&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 ------- 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. 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