  Model  1  is a mathe-
matically continuous  extention   to  Horst's   (1975)  "equivalent adults"
approach, which considers the probability of survival from the start of
an entrained  or  impinged life stage  to "adulthood"  (usually the first
year  of maturity).   Horst's  model  is  open-ended  in that  it assesses
entrainment  and  impingement  losses  of a  single  cohort  in  terms  of
"equivalent adults" and  does  not provide for any  feedback  in the form
of  a  reduced  number  of adults  spawning  a  reduced  number  of  eggs.
Patterson's model  1,  however,  closes this  loop  by describing the rate
of  change in abundance  of a  subpopulation of  entrained  and impinged
yellow  perch  as a  differential  equation  [Patterson 1979:  Eq.  (21)],
which  includes   a   term  for  the  lost  reproductive potential.   Each
entrained or impinged life  stage  is  adjusted  to  the  age of  recruitment
(age  class  2),  similar to  Horst's   "equivalent  adults."    Both  the
"equivalent adults"  approach  and  model 1  assume  no density-dependent
mechanism.
     Patterson presents  ranges  of  estimates for  several  life-history
parameters:   (1 )  fraction  of larvae  surviving  to   the  young-of-year
stage,  e  (0.02,  0.1);  (2)  fraction of young-of-year surviving  to age
class  1;  s  (0.346,  0.575); (3)  annual  instantaneous fishing mortality
rate,  f  (0.22, 0.95);  (4)  annual  instantaneous  natural  mortality rate,
m (0.22, 0.29); and  (5)  mean  annual  rate of larval production, Y (375,
3600).  These parameters and numbers of fish entrained and impinged are
the  sole  inputs to  model   1.   The range of  the  parameter  e was  not

-------
                                   20
estimated,  but judged  reasonable  in Patterson's  Sect.  4.   The  para-
meters  s,  f,  and m  are obtained from the  literature.   The last  para-
meter, y ,  is  calculated from the product of  hatching success (0.25 to
0.5),  number   of  eggs  spawned  per  sexually  mature  female  (10,000 to
30,000),  and   the  proportion of  sexually  mature  female  spawners per
individual  (0.15  to  0.24).   Patterson assumes  a  sex ratio  of 1:1.  A
stable  population age  structure  is  assumed  in estimating the number of
eggs spawned and the fraction of mature  spawning females.
     Model 1 [Patterson 1979, Eq.(2D] can be  written  in  the  form

                       N = a(a + b-N)  - (m + f)N  ,                (5.1.1)

where  a is the  number of entrained  and  impinged  yellow  perch  as age
class   2  "equivalent   adults,"   b  is  the  reproductive  potential
         ™n
(Y.e.s.e ') of age  class  2  fish,  a is  the  proportion of  age  class
2 fish that will die due to natural and fishing causes after six years,
m and  f are the  natural and  fishing annual   adult  instantaneous mor-
tality  rates,  respectively,  and  N  and N are  the  rate  of  change and
absolute  adult  population   size  respectively.   In  the  absence  of
entrainment and impingement  (a  =  0),  model  1  is stable  if  and only if
production perfectly  matches mortality (ab =  m  +  f).   In  this case
the "impacted" population is  nonexistent  (N =  0).   An artifact  of the
entrainment and impingement  terms appearing  as constants  in Eq. (5.1.1)
allows  for  stability  when   (m  +   f)  exceeds (a-b).   Thus,  in  an
essentially density-independent model, we have feedback  which controls
the "impacted" population so  as to allow for an  "equilibrium" popula-
tion size.   However, the range  for  (m  +  f)  is  0.44  to  1.24,  while

-------
                                    21

 the  range for  (a-b)  is 1.23  to 159.86.   Since  (m  +  f) is  generally
 less  than  (orb),  model  1  will  generally  not  approach  an  equilibrium
 population  size (Patterson  1979).   To  obtain  equilibrium  conditions,
 Patterson has resorted to  assuming  a value  for y  of 15  (Patterson  1979,
 Table  24).   Patterson assumes  this  value  after he has already  presented
 a  range of  "reasonable" parameter values  that  does  not include  this
 value.   Since  early  life  history survivals  (s and  e)  are  usually
 poorly  known for highly fecund fish,  these values  rather than y should
 be  adjusted if  "equilibrium"  conditions are  to  be maintained.   Also
 note  that  a value for m of 0.29 was  used compared  with its  range  of
 0.22 to 0.29, and values for f of 0.52 and  0.95 compared  with  its  range
 of 0.22 to 0.95  (Table 24, Patterson 1979).
     Horst's  (1975)  "equivalent adults"  methodology assumes  that all
 entrained  larvae  (or  any  other  life  stage   that is   entrained  or
 impinged)  have  just  entered  that  life  stage  (C.  P.  Goodyear 1978).
 However,  larvae  that have  survived 24 days  of  the  larval  life  stage
 have a much  higher probability of  surviving to  sexual maturity  than  do
 recently hatched larvae.   Thus a serious  underestimate of the  potential
 impact  due  to   entrainment  by  a   power  plant  can  result from   this
 approach  without correcting  for this  source  of  bias   (Adams  et al.
 1979).   In  Patterson's model  1  he has  adjusted  larval  survival   (e1)
 assuming that the mean  age of  a larva that is entrained  is five  days.
 He  (Patterson 1979,  Appendix  H) provides some justification  for   this
assumption based on  a  probabilistic argument.   An alternative approach
would be to  assume  an exponential   mortality  [Eq.  (A-2)]  and  calculate
the expected age of an individual larva  (t):

-------
                                    22
                                  e"25p[25+
                                                     '
 where  25 is  the  duration of  the  larval  stage  in  days,  and  p is  the
 daily  instantaneous natural mortality rate.  From Eq.  (5.1.2)  we obtain
 as   the  expected  age  of  an  individual  larva  values   ranging   from
 6  (p = 0.16)  to 8  days  (p  =  0.09), compared  with Patterson's (1979,
 Appendix H) estimate of 5 days.   The expected  survival  for  an  "average"
 larva  to the young-of-year stage would  then be  given by

                               e1  = e-tP .                      (5.1.3)
 Patterson  (1979,  Table 24) obtains as  a  range for  e1  of  0.08 to  0.13
 and  uses  this  range  in place of  e in his  analysis.   This adjustment
 of e might  be an acceptable approximation if  the  larval  life stage  is
 the only entrained or  impinged life  stage as a result of the  operation
 of  the  Monroe  Power  Plant;   however,  later  life  stages   are   also
 entrained and/or impinged.  Thus these  later life stages (young-of-year
 and age classes 1  through 7)  should have their survival rates adjusted
 similar  to  that  in Eq.  (5.1.3).   Since  model  1  considers  only  the
 entrained and impinged portion  of  the  population,  and  those yellow
 perch  lost  to  entrainment  during  the  larval  life  stage cannot be
 entrained and/or  impinged  again  during  a  later  life stage,  larvae
 should experience  the unadjusted  survival  when entrained  or impinged
 during a  later  life stage.  Hence model   1  needs  to  track separately
each life stage entrained or impinged.   In  its present form,  model  1
cannot account in an adequate manner for  power plant impacts occurring

-------
                                   23
other  than  at the start of  a  life stage  and can  adjust  only  for the
first life stage entrained or impinged.
     In  Table  24 (Patterson  1979), identical  ranges  for  the number of
age  class 1  (IA1)  and age  class 2+  (age  class 2  and  older)   (IN)
impinged  yellow  perch  are  given  (50,000  to  100,000).    Since  most
impinged fish are younger than those in age class  2 (Sharma  and Freeman
1977),  the  assumption of uniform  impingement over all age classes of
yellow  perch  would  lead  to  an  overestimate  of  the  potential  impact.
Tables 34 and 35 in  Patterson (1979) provide weightings used to combine
estimates  of impacts  to  the  yellow  perch  fishery   (Patterson  1979,
Tables 25 to 32) based on the range of parameter values and  entrainment
and  impingement  impacts presented in his  Table  24.   These weightings
appear  highly  subjective, and  there is  no  attempt  in  the Patterson
report  to  justify  them.   These  weightings  imply  entrainment  and
impingement losses in larvae to  be 11,000,000, young-of-year losses to
be  100,000,  and losses  to  both  age  classes  1  and  2+  to  be  67,500.
C. D. Goodyear (1978) suggests that the best estimates for these losses
are  1,351,886   (larvae),   3,614,391    (young-of-year),   181,668    (age
class 1), and 441,159 (age class  2+).

                              5.2  Model  2

     Model  2  considers  the  entire  yellow  perch  population  in  Lake
Erie.  Patterson maintains the "equivalent adults" approach through his
coefficients  for   E,,   (s-e),   E    +  I    (s),   and   I.   (1)   [see
                    —•*           y      y                "
Eqs.   (21.1)   and   (23)].    However,   model   2   now   introduces   a

-------
                                    24
 density-dependent   factor  into  his  model   in  a  logistic   fashion
 (Patterson  1979);  from  Eq.  (23);
                                                                 (5.2.1)

 where  K  is the  habitat carrying  capacity  (number  of  adults).   This
 model  results  in an  S-shaped  growth  curve  of the  population  which
 approaches  an  equilibrium  population  size   (regulated  by  its  life-
 history  parameters  and K).  The  levels  of entrainment and  impingement
 will  reduce the  equilibrium  population.   Patterson  (1979,  Table  36)
 assumes  a  value  of  5  x  107  for  K without any  justification  pre-
 sented.   However,  Patterson  uses  model   2  primarily  to  explore  the
 effects  of  varying  y on  population  losses,  rather  than  to  provide
 specific impact predictions.
     Since  the coefficient of  the  density-dependent  term  in  model  2
 (S'E-Y)  adjusts  the  population  to  age  class  1  "equivalent  adults,"
 density  dependence  is allowed  to occur  throughout  the  first  calendar
year of  life  (spawning  to  December  31).  The timing of  the  density-
 dependent mechanism  can affect  the  resultant  predicted losses  to  the
 fishery.   Power-plant  impacts  that  occur  prior  to  or  concurrent with
 the density-dependent  mechanism  will result  in  less  of  a  reduction to
 the fishery than  would comparable impacts  occurring  after  the density-
dependent mechanism  (Christensen  et  al. 1977).  Allowing  the timing of
the  density-dependent  mechanism   during   the  young-of-year  stage
moderates power plant  impacts  at this stage of  development.   However,
most  density-dependent  mechanisms   are  thought   to  occur  during

-------
                                   25

critical periods, the most important of these  is during the  larval  life
stage  when  the  yolk  sac   is  consumed  (Gulland  1965).    If  density
dependence  occurs  during the  larval  stage,  model  2 will   tend  to
underestimate  the actual  impact  for a  specified  set  of  life  history
parameters and entrainment and impingement estimates.
     In its present form, model 2  suffers from the same duality in  the
life-stage survival rates that  model  1  suffers from.  Either  the life-
stage  survival  values  represent  an  average  survival   (for  an  average
aged individual),  or  they represent  survival  through  the  entire  life
stage  for  those that are  impacted in  a  later  life stage.   Patterson
uses a different range for  Y in  model  2 than  he did in  model 1.  A
value of 15 was used in  model 1, while the values  50, 75, 300, and  1500
were used  in  model  2 (Table  26,  Patterson 1979).   This  compares  with
his original  estimated range  of  375 to  3600.

