A Mathematical Model of Coevolving Populations

Abstract

We present a general model for coevolution of two interacting populations which we believe can serve as a framework for theoretical investigations in this area. The model describes coevolution of two competitors, exploiter and victim. or two mutualistic species. The conditions on the parameters of the model which are required for local stability of various equilibria are derived. In order to simplify analysis the stability conditions are separated into those which are independent of the nature of the time response of the system (intrinsic conditions) and those which depend on the time response. For the most part, only intrinsic conditions are discussed. Because of the nature of the model the results of our analysis are stated only in general terms. Much of their significance will depend on the particular coevolutionary systems being analyzed. The results of our analysis indicate that coevolution may result in stable coexistence of two populations even when the stability conditions derived from generalized versions of the Lotka-Volterra equations are violated; that coevolution may also destabilize what appears to be a stable interaction between populations; and that coevolution may result in stable disruptive selection regimes. Local stability has been the focus of this paper, but the local instability, say, of polymorphic equilibria does not exclude the possibility of time-dependent polymorphic solutions. We shall discuss such solutions in further detail in a subsequent paper.