United States
Environmental Protection
Agency
Environmental Research
Laboratory
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).
Introduction
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-
eters.
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.
Summary
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.
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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
Agency
Center for Environmental Research
Information
Cincinnati Ori 45268
Official Business
Penalty for Private Use $300
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