-------
                                   26

                             6.  DISCUSSION

     The purpose of  this  section  is  to summarize some of the  important
weaknesses and  strengths  of Patterson's approach from  Sects.  4 and  5,
to  introduce   an   age   structure   approach  which   is  compared  with
Patterson's approach, and  to describe briefly  future  data  needs which
would  improve any  further  analyses  of long-term impacts of entrainment
and impingement of yellow perch at the Monroe Power  Plant.

          6.1   Weaknesses and Strengths of  Patterson's  Approach

     The approach  in the  Patterson  (1979)  report consists primarily  of
two parts.  The first is his materials balance model,  which is used  to
estimate  the  larval production parameter,  h.  Values  for  the  daily
instantaneous natural  mortality  rate, p,   are given  without  adequate
justification.  Larval   abundance  in Michigan  and  Ohio waters  in 1975
and 1976 are  obtained  by Eq.  (A-4).   The  estimate  for Michigan waters
in  1975 is  suspect,  since the  peak  spawning period was apparently
missed.  The abundance  estimates obtained  from this model  are  not used
in model 1  (or in model  2).
     Model  1  is primarily  used to  address the question  of  long-term
impacts on the  yellow  perch fishery  due to entrainment and impingement
mortalities of the Monroe Power Plant.  The use of model 1 by  Patterson
has several  flaws,  including   (1) general  lack of stability  for  the
range in the life-history parameters presented by Patterson and (2) the
duality problem with life  stage survival estimates  as presently incor-
porated into the model.   This second  problem also applies to model 2.

-------
                                    27

                  6.2  Alternative Age-Structure Approach

      A detailed  description  of an  age-structure  approach  to  modeling
 the effects  of entrainment  and  impingement  at  the  Monroe Power  Plant on
 the yellow perch  fishery  is presented in  Appendix  A.   This approach was
 selected  primarily because  the impacts in  question are not  uniform in
 effect over  the  life  span  of  yellow perch.   As  fish  grow  longer  and
 swimming  capabilities increase, they become less  susceptible to  power-
 plant impacts.   Some  life stages are more  susceptible  to  plant passage
 entrainment  mortality  than  other   life   stages.   Younger  fish   are
 impinged  more  than older  fish,  both  because of  their  greater abundance
 and greater  susceptibility.
      The   age-structure  model  in   Appendix  A  uses   entrainment   and
 impingement  estimates  from   Patterson.    Our  model    incorporates   a
 density-dependent  mechanism only during the larval life stage,  provid-
 ing  for  a  density-independent  effect to  later  life  stages.  Three
 levels of  density  dependence  are considered as well as  complete density
 independence.   Table  8  (in Appendix A)  presents  a  range  of potential
 annual  reductions  to  the fishery  for one  year  and over  six years  of
 adult  impingement.  High density dependence implies losses ranging  from
 0.2  to 2.9%, while low  to zero  density dependence implies losses rang-
 ing  from 6.5 to  21.6%.  Comparable  estimates  of  losses have  since  been
obtained using  the entrainment  and  impingement  estimates  suggested  by
C. D. Goodyear (1978):  4.3 to  27.4%  (high  density dependence)  and  55.2
to  83.5%  (for  low to  no density dependence).  C.  D.  Goodyear's  (1978)
estimates of impingement  losses in later life stages  are considerably

-------
                                    28
 higher   than  Patterson's  estimates  (based  on  his  weightings);  for
 example,

                           Entrainment and/or impingement estimates
      Life stage            Patterson (1979)   C. D.  Goodyear (1978)

      Larvae                   11,000,000            1,351,886
      Young-of-year               100,000            3,614,391
      Age class 1                  67,500              181,668
      Age class 2+                 67,500              441,159

 Larval  concentrations  estimated  for  the  U.S.  waters of  western  Lake
 Erie  are unadjusted for gear avoidance.   This  bias will carry  through
 to  the  abundance  estimate  of  later  life stages.   If entrainment  and
 impingement  estimates  are  not  biased   in  the  same manner,  then  the
 subsequent impact  predictions  based on  relatively  low abundance  esti-
mates will be too high.  Impingement estimates,  which may  account  for
most  of the  predicted  losses to  the  fishery,  are  not  subject to  gear
bias,  since   they  are  based  on  actual  counts,  while  the  abundance
estimates for impingeable life  stages are biased  low.   Therefore,  the
projected  losses  to  the  fishery  based on  C.  D.  Goodyear1s   (1978)
entrainment and  impingement  estimates  are believed to be too  large.
      In our age-structure model  we  have  adjusted survival  rates  during
the first year  of life  so  as  to  maintain the  population  at equilib-
rium.    Impacts   are  introduced  as  conditional   power-plant mortality
rates  (Appendix  A, Sect. 2.3), which removes  the  duality  problem of
life-stage survivals from which  Patterson's models  1  and 2  suffer.  In
addition, the density-dependent  mechanism is inserted during the larval
life stage, when it is more  likely to  occur.

-------
                                    29
      Section 4  in  Appendix  A gives a detailed discussion of the assump-
 tions and sources  of  error to  which our  age-structure model   is  sub-
 ject.  These errors enter into our analysis via (1) natural variability
 in  the  environment,  (2)  errors in  model  parameters  and inputs,  and
 (3) errors  in  model structure.  Patterson's models 1 and  2 are subject
 to  many of these  same assumptions  and  sources of  error.   It   is  felt
 that  the Patterson's models 1  and 2 introduce considerably more  error
 than  does  our  age-structure model with  respect  to the  third source  of
 error.

                          6.3  Future Data Needs

      Models are useful  in highlighting  areas where more or better  data
 are  needed.   The  precision and  accuracy  of model  predictions can  be
 improved by emphasizing  the collection  of data where model assumptions
 are  felt to be less valid  and  model predictions  are  most  sensitive.
 The  impact  of the  Monroe Power  Plant,  using a Leslie  matrix  or  age-
 structure approach,  on  the yellow perch  population depends greatly  on
 the  conditional  mortality  rate  by  life  stage.   The  conditional   mor-
 tality  rate is  obtained  from  the  exploitation  rate   and  either  the
 natural or total mortality  rate by life stage  (Sect. 2.3,  Appendix A),
 and the  exploitation rate is obtained from the ratio  of the number  of
yellow  perch killed as a result of  entrainment  and/or impingement  to
 the  total  number  produced.   Inaccuracies  in  any of  these  component
 parts  will   result  in  inaccurate  model   predictions.    It  would  be
 desirable to have  better estimates  for  all  of the components of  our
model,  including  (1)  yellow perch  life-history data for  western   Lake

-------
                                   30
Erie  based  on current  environmental  conditions  (natural  mortality by
life  stage  and adult age  classes,  age-specific  fecundities,  and  age-
specific  sex   ratios);  (2)  estimates  of  age-specific  commercial  and
sport  fishing mortality  rates  for  the  western basin  of  Lake  Erie;
(3) density estimates for each life stage, rather than just larvae, for
the entire western  basin of Lake Erie;  (4) estimates of gear avoidance
for the  sampling  gear used  in western  Lake  Erie and  the  Monroe  Power
Plant; and  (6) estimates  of entrainment and  impingement by life  stage
and adult age class.   The  desired  data  listed  above  would optimally
improve model  predictions,  but would  be  impossible to collect.
     C. P. Goodyear (1978) has pointed out that,  though  greater numbers
of young  fish  are  often entrained or impinged, most of  the impact may
be associated  with  the  older entrained or impinged fish.   If  we  con-
sider  C.  D.  Goodyear's (1978)  entrainment  and  impingement estimates
(Sect.  6.2),  the  losses   due   to   entrainment  and   impingement  of
young-of-year  (3,614,391)  are considerably  larger than  those due to
entrainment of larvae (1,351,886).  Losses to young-of-year stage  would
also  occur  after  the occurrence of  density-dependent  mortality  that
might  take  place  during the larval  stage.  Therefore,  we  would  argue
for the following  list of  data needs:
1.  density estimates  for  young-of-year  yellow  perch  for  at  least
    Michigan and Ohio waters of western  Lake  Erie,
2.  continued    frequent   collection   of   concentration   data    for
    young-of-year  entrained or impinged  at  the  Monroe Power  Plant,
3.  estimates  of entrainment and  impingement  survival for young-of-year,
4.  estimates  of gear avoidance  (relative  to  a  high-speed  sampler) of
    young-of-year  for sampling gear  used  in western Lake  Erie  and the
    Monroe Power Plant,

-------
                                   31
5.  yellow  perch  life  history data  for  western Lake  Erie  based  on
    current environmental conditions  (especially  the  natural mortality
    rate for young-of-year).
Item 1 is  especially  important for obtaining accurate estimates of the
abundance of young-of-year yellow  perch.   Underestimating their abund-
ance will  lead  to an  overestimate of the  impact of the  Monroe Power
Plant.   A  mark-recapture study  for young-of-year  yellow  perch would
provide an  alternative  approach  for item  1  and would  reduce  the data
needs in item  4 to just  estimates of gear  avoidance of young-of-year
for sampling gear  used at the  Monroe Power  Plant.

-------
                                    32

                                REFERENCES

 Adams,  S.  M., H. A.  McLain,  D. S.  Vaughan,  G.  F. Cada, D.  Kumar,  and
     S.  G.  Hildebrand.  1979.  Analysis  of  the  Prairie  Island  Nuclear
     Generating  Station,  Intake Related  Studies.   Report  to  Minnesota
     Pollution   Control   Agency.    Oak   Ridge    National   Laboratory,
     Oak Ridge,  Tennessee.  235 pp.
 Bjorke,  H., 0.  Dragesund,  and 0.  Ulltang.   1974.  Efficiency  test  on
     four    high-speed   plankton    samplers.     pp.    183-200.     IN
     J. H.  S. Blaxter   (ed.),  The  Early   Life   History   of   Fish,
     Springer-Verlag, Berlin.
 Boreman, J. and  C.  P.  Goodyear.   In press.   Biases in  the estimation  of
     entrainment mortality.   IN L.  D. Jensen (ed.), Proceedings of the
     Fifth  National Workshop  on Entrainment  and  Impingement,  Ecological
     Analysts, Inc., Melville, New York.
 Christensen,  S.  W.,  D.   L.   DeAngelis,  and   A. G.   Clark.    1977.
     Development  of a  stock-progeny model  for  assessing  power  plant
     effects  on  fish  populations.   pp.   196-226.  IN  W.   Van  Winkle
     (ed.), Assessing  the  Effects  of Power-Pi ant-Induced  Mortality  on
     Fish Populations, Pergamon Press, New York.   380 pp.
Cole, R.  A.  1977.   Entrainment  at  a  Once-Through Cooling  System  on
     Western  Lake  Erie,  Vols.  I  and II.  Institute of Water  Research
     and  Department   of    Fisheries   and   Wildlife,   Michigan   State
     University, East Lansing, Michigan, January.
Detroit Edison  Company.   1976a.   Monroe  Power  Plant  Study  Report  on
     Cooling Water Intake.   September.

