Adaptive Dynamics Based on Ecological Stability

An important step in coevolution occurs when a new mutant clone arises in a resident population of interacting individuals. Then, according to the ecological density dynamics resulting from the ecological interaction of individuals, mutants will go extinct or replace some resident clone or work their way into the resident system. One of the main points of this picture is that the outcome of the selection process is determined by ecological dynamics. For simplicity, we start out one from resident species described by a logistic model, in which the interaction parameters depend on the phenotypes of the interacting individuals. Using dynamic stability analysis we will answer the following purely ecological questions: After the appearance of a mutant clone,

(1)

what kind of mutant cannot invade the resident population,

(2)

and what kind of mutant can invade the resident population?

(3)

what kind of mutant is able to substitute the resident clone,

(4)

and when does a stable coexistence arise?

We assume that the system of mutants and residents can be modelled by a Lotka-Volterra system.We will suppose that the phenotype space is a subset of R

n

and the interaction function describing the dependence of the parameters of the Lotka-Volterra dynamics on the phenotypes of the interacting individuals is smooth and mutation is small. We shall answer the preceding questions in terms of possible mutation directions in the phenotype space, based on the analysis of ecological stability. Our approach establishes a connection between adaptive dynamics and dynamical evolutionary stability.