Kinetic Descriptions of Chemical and Biological Systems:

On selection dynamics with nonlocal competition

Hailiang Liu

Iowa State University


In this talk, we will present some nonlocal PDE models of selection dynamics and their numerical approximations. The models describe the evolution of a population structured with respect to a continuous trait, subject to nonlocal (linear or nonlinear) competition, leading therefore to selection, with or without mutation. In these models steady state is not unique, which one is selected and globally stable is important at both continuous and discrete levels Here we discuss the discrete dynamics of some modes. Various time-asymptotic convergence rates towards the discrete evolutionary stable distribution (ESD) are established. For some special ESD satisfying a strict sign condition, the exponential convergence rates are obtained for both semi-discrete and fully discrete schemes. Towards the general ESD, the algebraic convergence rate that we find is consistent with the known result for the continuous model. The numerical solutions of the model with small mutation are shown to be close to those of the corresponding model with linear competition.