-------
                                   33
Detroit Edison Company.  1976b.  Monroe Power Plant Data Sheets on  1976
     Larval  Entrainment.
Goodyear, C.  D.   1978.  Evaluation  of 316(b)  Demonstration  - Detroit
     Edison's Monroe Power Plant.  Great Lakes Fishery Laboratory,  U.S.
     Fish and Wildlife Service, Ann Arbor,  Michigan.   245 pp.
Goodyear,  C.  P.   1978.    Entrainment   impact  estimates  using  the
     equivalent  adult  approach.   FWS/OBS-78/65,  Fish  and  Wildlife
     Service, Department  of the Interior.   14 pp.
Gull and, J.  A.  1965.   Survival of the youngest stages of fish, and its
     relation to year-class  strength.  Int.  Comm.  Northwest At!.  Fish.
     Spec. Publ., (ICNAF) 6:363-371.
Hartman, W.  L.   1973.  Effects of exploitation, environmental changes,
     and new  species on  the  fish habitats and  resources of Lake Erie.
     Great Lake Fishery Commission.  Technical Report  No. 22, pp.  1-37.
Hemmick, W.,  J.  Schaeffer, and R.  Waybrant.   1975.  Larval  Fish  Survey
     in Michigan Waters  of  Lake Erie,  1975.   Great Lakes Studies Unit,
     Aquatic  Biology   Section,  Bureau  of  Environmental   Protection,
     Michigan Department  of Natural Resources.
Herdendorf,   C.  E.,  C.  L.  Cooper,  M.  R.  Heniken, and  F.  L.  Synder.
     April   1976.   Western   Lake   Erie   Fish  Larvae  Study   -   1975
     Preliminary  Data  Report.   CLEAR Technical  Report No.  47,  The Ohio
     State  University  Center  for  Lake Erie Area  Research,  Columbus,
     Ohio.

-------
                                   34
Herdendorf,  C.  E.,  C.  L.  Copper,  M.  R.  Heniken, and  F.  L.  Synder.
     March  1977.   Western   Lake   Erie  Fish  Larvae   Study   -  1976
     Preliminary Data Report, CLEAR Technical Report No.  63.   The Ohio
     State  University  Center for  Lake  Erie Area  Research,  Columbus,
     Ohio.
Horst,  T.  J.    1975,   An assessment  of impact due to  entrainment of
     ichthyoplankton.   pp. 107-118.   IN S.  B.  Saila  (ed.),  Fisheries
     and    Energy    Production:  A   symposium.     Heath,   Lexington,
     Massachusetts.
Hubbell,  R. M.  and C.  E.  Herdendorf.  September  1977.   Entrainment
     Estimates  for  Yellow  Perch in Western  Lake  Erie  1975-76.   CLEAR
     Technical  Report No. 71, The  Ohio  State University Center for  Lake
     Erie Area Research,  Columbus,  Ohio.
Leithiser,  R. M., K. F.  Ehrlich, and  A. B. Thum.   1979.   Comparison of
     a  high volume  pump  and  conventional plankton  nets  for collecting
     fish larvae  entrained in power  plant cooling systems.   J. Fish.
     Res. Board Can. 36:81-84.
Michigan   DNR.    1975.    Computer   Listing  of   1975   Larval    Fish
     Concentrations   Sampled   in   the   Western  Basin   of  Lake  Erie.
     Michigan Department of Natural Resources.
Michigan   DNR.    1976.    Computer   Listing   of   1976   Larval    Fish
     Concentrations  Sampled   in   the   Western  Basin   of  Lake  Erie.
     Michigan Department  of Natural Resources.
Muncy,  R. J.  1962.   Life history of the yellow perch,  Perca flavescens,
     in  estuarine waters of  Severn River,  a  tributary  of Chesapeake
     Bay, Maryland.   Chesapeake  Sci. 3:143-159.

-------
                                   35
Murphy, G.  I., and  R.  I.  Clutter.  1972.  Sampling anchovy larvae with
     a plankton purse seine.   Fish.  Res.  Bull.  70(3):789-798.
Nepszy,  S.  J.   1977.    Changes  in  percid  populations  and   species
     interactions in Lake  Erie.   J.  Fish. Res.  Board Can. 34:1861-1068.
Ney,  J.  J.   1978.   A  synoptic  review  of  yellow  perch  and   walleye
     biology.  Am. Fish.  Soc.  Spec.  Publ. 11:1-12.
Ney, J. J., and L. L. Smith,  Jr.   1975.   First-year growth of  the yellow
     perch, Perca  flavescens, in the Red Lakes, Minnesota.   Trans. Am.
     Fish. Soc. 104:718-725.
Noble, R. L.  1975.  Growth of young yellow  perch  (Perca flavescens) in
     relation   to   zooplankton   populations.    Trans.   Am.   Fish.
     Soc. 104:731-741.
Patterson,  R.  L.   1979.   Production,  Mortality,   and  Power  Plant
     Entrainment  of  Larval  Yellow  Perch    in   Western   Lake  Erie.
     EPA-600/3-79-087.    Environmental    Research   Laboratory,   U.S.
     Environmental Protection  Agency,  Duluth, Minnesota.  187  pp.
Schnack,   D.   1974.   On  the  reliability of methods  for  quantitative
     surveys of fish  larvae,  pp.  201-212,   IN J.  H.  S.  Blaxter (ed.),
     The  Early Life History of Fish,  Springer-Verlag, Berlin.
Schneider,  J.  C.   1973.   Density  dependent  growth  and mortality  of
     yellow  perch  in  ponds.    Fisheries Research  Report   No.  1765,
     Michigan Oept.  of Nat. Res.   18  pp.
Scott, W. B., and E.  J. Crossman.   1973.  Freshwater  fishes  of  Canada.
     Fish.  Res.  Board Can.  Bull.  184,  966 pp.

-------
                                   36
Sharma, R.  K.,  and R.  F.  Freeman III.   1977.   Survey of Fish  Impingement
     at  Power   Plants  in  the  United States.   Volume  I.   The  Great
     Lakes.    ANL/ES-56.    Argonne    National   Laboratory,   Argonne,
     II linois.   218 pp.
Smith, L.  L.,  Jr.  1977.   Walleye (Stizostedion  vitreum  vitreum) and
     yellow perch  (Perca flavescens)  populations and fisheries  of the
     Red   Lakes,   Minnesota,    1930-75.     J.    Fish.    Res.   Board
     Can. 34:1774-1783.
Tranter,  D.  J.  (ed.).   1968.    Zooplankton  Sampling.   Monographs  on
     Oceanographic Methodology 2.   The UNESCO  Press, Paris.   174 pp.
Van Meter, H. D.   November  1960.   The yellow  perch of Lake  Erie.  Ohio
     Conservation Bulletin,  pp.  22-23.
Wells, L.  1977.  Changes in yellow perch (Perca  flavescens)  populations
     of Lake Michigan,  1954-75.   J.  Fish.  Res. Board Can. 34:1821-1829.
Wolfert, D.  R.,  W. D.  N. Busch,  and  C.  T.  Baker.   1975.  Predation by
     fish  on  walleye   eggs  on  a  spawning reef  in western  Lake Erie,
     1969-71.  Ohio J.  Sci.  75(3):118-125.

-------
                                   A-l

                                APPENDIX A

                  AN  AGE  STRUCTURE MODEL OF YELLOW PERCH
                           IN WESTERN LAKE ERIE

                           Douglas S. Vaughan
                     Environmental  Sciences  Division
                      Oak Ridge National  Laboratory
                       Oak Ridge, Tennessee   37830

                                 ABSTRACT

     An  age structure model  based on  life history  data (i.e.,  early
life-stage  and  age-class survival  and age-class-specific fecundity)  is
developed  for  the yellow perch  (Perca  flavescens)  population  in  the
western  basin  of Lake Erie.   The  long-term impact  on  yellow perch  of
entrainment  and   impingement   at   the   Monroe  Power   Plant   (Monroe,
Michigan) is assessed.   Larval  abundance  estimates  are obtained from  a
modification of  a  "materials  balance  model," while  abundance  estimates
for  later  life  stages  are  obtained using  life-stage  survival   esti-
mates.   For each life stage,  power plant exploitation  rates are  esti-
mated  from  entrainment   and/or  impingement  and abundance   estimates.
Conditional power  plant  mortality rates are  calculated  from  exploita-
tion rates  and  life-stage survival  estimates and are incorporated  into
the age structure model.
     Using  side-by-side   projections   (with   and without power   plant
impact) of  the  age structure model, long-term  losses of fishery  yield
are estimated.  One  set  of projections  is made where the age  structure
model  includes  no  density-dependent  component.    Other   projections

-------
                                   A-2

consider the age structure model where  the survival  term for age class
0  depends  on  the  density  of eggs  produced  that  year.   The density
dependence is assumed to occur during the larval  life stage within  age
class  0.   Three levels  of compensation  during age  class 0  are  con-
sidered.  The unknown status  of  the yellow perch population in western
Lake  Erie  precludes  a precise specification  of the level of  compensa-
tion.   However,  heavy fishing pressure on  the yellow perch population
in  western  Lake Erie implies that  losses  in yield  are better  repre-
sented  by ranges based on  little or  no compensation (6 to 22%) than by
ranges  based on moderate to very strong compensation (0.2 to 2.9%).

Keywords:  age    structure    model,    density-dependent,     density-
            independent,   entrainment  and   impingement,   environmental
            impact,  fish population  analysis, Lake Erie,  Leslie  matrix
            model, Monroe  Power Plant, yellow perch

-------
                                   A-3

                            1.   INTRODUCTION

     A  difficult  problem  confronting  fishery  science  is   that   of
assessing  the  long-term effects of an  incremental  mortality on a  fish
population.   This  incremental  mortality might  arise through  increased
fishing pressure or through a change in the environment such as  a  power
plant operation.   In the latter  case, such mortality is usually  assumed
to  act  independently of  the density  of the  fish  population,  but  is
dependent  upon  the age (or size)  of  the fish.   Those  fish  drawn  into
the  cooling water intakes  of power  plants  and  small  enough  to  pass
through  the meshes of  the debris  screens  are  said to  be  entrained,
while larger  fish  entrapped on  the  surface  of the  debris screens  are
said to be impinged.   Mortality  results from these processes,  although
the actual  proportion of  fish  killed under a  particular  set of condi-
tions  (power  plant operation  and environmental)  has  been  questioned.
As a result of  the age dependence of  incremental  power plant mortali-
ties, the  need  for an  age  structure model  to  assess long-term  impacts
arises.
     An age structure (Leslie matrix) model  was independently  developed
by  Bernardelli  (1941),  Lewis  (1942),  and  Leslie  (1945)  in  order  to
follow a female population through time.  The Leslie matrix,  A(t), is a
square matrix consisting of age-class-specific  fecundity factors on  the
first  row,  P-jfji   and  age-class-specific  survivals  on   the   sub-
diagonal,  p.  (Fig. 1).   Usher   (1972)  suggested  introducing  the   sur-
vival element in the  lower right corner of the Leslie matrix to  account
for the survival of fish beyond  the  oldest  age explicitly modeled.   By

-------
                                   A-4
representing fecundity factors as  shown in Fig. 1,  it  is assumed  that
spawning occurs over  a  very brief period just prior to  the  end of  the
time  period (Emlen  1973).   The  age  structure  (a column  vector  con-
taining  age-class-specific  abundances),  n(t+l),  at  time  t+1  can  be
obtained by  premulti plying  the age  structure,  n(t),  at time  t by  the
Leslie  matrix,  A(t),  as  shown  in  Fig.  1.   Because  survival   and
fecundity estimates for fish are generally obtained on an  annual  basis,
a time  unit of one year is used.   As originally developed,  the  Leslie
matrix  elements were  assumed  to  remain  constant  over  time.   Three
mathematical situations can arise:   (1) the  population  will  grow  expo-
nentially without  bounds;  (2) the  population  will decay  exponentially
to  extinction;  or  (3) the  population is perfectly balanced.   Thus,  in
this  form  the  behavior of  the model  is as  limited  as  the  exponential
growth  model.   Any   incremental  mortality  would   ultimately  drive  a
population  in  the  third  situation  to  extinction,  unless  the  excess
mortality was removed before extinction  occurred.  Survival  through  the
first year  of life  is  often difficult  if not  impossible to  estimate
with  any precision  for  highly  fecund fish   populations.   Vaughan  and
Saila  (1976)  obtained estimates of  survival  through age  class 0  (the
first year  of  life)  from  a Leslie matrix  by  assuming  that  the popula-
tion was perfectly balanced  situation (3)  described  above.
     The  fully defined elements of  the  Leslie matrix permit one  to
project  the  age structure of a  population from one year  to  the  next.
The  age structure at time t,  n(t),  can  be converted  to the  fishery
yield, y(t), by premultiplying n(t)  by the diagonal  matrix,  D,  and  the
row vector, f.  Yield in  metric  tons (t) is  given  by the equation:

                      y(t) = f • D • n(t)  •  106  .                  (1)

