&ER&
United States
Environmental Protection
Agency
Environmental Research
Laboratory
Athens G A 30613
Research and Development
EPA-600/S3-82-030 May 1982
Project Summary
Persistence in
Model Ecosystems
T. C. Card
Mathematical models aid in under-
standing environmental systems and
in developing testable hypotheses
relevant to the fate and ecological
effects of toxic substances in such
systems. Within the framework of
microcosm or laboratory ecosystem
modeling, some differential equation
models, in particular, become tracta-
ble to mathematical analysis when the
focus is on the problem of persistence.
In this report, a hierarchy of microcosm-
related models, the top level of which
contains a nutrient-producer-grazer
food chain, and general food chains
are analyzed for persistence. The re-
sults, which take the form of inequali-
ties involving model parameters, spec-
ify necessary conditions and sufficient
conditions for continued presence of
the model components throughout
indefinite time intervals. These results
can serve as a basis for preliminary
evaluations of model performance.
This Project Summary was develop-
ed by EPA's Environmental Research
Laboratory, Athens. GA, to announce
key findings of the research project
that is fully documented in a separate
report of the same title (see Project
Report ordering information at back).
Introduction
Persistence in mathematical repre-
sentations of ecosystems is the analog
for survival of organisms and continued
presence of nutrients in the modeled
system. Although exact solutions of the
non-linear, differential equations de-
scribing growth-decay rates of sub-
stances or organisms in environmental
systems are impossible to obtain, quali-
tative solutions for persistence are
possible. The technique consists of
constructing auxiliary functions and
differential inequalities to determine
necessai^ardsufficientconcttionslbrpersistence
from model parameters.
This technique is applied to a variety
of generally accepted ecological models,
beginning with the basic chemostat,
including cycling components, and final-
ly including a top level predator. Neces-
sary and sufficient conditions for per-
sistence are obtained for the chemostat
and cycling models. Only necessary
conditions, however, are obtained for
the full system.
General Lotka-Volterra food chain
models are also investigated. Persis-
tence of a top level predator is estab-
lished for very general models. Persis-
tence at the top level in a Lotka-Volterra
model of general omnivory is also
developed.
All the results, which have the form of
inequalities involving model parameters,
specify conditions for continued pres-
ence of the model components for an
indefinite time. These results can serve
as a basisfor preliminary model evalua-
tion and experimental design.
Summary
The mathematical analysis focusing
on persistence in differential equation
models results in criteria taking the
form of inequalities involving model
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parameters. For the five-component,
microcosm-related model and its sub-
systems, these results are expressed in
terms of threshold values for the nutri-
ent input rate. In the case of the food
chain models, positivity of weighted
sums of the intrinsic growth-decay rates
constitute the conditions for persistence.
Generally, the nutrient input level, the
intrinsic growth rate of the food source,
or the rate of supplementation of inter-
mediate predator being sufficiently large
guarantees persistence. The methods
used consist of approximating the differ-
ential equations; in particular, lineariza-
tion and the auxiliary function-compari-
son principle techniques are the main
tools employed from the qualitative
theory of ordinary differential equations.
In each of these models, it has been
shown that solutions are bounded. A
consequence of persistence for auton-
omous (right hand side of equations
independent of t explicitly) systems, in
this case, is that any solution with
positive initial values must approach
asymptotically either an equilibrium or a
solution of oscillatory type in the interior
of the feasible region. Furthermore, and
possibly most significant, is the fact that
persistence implies the existence of a
recovery mechanism in the model. Solu-
tions exhibiting small component den-
sities at some time t tend to equilibrium
or oscillatory type stable density configu-
rations in the interior of the feasible
region. That these models do not take
into account stochastic effects contri-
butes to this result.
T. C. Gard is with the Department of Mathematics, University of Georgia, Athens,
GA 30602.
J. Hill IV is the EPA Project Officer fsee below).
The complete report, entitled "Persistence in Model Ecosystems, "(Order No. PB
82-196 916; Cost: $6.00, subject to change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Environmental Research Laboratory
U.S. Environmental Protection Agency
Athens. GA 30613
U S GOVERNMENT PRINTING OFFICE. 1982 — 559-017/0721
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
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