&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
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Information
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