-------
                                   A-5

The  row  vector,  £',  contains the proportion of  each  age class lost  to
the  fishery  or  age-class-specific  fishing  exploitation  rates.    The
diagonal matrix, J£,  consists of age-class-specific weights in grams  on
the main diagonal and zeros elsewhere.
     Since  competition during  early  life  stages  and  cannibalism  by
older  age  classes  on  younger age classes  can result  in the density-
dependent regulation of a  fish  population,  density-dependent  mechanisms
are  incorporated into population  models.    Gull and  (1965)  notes  that
"most  fish  have  enormous  fecundities, and  there  must be a correspond-
ingly  enormous  mortality  between  egg and  adult  stages."   He further
suggests that "most of this mortality  is believed to  occur in  the  first
few months of life," and  this stage is believed important because  "the
regulatory  (density-dependent)   effects...may  occur  in  this stage."
Survival through the first year  of  life  can be made  dependent upon  the
number  of  eggs   produced  at  the start of  that year  (DeAngelis  et  al.
1980).   However, there is an upper limit of stress  to which  a popula-
tion  may respond  in  a density-dependent manner,  beyond  this  limit a
population  may  respond   in  a  density-independent   manner   (Goodyear
1977).   Heavily  fished populations  may  be  assumed  to  be approaching
this limit closely.
     In  this paper an assessment is made of the  losses in yield to  the
yellow perch commercial and sport fisheries of western Lake Erie due  to
the  imposition   of  incremental  age-specific   density-independent   mor-
tality.  In particular, entrainment and  impingement mortalities  due  to
the  Monroe Power  Plant   (Monroe,   Michigan)  are  incorporated  into a
Leslie  matrix  model  having  density-dependent  survival  during   age

-------
                                   A-6

 class   0.    Population   projections  are   made   (with   and   without
 power-plant-induced  mortalities),  and  annual  losses  in yield  to  the
 yellow  perch fishery are estimated.   These  estimates  of annual  losses
 in yield  are based on no  density dependence  in age  class  0  survival  and
 on  three  levels of  density-dependent  age  class  0  survival.   Estimates
 of  losses in yield  based  on  the lowest level of density  dependence  and
 no  density  dependence  are  believed to  be more  representative  of  the
 annual  losses that result from the operation of the  Monroe Power  Plant.

                               2.  METHODS
     In  this section,  four  topics  are  discussed.   First,  a detailed
 description  is  given of  the methods used to estimate  the elements  in
 the  Leslie matrix from  life history  data.   Second,  a "materials  balance
 model"  developed by  Patterson  (1979)  is  modified  and  used  to  obtain
 estimates of the larval and later  life-stage abundances  of yellow perch
 (Perca  flavescens).  Third, conditional power plant mortality rates  are
 determined from  estimates of  entrainment and/or impingement  mortality,
 life-stage  abundance,  and  life-stage  survival,  using  a  methodology
 described by Barnthouse et al.  (1979).  Finally, the density-dependent
 Leslie  matrix developed  by   DeAngelis et  al.   (1980)  is  modified   to
 include incremental  density-independent mortalities (i.e.,  power  plant
 impacts).  These data and model  modifications  are  then  used  to  compare
 side-by-side  projections  (with  and  without power  plant operation)   in
order to  assess the  long-term impacts  of  the Monroe Power Plant on  the
yellow perch population in western  Lake Erie.

-------
                                   A-7
2.1  Life History Data.  The Leslie matrix model requires the following
information from yellow  perch life  history  data:   (1)  number  of eggs
produced  per  individual   for  each   sexually  mature  age  class,  f.;
(2) probability of surviving from  the  start  of one age  class  (i-1) to
the start of  the  next age class  (i),  p^;  and  (3)  total  number of age
classes, N (Fig. 1).   Once the Leslie matrix  is defined,  it can be used
along with  estimates of weight  at age  (both  sexes combined)  and age
specific fishing exploitation  rates  (proportion of population  removed
by the  fishery) to estimate the  loss  in  biomass of the population to
the commercial and  sport  fisheries.
     Estimates  of  the  number of eggs  produced  per  individual  for each
age  class  are obtained   from   the  following  sequence.    First,  a
von Bertalanffy  growth  function   (Table  1,   footnote  a)  is   used  to
estimate the  standard length  in millimeters  by  age class  for yellow
perch females [data  from  Table  7 in  Jobes  (1952)].  Lower  and  upper
bounds of the asymptotic 95% confidence intervals,  as well as the  point
estimates,   are   estimated   for   the   three   parameters   of   the
von Bertalanffy  growth  function  by  nonlinear  regression  techniques
(Table 1).  Standard lengths in millimeters for female yellow perch are
then generated  for age classes 1  through  7 for all  three cases  (lower,
middle, and upper)  (Table 2).
     Total  weight  in grams  for  female yellow perch  (W) is calculated
from standard lengths  in  millimeters (L)  using the  allometric equation
(Jobes 1952; p. 252),
                       W = 1.776 x 10-5 L3.015  §                   (2)

-------
                                   A-8
Table 1.   Coefficients of the von Bertalanffy growth function estimated
          by nonlinear regression for yellow perch (Perca flavescens) in
Lake Erie from data
Coefficient9
Females only:
L
00
k
t
0
Both sexes combined:
L
00
k
t
0
in Table 7 of
b
Lower

259.3

0.369
0.087


240.4

0.321
-0.052

Jobes (1952)
Middleb

269.9

0.424
0.200


260.3

0.437
0.182


Upper

280.5

0.480
0.314


280.2

0.553
0.416

aLt = L   [l-e~ *   o'] , where L. equals standard length (in millimeters)
  C    °°                         L                          -I
 at age t (in years), and L^ (in millimeters), k (in years" ), and t  (in
 years) are coefficients of the von Bertalanffy growth function.

bThe middle column is obtained from the point estimates of the parameters
 in the von Bertalanffy growth function, while the lower and upper columns
 are obtained from the lower and upper bounds, respectively, of the
 asymptotic 95% confidence interval.

-------
                                          A-9
Table 2.  Life history data for yellow perch (Perca flavescens) of western Lake Erie
Age
class
Standard
length3
(mm)
Total.
weight0
(9)
Fecundityc
(eggs/mature
female)
Fecundity
(eggs/individual)
M
Survival
"V
Fecundity
factor
Lower estimates:
              74
             131
             171
             198
             217
             230
             239
Middle estimates:
              78
             144
             188
             216
             235
             247
             255
Upper estimates:

 1            79
 2           156
 3           203
 4           233
 5           251
 6           262
 7           269
  7.7
 43.0
 95.0
148.6
195.5
233.2
261.8
  8.8
 56.9
126.0
192.9
247.5
288.3
317.3
  9.2
 71.8
160.
241.
303.
345.9
374.4
.5
.3
.0
           827
          5285
         12416
         20098
         27012
         32656
         37003
           960
          7147
         16828
         26621
         34821
         41044
         45508
 1003
 9180
21838
33891
43295
49940
54387
                   0
                   0
                4966
                8039
               10805
               13062
               14801
                   0
                   0
                7573
               11980
               15670
               18470
               20479
    0
    0
10919
16946
21648
24970
27193
             0.000308
                0.75
                0.50
                0.
                0.
                0.
    .29
    .29
    .29
                                                                    0.29
             0.000116
                0.77
                0.61
                0.43
                0.43
                0.43
                0.43
0.000034
   0.80
   0.73
   0.64
   0.64
   0.64
   0.64
    0
    0
 2483
 2331
 3133
 3788
 4292
                     0
                     0
                  4620
                  5151
                  6738
                  7942
                  8806
    0
    0
 7971
10845
13855
15981
17404
Calculated using the coefficients of the von Bertalanffy growth function from Table 1
 for females only.

°W = 1.776 x 10"5 L3'015, where W is the total  weight (g) and L is the standard
 length (mm).

CEG6S = 92.045 W1'077, where EGGS is the fecundity (eggs/female) and W is the
 total weight (g).

 Corrected for immature females, non-spawning mature females and sex ratio.

Survival  is from age class i-1 to age class i  (i  = 1,2	7).
 Fecundity factor = fecundity (eggs/individual)  x survival.

-------
                                  A-10
where W  is  the total weight  in  grams and L  is  the standard length  in
millimeters.   This  equation was applied  to  lengths given  for  each  of
the three cases in Table 2.
     The  fecundity,  expressed   as  the  number  of  eggs  produced  per
sexually mature female (EGGS), was calculated for each age class from a
weighted average  of coefficients from three  annual  regressions  of the
number of eggs on total weight  in  grams  (W)   (Sztramko and Teleki 1977,
and Table 1):
                         EGGS =  92.045 W1-077   .                     (3)

The  weighting  used  in  combining  the regression  coefficients  was  by
sample size.
     The number of eggs produced per individual of a given age class  is
estimated from the  product of  (1)  the  fraction  of females  mature  in
that  age class,  (2)  the  probability of  a  mature  female  spawning,
(3) the age-class-specific  sex  ratio,  and (4) the  number  of eggs  pro-
duced per mature  female  of that age  class  (Table  2).   No females are
assumed to  be  sexually mature prior to age class  3,  while all   females
of age class 3 or greater  are assumed to be   sexually mature  (Patterson
1979).  Patterson (1979) indicates  a  range of 0.8  to  1.0  for the  pro-
bability of a  mature  female spawning.  The middle  of  this range,  0.9,
is used  in   the  middle case,  while  the  lower  (0.8)  and  upper (1.0)
bounds are  used  in  the lower and  upper cases,  respectively.   Also,  a
sex ratio of 1:1  was assumed for all age classes  (Patterson 1979).
     In this analysis each  age class is considered  to extend from May  31
to the  following May  31.   In  contrast,  Patterson (1969),  from whom

-------
                                  A-ll
natural and fishing survival are obtained, considered each age class  to
extend from January 1 to  the  following  January 1, except for age  class
0 which extends from late May to January 1.  Annual  natural and  fishing
survival  are from 0.75 to  0.80  and  0.39 to 0.80, respectively,  for age
classes 1  through  7 (Patterson 1979).   Fishing  pressure is  introduced
in January to age class 2,  so  age  class 1  survival   reflects  no  fishing
mortality, age class 2  reflects only five months of fishing  mortality,
and age  classes  3  and  older reflect the  full amount  of  fishing mor-
tality.  The overall annual survival for age classes 1  through 7 is the
product of  annual  natural  survival  and  the  appropriate annual  fishing
survival  (Table 2).
     The final  element needed for the Leslie matrix  is  survival  through
age class 0.   Because  of  the difficulties  in obtaining   reasonable
estimates of this parameter  from  field  data,  estimates are obtained  by
assuming  a  stable  population  and  using  Eq.   (6)  in DeAngelis  et al.
(1980).   These  estimates  range   from   3.4  x  10"    (upper  case)   to
        -4
3.1 x 10    (lower case)  with the  middle case yielding  an  estimate  of
1.2 x 10"4 (Table 2).
     The stable age structures  presented in Table  3 are calculated  by
first assuming one  million eggs were produced and  sequentially multi-
plying the number in one  age class by the survival  value  for that age
class  to obtain the  number in the  next age  class.
     Fishing yield  (in metric  tons) is  calculated  using Eq.  (1).  The
age-class-specific fishing exploitation  rates  are calculated  using the
estimates of  natural  and  fishing  survivals in  Table   2 (Ricker  1975;
p. 11).  The age-class-specific weights  in grams are  calculated  using

