Model stability, resilience, and management of an aquatic community

A community model may be considered stable when, in the absence of exogenous variation, all population trajectories encircle or asymptotically approach equilibrium. In this paper, community models in which, in the absence of analytical indications of stability, all populations either 1. exhibit trajectories toward equilibrium or 2. possess properties such that departures from equilibrium are inhibited will be defined as resilient. The necessary properties include appropriate sensitivity (i.e., the total derivative, df _i /dV _j , of the i ^th species function, f _i =dN _i /dt, with respect to the j ^th variable) to exogenous variables. A real, though simplified, ecological system consisting of Daphnia galeata and its algal food source in an oligotrophic lake appears to be generally resilient in that changes in the exogenous factors nitrate concentration and temperature of the lake water consistently restrain the departure of predicted population densities from equilibrium.Each population in the community is represented by the Verhulst-Pearl logistic model of population growth augmented to include environmental effects on rate of increase, r; carrying capacity, K; and the effects of predation on population density, N; and therefore the population rate of change, dN/dt.It is suggested that such community submodels and sensitivity analysis represent logical and appropriate amplifications in the use of mathematical models in the management of populations.