On nonlinear population waves

摘要

We discuss a model system of partial differential equations for description of the spatio-temporal dynamics of interacting populations. We are interested in the waves caused by migration of the populations. We assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations. For the particular case of one population and one spatial dimension the general model is reduced to analytically tractable PDE with polynomial nonlinearity up to 4th order. We investigate this particular case and obtain two kinds of solutions: (i) approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and (ii) exact solutions which describe nonlinear kink and solitary waves. In an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node. Finally we derive conditions for the asymptotic stability of the obtained solutions.