-------
                                   A-12
Table 3.   Stable age structures and fishing weights and mortalities for
          yellow perch (Perca flavescens) in western Lake Erie

Age Age
class structure
(numbers of fish)
Lower estimates:
0 1 000
1
2
3
4
5
6
Middle estimates:
0 1 000
1
2
3
4
5
6
Upper estimates:
0 1 000
1
2
3
4
5
6

000.0
307.5
230.6
115.3
33.4
9.7
2.8

000.0
116.2
89.5
54.6
23.5
10.1
4.3

000.0
34.4
27.5
20.1
12.8
8.2
5.3
Fishing
weights
(g)

_
_
37.7
73.9
111.2
145.1
173.6

_
_
70.9
135.6
193.6
238.8
271.5

_
_
108.9
208.5
287.3
340.6
374.1
Fishing .
mortalities
(u)

0
0
0.079
0.178
0.178
0.178
0.178

0
0
0.192
0.396
0.396
0.396
0.396

0
0
0.287
0.544
0.544
0.544
0.544
 Calculated using coefficients of the von Bertalanffy growth function
 from Table 1  for both sexes combined, Eq. (2). and weighted by season.

Wishing exploitation rates from Patterson (1979).

-------
                                  A-13
the von  Bertalanffy  coefficients  in Table  1  based on both  sexes com-
bined  and  Eq.  (2).   Although  a  year-round  fishery  for yellow perch
exists,  most  fishing  occurs  between  spring  and fall.   A  weighted
average of the fish weights is  calculated  for each age  class such that
spring  (t)  and fall  (t+1/2) fish  weights  each receive  a  weighting of
0.25,  while  summer  (t+1/4)  fish  weights  receive  a weighting  of 0.5.
The values  calculated  for the fishing exploitation rates  and  the fish
weights by age class are  presented  in  Table 3.
     The particular  life stage or  age class  which  is   stressed  has a
great effect on the long-term impact on the population.   The early life
history of the yellow perch  is  broken into four life stages (Patterson
1979):   (1)  eggs  (6-12 days),  (2)  larvae  (25 days),  (3) young-of-year
(6 months),  and (4) yearlings  (1  year).   The first three  stages  and
approximately  5 months of the  fourth  stage occur during  age  class 0.
Ranges of these life stage survivals are from  Patterson  (1979) as  shown
in Table 4.   The  geometric mean  survivals of  these  four  stages  are
0.35,  0.045,  0.44,  and   0.90.   These ranges  in  life  stage survivals
imply  a  range  in survival for age  class  0  of 0.002 to 0.026 (geometric
mean = 0.006).
     The  survival  rates   of  these  early life stages are  difficult to
estimate.   Since  the range  in survivals  obtained from the  four life
stages is considerably higher than  the range  in survivals for age  class
0 based  on  a stable population, Patterson's  (1979) life-stage  survival
estimates are  reduced  proportionately  so as to maintain the population
at equilibrium (Table 4).   This  assumption of a  stable population is
probably not true in the  light of declining catch-per-unit-effort (CPUE)

-------
                                    A-14
Table 4.  Early life-stage survivals adjusted to maintain the yellow perch
          (Perca flavescens) population at equilibrium
Case
Lower
Middle
Upper
Egg
survival
[6 days] a
0.214 (0.50)b
0.156 (0.35)
0.112 (0.25)
Larval
survival
[25 days]
0.006 (0.10)
0.004 (0.045)
0.002 (0.02)
Young-of-year
survival
[6 months]
0.297 (0.58)
0.234 (0.44)
0.181 (0.34)
Yearling
survival
[5 months]
0.815 (0.91)
0.827 (0.90)
0.826 (0.89)
 Duration of life stage in brackets.

3Life stage survivals from Patterson  (1979) prior to adjustment in
 parentheses.

-------
                                  A-15
experienced  by  the  Canadian  fishery  for  yellow perch  in  Lake Erie
(Nepszy 1977).   In  fact,  the declining  CPUE's  would  indicate that age
class 0 survival assuming a stable population is overestimated based on
current environmental  conditions in Lake  Erie.

2.2   Life  Stage Abundance  Estimates.    Larval   yellow  perch  concentra-
tions are  available for the  Ohio and Michigan waters  of  the western
basin in Lake Erie  in 1975  and  1976 (Patterson  1979;  Herdendorf et al.
1977a,  1977b)  (Fig.   2).    A  modification of  a  "materials  balance
formulation"  (Patterson  1979),  described in the  Appendix,   is  used to
obtain  estimates  of larval  yellow perch  production  for the Ohio and
Michigan waters in  1975  and  1976  (Table  5).  United  States   waters
comprise about  41% of  the volume  of the  western  basin of Lake  Erie
(Nepszy 1977).   Assuming that  the Canadian waters  are  of equal pro-
ductivity to the U.S. waters  for yellow  perch larvae, the production of
yellow  perch  larvae for  the entire western basin of  Lake Erie is  2.45
times  the   combined production  of  yellow  perch  larvae  in  Ohio and
Michigan waters of the western basin.
                                                                       g
      In  the  western  basin  larval   production  is   from  13.8   x 10
(larval  survival  = 0.006)  to  17.2  x 109  (larval  survival  = 0.002),
                          Q
mid-estimate  is  14.9  x 10   (larval  survival  = 0.004).   The number of
recruits to the young-of-year stage   Eq.   (A-2)   is  from  3.4  x 10
(young-of-year   survival   =  0.181)   to  8.3  x  10    (young-of-year
survival =   0.297),  mid-estimate   is   6.0   x    10    (young-of-year
survival =  0.234).  Yearling  production  estimates  are then  obtained by
multiplying  the  number  of young-of-year recruits  by  the corresponding

-------
                                 A-16
  Table 5.   Estimates of  larval yellow  perch  (Perca  flavescens)  production
             by state, year, and larval  survival  in Ohio  and  Michigan
             waters of western  Lake  Erie
State Year
Ohiod 1975
(for Ohio
waters)
1976


Average6


Michiganf 1975
(for Michigan
waters)
1976


Average6


Larval
survival
0.002
0.004
0.006
0.002
0.004
0.006
0.002
0.004
0.006
0.002
0.004
0.006
0.002
0.004
0.006
0.002
0.004
0.006
hb
9.5
8.5
8.0
7.9
7.0
6.6
8.7
7.75
7.3
34.4
25.6
21.5
2.6
2.4
2.2
18.5
14.0
11.85
Production0
(x 109)
6.246
5.589
5.260
5.194
4.603
4.340
5.720
5.096
4.800
2.390
1.783
1.498
0.181
0.167
0.153
1 . 289 •
0.975
0.826
 See  larval survival column  in  Table 4.
 Method of calculation  is  described  in Appendix;   h  is  the  mean  total
 number of larvae per 100  m   of water in  the  reference  volume.
cProduction = d-B-h, where d  equals 7 or  14 for  Ohio or  Michigan
 waters, respectively,  and B  equals 9.393 x 107  or 4.976 x  106 for
 Ohio or Michigan waters,  respectively,   d  is  the  number of days in  each
 time period, and h  is  the reference volume in units of  100 m  .
 Source data  for Ohio from Herdendorf et al. 0977a  and 1977bX
Arithmetic average of 1975 and  1976 values.
 Source data  for Michigan  from  Patterson  (1979).

-------
                                   A-17

young-of-year  survival.   Yearling  productivity  Is  from  0.6  x  107
 (annual  yearling  survival   =  0.727)  to  2.5 x  107  (annual   yearling
 survival   =   0.688),   mid-estimate  1s   1.4  x  107   (annual   yearling
 survival  = 0.713).  Adult recruits [Patterson's  (1979) age  class  2]  are
 then calculated  from the product  of yearling  production and correspond-
 ing annual yearling  sur-  vival.    These estimates range  from 0.5  x  107
 to  1.7 x 10  , mid-  estimate 1s  1.0 x 107.   These life-stage  produc-
 tion estimates assume that the population 1s stable.

 2.3  Conditional  Mortality Rates.    Natural  mortality  may  be  expressed
as  either the  expectation  of  natural  death  (the proportion  of  the
population which  dies  due to  natural  causes  in the  presence of com-
 peting  sources of mortality)  or  the conditional  natural  mortality rate
 (the proportion  of the population which dies  due to natural causes in
the absence of competing  sources  of mortality) (Ricker 1975).   Fishing
mortality may be expressed analogously  as  either the rate  of  exploita-
tion or the  conditional  fishing  mortality  rate.   If fishing  mortality
is subsumed into natural mortality, then  the "power plant  exploitation
rate" or  "conditional power plant  mortality rate,"  respectively, may be
discussed.  However,  it is in the  form of the power  plant survival rate
 (1  -   conditional  power  plant  mortality  rate)  that entrapment  and
Impingement estimates are  incorporated into  the Leslie matrix model.
     Because  the conditional  power plant mortality  rate  is calculated
from  the power  plant  exploitation rate,  it  is  first   necessary  to
calculate the power  plant exploitation  rate.   This mortality  rate  is
equal   to the  ratio  of  the  number of  individuals  of   a  life   stage

-------
                                  A-18

entrained  and/or  Impinged by  the  power plant  to the  number  of  indi-
viduals  of   that  life  stage.    Production   estimates  for   larvae,
young-of-year, yearlings,  and  adults  are  given  in  the  section  2.2.
Entrainment and impingement estimates  for  these four stages  (Patterson
1979) are presented in Table 6.  Weightings for  the ranges of  estimates
of entrainment and  impingement for the four life  stages (Table 6) are
based  on  Patterson's  (1979)   subjective  appraisal of   the  numbers of
larvae, young-of-year, yearlings, and  adults  that are entrained  and/or
impinged  by   the  Monroe  Power Plant.   These  weightings  are  used to
combine  estimates  of  the conditional  power plant mortality  rate  [see
Eqs. (4) and (5)].
     The conditional  power  plant  mortality rate,  m,  is related to the
power plant exploitation rate,  u,  by the equation
                           m
= 1  . (l-A)U/A  ,                        (4)
where A  is the  total  fraction of  the  population removed  due  to both
"natural"  (natural and  fishing mortalities)  and  power plant sources of
mortality  (Barnthouse et  al.  1979).  Assuming that  the survival esti-
mates in  Tables 2 and  4 reflect both  "natural" and  power plant mor-
tality,  then A equals one minus the survival estimate,  and  the solution
of Eq. (4)  is  straightforward.   If, however, the survival  estimates in
Tables 2 and 4 are assumed to  be  independent of power  plant mortality,
then  these  survival  estimates  are  conditional   "natural"  survival
rates.  According to  Ricker (1975),

                        1  - A = (1  - m)(l  -  n)   ,                    (5)

-------
                                     A-19
Table 6.  Numbers  of yellow perch entrained  and/or  impinged  annually  at  the
          Monroe Power  Plant with subjective weightings  from Patterson  (1979)
  Life stage  (age)
Estimates3
Weightingsb
Weighted mean
    of the
  estimatesc
Larvae (6-12 days)
Young-of-year (1 month)
Yearlings (7 months)
Adults (19 months)
2 000 000
10 000 000
20 000 000
40 000 000
100 000
50 000
100 000
50 000
100 000
0.18
0.62
0.18
0.02
1.00
0.65
0.35
0.65
0.35
11 000 000
100 000
67 500
67 500
aPages 131-132 in Patterson (1979)

bTables 28-29 in Patterson (1979).  These weightings are used to combine
 estimates of the conditional power plant mortality rates for the range of
 entrainment and/or impingement estimates in this table.

cThe weighted mean of the estimates is the sum of the product of the
 estimates and their corresponding weightings.

