United States
                    Environmental Protection
Environmental Research
Athens GA 30613
                    Research and Development
EPA-600/S3-84-081  Aug. 1984
c/EPA          Project  Summary
                    Analysis  of  Persistence  in
                    Model  Ecosystems:
                    Deterministic  and  Stochastic
                    Food  Web  Models
                    Thomas 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. With-
                    in the framework of microcosm  or
                    laboratory ecosystem modeling, some
                    differential equation models, in partic-
                    ular, become tractable to mathematical.
                    analysis when the focus is on the
                    problem of persistence.
                      In this report, a microcosm-related,
                    nutrient-producer-grazer, chemostat-
                    chain model  and general food web
                    models  are analyzed for  persistence.
                    The results, which take the  form  of
                    inequalities involving model param-
                    eters, specify  sufficient conditions for
                    continued presence of the model com-
                    ponents throughout indefinite time in-
                    tervals.  These results  can serve as a
                    basis for a preliminary evaluation  of
                    model performance.

                      This Project Summary was developed
                    by EPA's Environmental Research Lab-
                    oratory, 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 order-
                    ing information at back).

                      Persistence m mathematical represen-
                    tations of ecosystems in the analogue for
                    survival  of organisms  and continued
                    presence  of nutrients  in  the modeled
                    system. Although exact solutions of the
                    nonlinear differential equations describ-
ing growth-decay rates of the substances
or organisms are impossible to obtain,
qualitative investigations for persistence
are possible. The technique consists of
constructing auxiliary functions and dif-
ferential inequalities to determine condi-
tions for persistence from model param-
  This technique is applied to a variety of
generally  accepted ecological models.
Specifically, previous results for a nutrient-
producer-grazer chemostat-chain have
been improved; sufficient  as well as
necessary conditions for persistence are
obtained.  General Lotka-Volterra type
deterministic and stochastic food web
models are investigated. Sufficient condi-
tions for persistence of a top-level preda-
tor are  established for models that can
represent  arbitrary numbers of trophic
levels and species per trophic level and
arbitrary degrees of intraspecific compe-
tition and omnivory.
  All the results, which take the form of
inequalities involving model  parameter
bounds, specify conditions for continued
presence of the model components for an
indefinite time.  These results can serve
as a basis  for preliminary model evalua-
tion and experimental design.

  The mathematical analysisfocusing on
persistence in differential equation mod-
els of current and potential interest to the
U.S. Environmental Protection Agency
results  in  criteria taking the form of
inequalities involving model parameters.

   For the chemostat-chain model, the main
   result is expressed in terms of a threshold
   value for the nutrient  input rate. In the
   case of deterministic food web models,
   positivity of  minimum  weighted net
   growth rates constitutes the persistence
   conditions. For stochastic food web mod-
   els,  these net  growth rates must exceed
   corresponding weighted sums of the
   random fluctuation intensities. Generally,
   the  nutrient input level or a sufficiently
   large total intrinsic  growth rate  of the
   food sources guarantees persistence.
    The methods used consist of approxi-
   mating  the differential  equations;  in
   particular, the auxiliary function-compar-
   ison principle techniques are the main
   tools employed from the qualitative theory
   of ordinary and  stochastic differential
   equations. The extension of the results
   previously reported to the case of general
   food web models demonstrates the robust
   nature of this approach to the problem of
   ecosystem model stability.
         Thomas C. Card is with University of Georgia. Athens. GA 3O602
         Harvey W. Holm is the EPA Project Officer (see below)
         The complete report, entitled "Analysis of Persistence in Model Ecosystems
           Deterministic and Stochastic Food Web Models," (Order No PB 84-226 984
           Cost: $7.00, 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 30614
                                             US GOVERNMENT PRINTING OFFICE, 1984 759-O1 5/7780
United States
Environmental Protection
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