-------
                                  A-20

where  n  is  the  conditional   natural   mortality  rate.   Substituting
Eq. (5) into Eq. (4), m now occurs on  both  sides  of  the equation.  The
resultant equation is transcendental  and must be solved by an iterative
technique (Barnthouse et al.  1979).
     Estimates  of  n  (or  A)  are  needed for  the  duration of  the life
stages  for  which  conditional  power  plant  mortality  rates  are  to  be
estimated.  Larval, young-of-year, and yearling survival  rates  (1-n  or
1-A)  are  given  in the  previous section with  the  corresponding pro-
duction estimates.  To obtain  adult  survival  rates,  two cases of adult
impingement are  considered.   First,  adults  are assumed susceptible  to
impingement only  during  the first year of that  life  stage.   In this
case,  the  one-year survival  rates for adults  (age  1.6 to  2.6)  range
from 0.748  for  an  adult  production  of 1.7 x  10  to  0.803  for an adult
production  estimate of  0.5  x 10   with  a  geometric  mean  of  0.775 for
an  adult  production estimate of  1.0 x  10  .   The second  case  assumes
that impingement occurs uniformly throughout  the  adult life  stage (age
1.6  through age class  7).   Survival  estimates  for  this  period  range
from  0.0009 for  an adult  production  of 1.7 x  10    to  0.074 for  an
adult  production of 0.5  x 10 ,  with  a geometric mean  of  0.0082 for  an
adult production of 1.0  x 10 .
     For all  four  life  stages the two sets  of estimates  (survival  as
1-n or 1-A) of  the conditional  power plant  mortality rates were  nearly
identical.  Weighted  averages  of estimates  of  conditional  power plant
mortality rates for the  four life stages are  presented in Table 7 using
the weightings in Table  6.

-------
                                     A-21


Table 7.  Conditional power plant mortality rates by life stages and
          power plant survival rates used as inputs to the Leslie
          matrix model

Conditional power plant mortality
Larvae (6-12 days old)3
Young-of-year (1 month old)
Yearlings (7 months old)
Adults (19 months old)
(1) One year impingement
(2) 6.4 years impingement
Power plant survival rates:
eo
el
One -year impingement:
e2
e3
6.4-year impingement:
e2
e3
Lower
rates:
0.0041
0.0021
0.0033
0.0046
0.0277
0.9959
0.9965

0.9962
0.9973

0.9963
0.9956
Middle
0.0040
0.0032
0.0057
0.0077
0.0326
0.9960
0.9944

0.9935
0.9955

0.9946
0.9948
Upper
0.0039
0.0061
0.0127
0.0166
0.0416
0.9961
0.9886

0.9857
0.9903

0.9899
0.9934
aAges in parentheses represent age at start of life stage.

 Power plant survival rates eQ-e3 (see discussion in text)  represent the

 form in which power plant impact is incorporated into the  Leslie matrix
 model.

-------
                                  A-22

2.4   Incremental  Density -Independent Mortalities.   Density dependence
is expressed  in  the Leslie matrix  model  for yellow  perch through the
survival term for age class 0;  that is,

                       Pl(t) =  Ci/[l  + C0n0(t)]   ,                   (6)
where  CQ  and  C1  are  positive  constants  (De Angel is  et  al .  1980).
When the population is at equilibrium,

                         Pie = Ci/M  +  Conoe)   ,                      (7)
where  ple  and  nQe  are the  equilibrium survival and  equilibrium size
(in  numbers)  of  age  class 0.   ple  is calculated  from  Eq.   (6)  in
DeAngelis et  al.  (1980) and nQe is set  to  one million.  Thus,  solving
Eq. (7) for C] and substituting this  into Eq.  (6),

                Pl(t) = (1 + C0n0e)   Ple/[l  + C0n0(t)]   .            (8)
Note   that   CQ  equal   to   zero   implies   no   density  dependence
(PJ = P]e)-   A  measure   of   compensation,   XQ   (equal   to   Cnn0e^'
is allowed  to range  over low  (0.01),  moderate  (1.0),  and very strong
(100) levels of assumed compensation.
     The conditional  power  plant  mortality  rates by life  stage cannot
be used  directly in  the  form  presented in  Table  6.    In  the  present
application of  the Leslie matrix  model, density  dependence  is  assumed
to occur during the  larval  life  stage.   Age class 0  survival, p-,(t),
including power plant impact,  is expressed as
                  p(t)  - ei(l  + C0n0e)  ple/[l + e0CQn0(t)]   ,        (9)

-------
                                  A-23
where  eQ  is the  power  plant survival rate  for the  larval  life stage
and  e,  is  the power  plant  survival  rate  for the  remainder  of  age
class 0 [young-of-year life stage and first five months of the yearling
life  stage  (Table  6)].   Power plant  impact  is  introduced  into  age
class 1 by

                             p2 = 62 • P2  .                         (10)
where e2  is  the  power plant survival rate for  age class  1  (the last 7
months of the yearling life stage and  first  5 months  of the adult  life
stage)  (Table  6).   Finally,  power plant  impact is  introduced  into the
adult age classes (2-7)  by

                             Pi  = 63 .  PJ   ,                         (11)

where e.,  is  the  power plant  survival  rate for age class  i  (Table  6).
The  index  i  is set to  3  for  the case where  impingement  is  assumed to
occur only during  the first year of the adult life  stage  (hence only
seven months of impact  through  impingement occurs  during  age class 2).
In the  case  where impingement  is assumed to occur uniformly  over  the
entire adult life  stage,  i  is allowed to run over all  integers from 3
through 8 (e^ represents annual  power plant survival).
     Equation  (9)  is modified  for  the  case  where no  compensation is
assumed; that is,

                           PI = ei • e0 • Pie                       (12)
where  p1, the  proportion  surviving  through  age  class   0,   is  inde-
pendent of time and  density.   Equations  (10)  and (11) remain unchanged
for the density independent case.

-------
                                  A-24

                               3.   RESULTS

     Projections of the yellow perch population for 100 years were made
with and  without the  power  plant impact incorporated  into  the Leslie
matrix model.  The projections with the power plant impact included the
power plant survival rates for the first 40 years  (typical duration for
power  plant operation)  followed  by  60 years  without  the  power plant
survival rates.  Projections were made for the four levels of compensa-
tion (XQ  =  0,  0.01, 1.0, 100)  and the three cases based on ranges of
life history  data.  The  relative losses  (in  %)  to  the  yellow perch
fishery for the two cases of adult impingement are  presented  in Table 8.
     Figure 3  illustrates the reduction in  yield to the yellow perch
fishery based  on the middle case  with one year  of  adult impingement.
During  the  first  40 years,  when the  power plant  is   operating,  the
reduction in yield  is  monotonic  increasing.   As  the  compensation term,
XQ, becomes smaller,  the magnitude  of the reduction in yield  becomes
greater.  During the  remaining  60 years of  the  projection,  when  the
power plant is not operating, the compensation term affects  the  rate of
return  of  the population  to its  equilibrium  value   (DeAngelis  et  al.
1980).   When  the compensation  term  (XQ) is  equal to  zero,  i.e.,  no
compensation,   the  population does not return to  its  old  equilibrium
value,  but  stabilizes  at the population level attained after approxi-
mately  40  years,  when   the  power  plant  ceases  operation.    For  XQ
greater  than  zero,   different   response  times   are  shown.    As   XQ
increases,  so  does  the  time  of  return  to  the  original   equilibrium
population.

-------
                                   A-25
Table 8.  Relative annual  loss in yield (%)  to yellow perch  fishery
          (both commercial  and sport)  based  on 40 years  of power
          plant impact followed by 60  years  of no power  plant impact
Level of compensation3
(XQ)
One-year adult impingement:
0
0.01
1.0
100.0
6. 4 -year adult impingement:
0
0.01
1.0
100.0
Lower

10.4
6.5
0.6
0.2

11.7
7.8
0.8
0.4
Middle6

13.4
9.9
1.1
0.5

14.9
11.4
1.3
0.6
Upper

19.5
16.8
2.6
1.2

21.6
18.9
2.9
1.5
aThe level of compensation ranges from no compensation (XQ = 0; density

 independent), to slight compensation (XQ = 0.01), to moderate compensation
 (XQ = 1.0), to very strong compensation (XQ = 100).

 See Fig. 3 for annual projection of middle case with one year of adult
 impingement.

-------
                                  A-26
     In Table 8  it  is  seen that the impact resulting  from  one year of
adult  impingement is smaller  than the impact  resulting  from 6.4 years
of adult  impingement,  even though equal  numbers  of  adults  are assumed
to be  impinged  in  both  cases.   In  the  latter case,  individuals with
higher reproductive value  are impinged.   In general,  younger fish are
more likely  to  be  impinged  than older  fish.   Thus,  annual  losses in
yield  should  lie somewhere between  the  estimates,  based on these two
extreme assumptions  concerning adult  impingement.
     Impacts are seen to be small when a moderate or very strong  amount
of compensation  is  assumed   (0.2  to 2.9%  loss  in  yield)   (Table 8).
Impacts are more  substantial  when  little or no compensatory  reserve is
assumed  (6.5  to  21.6%  loss   in  yield).   Because  yellow perch  in the
western basin of Lake  Erie are  heavily  fished (Patterson 1979,  Nepszy
1977), it appears that the assumption of little or no  compensation will
result in more  realistic  estimates  of  long-term  losses  to the  yellow
perch  fishery,  as compared with  estimates  based  on  the assumption of
moderate to very strong compensation.

                              4.  DISCUSSION

     Several   sources  of  error   enter   into   this   analysis  through
(1) natural variability  in the  environment,   (2)  errors in  the   esti-
mation of both model parameters and  inputs,  and (3) errors  in  the model
structure.  Natural  variability  results  from both temporal  and spatial
heterogeneity in the environment.  Errors in estimation result not  only
from natural  variability in the  data used to estimate model  parameters
and  inputs but  also from  biases or inaccuracies in  the  data.   Since

-------
                                  A-27
models are a simplification of the  system modeled,  errors  in the model
structure can also lead to errors  in the analysis.
     This  modeling  exercise  assumes  no   environmental   variability
(spatially or temporally) in the western basin  of  Lake Erie during the
100 years of the model projection.  Only in the three  density-dependent
cases  (Xg  =  0.01,  1, 100), where  age class 0  survival  depends  on the
eggs  produced  that  spawning  season,  does  any  parameter  value  change
with time.  Because  the  results  of this assessment are  presented  as a
comparison of projected yields with and without power plant impacts, it
is  believed  that  the  effect  of  the assumption  of  no environmental
variability is minor.
     Errors in the  data  and  simplifying assumptions necessary for some
estimates  to  be  made can cause  estimation errors  in the  model  para-
meters and  inputs.   The data used  to estimate the  life history para-
meters were  obtained for yellow perch  in  Lake Erie.  The  growth data
(length  and  weight) are  based  on  data  collected  prior to  1952 when
environmental conditions  may  have differed markedly  from  current con-
ditions  (Jobes  1952).   Similarly,  the  fecundity  relationship obtained
from  yellow  perch  in  the  eastern basin  of  Lake  Erie  (Sztramko  and
Teleki 1977) may  not be representative of yellow  perch  in the western
basin  of  Lake  Erie.   Overestimates  of   age-specific  egg  production
(resulting from overestimates of either length, weight, or  fecundity)
will  cause an underestimate of  ple>  which in  turn will result  in an
overestimate of  the power plant  impact (loss  in  yield to the yellow
perch  fishery).   Conversely,  underestimating  the   age-specific  egg
production will result  in an  underestimate  of the power plant impact.

-------
                                  A-28

Similarly, overestimating adult survival  (natural  and fishing survival)
will result in an overestimate of  the  power plant impact, while under-
estimating adult survival will result  in  an underestimate of the power
plant impact.  Ranges in  growth  and  adult survival  (lower, middle, and
upper cases  in Tables  3, 4,  and  5)  are intended to  reflect  natural
variability and errors  in estimating  life  history  parameters.   Ranges
in estimates of losses  of yield  to the yellow  perch  fishery (Table 8)
account in part for these uncertainties.
     Errors  in  estimating larval  concentrations  in  Ohio  and Michigan
waters can result from  gear  avoidance  and/or extrusion  by yellow perch
larvae.   Underestimates of  larval  concentration will result in under-
estimates of larval and  later  life-stage  abundances.   Overestimates of
the  conditional  power plant mortality rates will  result,  followed by
overestimates of the  power plant impact.   Another weakness in the data
concerns  the larval concentration data in Michigan waters in 1975.  The
start of  the spawning season was  missed.   This apparently resulted in
                                             _0
very low  estimates  of q  (on the order of  10   )  and  high estimates of
h.   Overestimates of  h  (and larval  abundance)  will  lead to underesti-
mates of  power plant impacts.
     Entrainment  estimates  will  be  biased   as  a   result  of  gear
inefficiencies and/or extrusion.  Errors  in entrainment  estimates are
also  introduced  by  inaccuracies   in  estimating  the  proportion  of
entrained  organisms  killed   by  the  power plant.   Patterson   (1979)
calculates a  fraction of 0.7  killed  for yellow  perch  larvae,  and he
uses this proportion  to reduce  the  number  of  larvae entrained  to the
number  of  larvae   killed due   to  entrainment.   Likewise,  errors  in

-------
                                  A-29

estimates of the number of  a  life stage killed due  to  impingement are
also introduced  as  a result  of  sampling inefficiencies and  errors  in
impingement survival.  Obviously, overestimates  of the number of a life
stage killed  due to entrainment  and/or impingement will  result  in  an
overestimate of  the  power plant  impact.   Conversely,  an  underestimate
of entrainment and/or impingement mortalities will  result  in an under-
estimate of power plant impact.
     Barnthouse  et  al.  (1979)  point out  two  assumptions  inherent  in
calculating  conditional  mortality  rates:   (1)  the  natural  causes  of
mortality and  power  plant impacts act independently of each other, and
(2) mortalities due  to  natural  causes and  power plant impacts are con-
stant during  the life stage  for  which they  are  estimated.  The  first
assumption is  likely to be  true,  while the latter assumption is  likely
violated  to  some extent.   Because  of the  seasonality often  associated
with  impingement,  power  plant  impacts  associated with   a  constant
impingement mortality over a  life stage will  introduce  error  of unknown
magnitude and  direction.   Another assumption  which introduces  error of
unknown  magnitude  and direction is  that of a  stable population,  which
was  assumed  twice  in this assessment.  First, the age class  0  survival
was  estimated  using this  assumption,  and,  second,  later  life-stage
abundances were  calculated  from estimates  of larval abundance  based on
this assumption.   Patterson's (1979)  weighting  factors are  assumed to
provide  reasonable  estimates  of  the  life-stage-specific  conditional
power  plant  mortality  rates based  on  ranges  of  entrainment  and/or
impingement  mortalities.    If the weightings cause  an overestimate of
the  conditional  power  plant mortality rate,  then  the power plant impact

-------
                                  A-30
will be  overestimated.   Conversely,  if the weightings  cause  an under-
estimate of  the conditional  power  plant mortality rate, then the power
plant  impact will  be underestimated.   The range  in  conditional  power
plant  mortality rates  (Table 7) based  on  ranges  in  life  history data
will hopefully give a reasonable range of power  plant  impacts.
     The  age  structure  approach  to  modeling  populations  provides  a
means  of assessing the long-term impacts  of power plants  (or any other
source   of  density-independent   incremental   mortality).    This   is
especially true when the  impacts  are not  uniform over  the entire life
span of  the  population.   The major  problem in  lending  realism to age
structure modeling is concerned with the controversial issue of density
dependence  (non-age  structure models  are not  free  of  this problem).
There  is  no  doubt  that  populations  in  their  natural   state,  when
moderately stressed, are able  to  respond in a compensatory manner.  It
is  also  true  that populations  under large  stresses  can  respond  in a
depensatory  manner resulting  in  a  population   "crash."   However,  the
main difficulty lies  in  estimating the  coefficients  which are used to
express  the  amount of  compensation.   In  this paper  a  range of values
for  the  coefficient  of compensation  is  used,  including the assumption
of  no  compensation.   Heavy  fishing  pressure  on  the  yellow  perch
population in  western  Lake  Erie implies that the estimates of loss in
yield  are better  represented  by  the  ranges  based  on  little or  no
compensation  (6 to 22%)  than by  the ranges based  on moderate  to very
strong compensation  (0.2 to  2.9%).   However,  the unknown  status of the
yellow perch  population   in western  Lake  Erie  precludes a  precise
specification of XQ.

-------
                                  A-31

                             ACKNOWLEDGMENTS

     Research  was  supported  by  the  U.S.   Environmental   Protection
Agency, Region V, under Interagency Agreement 40-650-77  (Task Order  No.
ORNL/EPA-V8)  with  the   U.S.   Department  of  Energy   under   contract
W-7405-eng-26 with  Union  Carbide Corporation.   I  would  like  to  thank
Dr. Lawrence Barnthouse,  Benjamin  Parkhurst,  Dr.  Vincent Gallucci,  and
Terrance Quinn for their comments on  this  manuscript.

-------
                                  A-32

                            LITERATURE CITED

Barnthouse, L. W., D.  L.  DeAngelis, and S.  W.  Christensen (1979).   An
     empirical  model   of   impingement  impact.    ORNL/NUREG/TM-290  and
     NUREG/CR-0639.     Oak   Ridge   National   Laboratory,   Oak   Ridge,
     Tennessee.
Bernardelli, H.  (1941).  Population waves.   J.  Burma Res.  Soc.  31:1-18.
DeAngelis, D. L., L.  J.  Svoboda, S. W.  Christensen,  and D. S.  Vaughan
     (1980).   Stability  and  return  times  of  Leslie  matrices  with
     density-dependent  survival:   Applications  to  fish  populations.
     Ecological  Modelling  8:149-163.
Emlen,  J.  M.  (1973).  Ecology:   An  Evolutionary  Approach.    Addison-
     Wesley Publishing Co., Reading, Massachusetts.   493 pp.
Goodyear,  C.  P.  (1977).   Assessing the impact of power plant  mortality
     on  the  compensatory  reserve  of  fish  populations,   pp.   186-195.
     IN:    W.    Van   Winkle    (ed.),   Assessing    the    Effects    of
     Power-Plant-Induced   Mortality  on  Fish   Populations.    Pergamon
     Press, New York.  380 pp.
Gull and, J.  A.  (1965).  Survival  of the youngest  stages of  fish,  and
     its relation to year-class  strength.   Special  Publication  of  the
     International  Commission  for  the  Northwest  Atlantic   Fisheries
     (ICNAF)  6:363-371.
Herdendorf,  C.  E.,  C. L.  Cooper,  M.  R.  Heniken,  and F. L.  Snyder.
     (1977a).  Western Lake Erie Fish  Larvae  Study  -  1975 Preliminary
     Data  Report.  CLEAR  Technical  Report  No.  47  (revised).   The  Ohio
     State  University Center  for  Lake Erie  Area   Research,  Columbus,
     Ohio.

-------
                                  A-33
Herdendorf, C.  E.,  C.  L.  Cooper,  M.  R.  Heniken,  and F.  L.  Snyder.
     (19775).   Western  Lake  Erie  Fish Larvae  Study  - 1976  Preliminary
     Data  Report.    CLEAR  Technical   Report   No.  63.   The  Ohio State
     University Center for Lake Erie  Area Research, Columbus, Ohio.
Jobes, F.  W.  (1952).  Age,  growth,  and production  of  yellow  perch  in
     Lake Erie.  Fishery Bulletin  52:205-266.
Leslie,  P.  H.  (1945).  On  the use  of matrices  in  certain population
     mathematics.  Biometrika 33:183-212.
Lewis, E.  G.   (1942).   On  the generation  and growth of  a  population.
     Sankhya 6:93-96.
Nepszy,  S. J.  (1977).   Changes  in  percid  populations   and  species
     interactions  in Lake  Erie.   Journal  of  the  Fisheries  Research
     Board of Canada 34:1861-1868.
Patterson,  R.   L.    (1979).   Production,  mortality, and   power plant
     entrainment  of larval   yellow  perch   in   western   Lake  Erie.
     EPA-600/3-79-087.    Environmental   Research    Laboratory,    U.S.
     Environmental  Protection Agency, Ouluth,  Minnesota.  187 pp.
Ricker,  W.  E.  (1975).   Computation  and  Interpretation  of Biological
     Statistics of  Fish Populations.   Bulletin  191.   Department of the
     Environment, Fisheries and Marine Service,  Ottawa,  Canada.   382 pp.
Sztramko,  L.,   and   G.  C.  Teleki.   (1977).   Annual   variation  in  the
     fecundity   of  yellow  perch   from  Long  Point  Bay,  Lake  Erie.
     Transactions of the American  Fisheries Society 106(6):578-582.
Usher,  M.  B.   (1972).    Developments  in   the  Leslie  matrix  model.
     pp.   29-60.   IN:  J.  N.  R.  Jeffers  (ed.),  Mathematical  Models  in
     Ecology.   Blackwell  Scientific Publications,  London.   398  pp.

-------
                                  A-34
Vaughan, D.  S., and  S.  B.  Sail a.   (1976).   A method  for determining
     mortality  rates  using  the  Leslie  matrix.   Transactions  of  the
     American Fisheries Society 105(3):300-383.

-------
                                  A-35

                                APPENDIX

                     ESTIMATION OF LARVAL ABUNDANCE

     Estimates of larval yellow perch  concentrations  are  available for
the Ohio and Michigan waters of the western basin  of  Lake Erie in 1975
and 1976  (Patterson 1979;  Herdendorf  et  al.  1977a,  1977b)  (Fig.  2).
Patterson  (1979)  uses  a  "materials balance  formulation"  to  estimate
larval yellow perch abundance for the  Ohio and  Michigan waters in 1975
and  1976.    He  describes  the  net  daily  rate of  change  in  larval
           •
abundance,  Q(t), for a  specified volume  as a  linear function of inputs
and outputs.   The  sole input  considered  is the  daily  production  of
larvae within the reference  volume,  h(t).   Outputs considered include
(1) daily net emigration  rate,  v(t);  (2)  daily rate  of recruitment to
the young-of-year  life  stage, r(t);  (3)  daily  rate  of larval natural
mortality,  m(t);  (4)  daily  loss rate  of  larvae  to  water  usage  other
than the Monroe  Power Plant,  L(t);  and (5) daily loss rate of  larvae to
entrainment at the Monroe  Power  Plant,  E(t).
     Production  of  larvae  is  distributed over  the  spawning season,
h(t),   using  the  binomial  probability distribution  (Patterson  1979);
i .e.,
                 h(t) = B -  h  •  ()  •  qx(l-q)m'x   ,                 (A-l )
for  the  time interval TQ  + xd  to  TQ +  (x+l)d;  where x  = 0,1	m;
h(t) is the  daily  production  rate;  B is the  reference  volume  in units
of  100 m   (9.4 x  10  for  Ohio waters  and  5.0 x  10   for  Michigan

-------
                                  A-36

waters); h  is  the mean total  number  of  larvae produced per 100  m  of
water  in the  reference  volume;  m+1   is the  maximum  number  of  time
periods over which  production  occurs  (m =  5);  q is  a  parameter  which
controls the  shape  of  the  binomial  distribution; d  is the  number  of
days in each of the  time periods  (d = 7  for Ohio waters and d = 14 for
Michigan waters);  and  TQ is  the Julian calendar  day  on which  pro-
duction  begins  (TQ  equals  127 and  106 for  Ohio waters  in  1975 and
1976,  respectively, and 120 and  106  for Michigan  waters  in  1975 and
1976, respectively).
     Losses  due  to   natural   mortality  are   assumed   to  follow  an
exponential  decay with  a  daily instantaneous mortality  rate,  p,  based
on  the  larval  life-stage  duration of 25 days  (Patterson 1979).   Since
the  losses  due to  "natural" mortality  were  adjusted  to  maintain the
population at equilibrium, the parameter p reflects not only losses due
to  natural  sources  of  mortality, but also  losses due  to entrainment
mortality [E(t) and L{t)] and  net emigration  [v(t)].   Thus,  the  terms
E(t),  L(t)  and v(t) become  redundant in the  materials balance  model.
Further, recruitment of larvae to the young-of-year life stage after 25
days, r(t),  is given by,
                         r(t) = h(t-25)e-25p  .                    (A-2)
                                      /
     Hence,  the materials balance equation  of  Patterson (1979) reduces
to:

               Q(t)  +  p •  Q(t)  = h(t)  - h(t-25)  • e-25P   ,          (A-3)

where  Q(t)  is  the number  of larvae present in  the  reference  volume on
            •
day t,  and  Q(t) is  the net rate  of change  in  larval  abundance per day

-------
                                  A-37

in the reference  volume  on  day t.    Equation  (A-3)  is easily solvable
over ranges of constant  values  of  h(t)  [general  solution in Appendix F
of Patterson  (1979)].  Daily estimates  of  Q(t) are obtained from esti-
mates of  yellow  perch  concentrations in  Ohio and Michigan  waters of
western Lake  Erie  (Patterson  1979;   Herdendorf  et al.  1977a,  1977b).
Using estimates  of larval  survival   in  Table  4   and  the  relationship
p = -ln(survival )/25 , Eq.  (A-3) contains  only two unknowns (h  and q).
A  consideration  of  the  sum  of squares  generated by  comparing  model
predictions over  ranges  of  h  and q with the  larval  concentration data
collected in Ohio and Michigan waters in 1975 and 1976 provides a means
of estimating h and  q.   Patterson  (1979) assumes values  for  q,  but in
this paper both h and q are  allowed to vary.
     Estimates of h are  provided in Table  5  for both Ohio and Michigan
waters in  1975 and 1976 separately   and averaged  over  the  two  years.
Larval production estimates  for the spawning season are calculated from
estimates of h by the relationship,
                        PRODUCTION  = d •  B  • h   .                  (A-4)

-------
                                     A-38
          "
           0,1*1
^2^2
                       0


                       0
                        o     o    •   «/?N-i
                                                            ORNL-DWG 73-2999R5
0,1
                                                              "
                                                               l.t
                               "
                                N-M
                                                                     =A(t)*n(n
                                                                4  YEARS (AGE)
        nc
        f,
Fig. 1.   Leslie matrix  formulation  of life  cycle  population model.   n(t) and
          n(t+l) are the population  vectors  at  times t and t + 1, respectively,
          Aft)  is  the  Leslie matrix,  the  Pi's are  survival   rates,  and the
          f-j's are fecundities.  The  bottom  part of this  figure  is  designed to
          show  the  relationship  of  the  parameters,  Pi  and  ff,  to  the age
          class  numbers,  rifff   F°>*  example,   p]   is  the  survival  rate   from
          age class 0 to age class  1.

-------
                                       A-39
                                                                 ORNL-OWG 60-11092 ESD
        DETROIT
           • •
MONROE
 POWER
PLANT
 TOLEDO
                              CANADA
      OHIO    SANDUSKY''';'•'•"-
     WATERS
                                                         0  (0  20  30  40  50  MILES
  Fig. 2.    Drawing  of Lake  Erie, showing detail  of the  western  basin and
             the location of the  Monroe Power Plant.

-------
                                  A-40
                                                 ORNL-DWG 80-7555  ESD
                                                              90
100
Fig. 3.   The reduction in  yield to  the yellow  perch  fishery due  to
          power  plant operation based  on four levels of  compensation.
          The  level   of  compensation   ranges  from  no   compensation
          (X0 =  0;   density   independent),    to   slight   compensation
          (XQ =  0.01),  to moderate  compensation  ()fo  = 1.0),  to  very
          strong  compensation   0^,   =  100).    Power  plant  mortality
          occurs for the first  40 years, and no  power  plant  mortality
          occurs for the  remaining  60  years  of  the  projection.   The
          projections are  based on  the middle case with only  one  year
          of adult  impingement.

-------
                                   B-l

                               Appendix B
                    MISCELLANEOUS EDITORIAL COMMENTS
B.I  Main Text
     Page     Line
       3    Footnote 1
      79        13
      82        30
      83
20
83
83
116
121
131
135
29
30
19
7
36
14
(s/r = should be replaced  with)
      Comment
Appendix 8 s/r Appendix H
N(t) = s/r N(t) =
0 s/r 0.00001; and 365 s/r 169; followed by  a
new line:     0         169  < t _<  365
0 s/r 0.00001; and 365 s/r 148; followed by  a
new line:     0         148  < t <_ 365
0.0001 s/r 0.0022
0 s/r 0.0001;  and  365 s/r 204;  followed by  a
new line:     0          204  < t <  365
numer s/r number
a = 1 . e-(m+f) T for Eq.  (2i.i)
parmaters s/r parameters
the tables s/r Table 33
B.2  Tables
     Table
       1
      2-7

       8
  Comments
  Are the units:   no./100 m3?
  Source  of  data?   [page 37  is  apparently  from  Cole
  (1977)].
  Should  state that  Table 8 is summary  of Tables  2-7.
  Day of  year  in  Table  8 does not agree with  day/month
  system  in  Tables  2-7.   Means for  Day  142 are off  by
  less than 0.1% for rows 1,  2, 5,  and  6.

-------
                                  B-2
B.2  Tables (cont'd)

     Table        Comments

       9          Breakdown  by  stage of development unnecessary.

      10          Day  of year  system  in Table  10  does not  agree  with
                  day/month   system  in  Table   9   (see   footnote   to
                  Table  10).   Data from 6/14-29  in Table  9  apparently
                  not  used in  summary in Table 10.

      11          Not  needed.

     12A-B        Table  12B  not needed:   Sector  B of Table 12B - switch
                  values for 6/21-23 with  7/1-3.  Both  zones  A and  B
                  were  not  sampled on  5/12-14  (footnote to  a  Table
                  12B),  why  was zone A  estimated from zones  C-E  while
                  zone B was  set  to zero?   The following errors  were
                  found  in Table 12B for sector D:

                             5/12-14          2.32 s/r 3.48
                             5/22-25          9.24 s/r 9.57
                             6/1-4           3.98 s/r 4.06

      12C         Should reference Table  13 which  provided  weightings
                  for  combining of Sectors  (Table 12B).

      12E         Table  12E  not needed.

      12F         Under  data  column:    5/28 s/r  4/28.   The  following
                  errors were found for  Mean Concentration column:

                             4/21-23          2.2xlO-3 s/r  5.8xlO~4
                             5/30             2.13x10-'  s/r 9.33x1O'2

                  Should reference Table  13 which  provides  weightings
                  for  combining Sectors  (Table 12E).

      13          This table should be  referenced  from Section 3 where
                  it is  needed in  conjunction with  Tables  12A-F.

      14          This  table  should  be   deleted   or  referenced  from
                  Section  3  for comparison  purposes.

     15-16        Not used in analysis.

      17          This  extensive   table  could  probably   appear  as  an
                  appendix,  although  it is not used in analysis; should
                  be referenced on page  92.

-------
                                   B-3
B.2  Tables  (cont'd)
     Table
     18-21
     25-32
      33
     34-35
Comnents
Should be reduced to just Table 21
Should appear as appendix.
Not referenced from text (see C.I)
Should summarize as single table.
B.3  Figures and Diagrams
     Figures      Comments
     3-7, 9-30
      31-34
       45
Drop or put in appendix  (might  put 1  or 2 in  text  as
examples).  Figure  17  not  referenced  (should be  on
page 45).
Of marginal interest.
Marginal; denote as Fig. H.I
     Diagrams     Comments
        A         Discussion  in  Section   2   could  be  more  closely
                  organized about specific sources of error outlined in
                  this diagram.
        B         Subscript left off of E.
B.4  Appendices
     Appendix
         A
Comments
Where are the data for null hypothesis 2  (bottom  sled
tow);  with  different  sampling  gear  what  is   the
validity  of  this   test?    Missing  data  point   in
Table A.I  for Surface Day  sample on  5/23.

-------
                                   B-4
B.4  Appendices (cont'd)
         B
Sample sizes for depth zones 18 ft - 24 ft, 25 ft and
24 ft  - 30  ft  (15)  are  greater than  I  count  from
Table 9 (23 and  13,  respectively).   Are data missing
from  Table 9 for Apr.  26-29,  1976?

For  Day  118,  second  column:   n-j  s/r   i\2-   Also,
for Day 118  I count 51  sample  points  for depth zone
12 ft  - 30 ft  instead of  55.   Data  for  6/14-29 in
Table 9 is not worked  up.
                  Data are presumed  to come
                  with respect  to  Table  11,
                  deleted.  Not  called  out In
                            from  Table 9.
                             this  appendix
                            text.
As stated
 could  be
                  As  pointed  out by Patterson,  methods 3 and 4 are less
                  susceptible to biased estimates  than methods 1  and 2,
                  although  several  potential  sources  of  bias  which may
                  effect methods  3   and   4  have  not  been  addressed
                  (Boreman  and Goodyear 1980).  What is the  source of
                  the data  in example, how was  it collected,  and where
                  were the  intake and  discharge  sampling stations?

                  Michigan  Water, 1976 (using Eq.  10), ranges of t:

                                      189  s/r 190
                                      200  s/r 201
                                      214  s/r 215

                  Michigan  Water, 1975 (using Eq.  9),  range of t:

                                      189  s/r 190

                  Michigan  Water, 1975 (using Eq.  9 and 14), range of t:

                                      189  s/r 190

                  Delete since not  used in any analysis.

-------