Nonlinear travelling waves in a generalized model of interacting dense populations

Abstract

In this study we extend the generalized reaction-diffiusion model presented in [20],[21], which describes spatio-temporal dynamics of interacting populations. In more detail, we generalize the model system of differential equations for the interaction of three populations in which the growth rates and competition coefficients of populations depend on the number of members of all populations. The model describes several novel features of the interacting agents compared to the well-known classic models in population dynamics. Using particular case of the recently developed SEsM (Simple Equations Method) namely the Modified method of Simplest Equation [5]–[8] and one of its extended versions [8, 9], we obtain a new traveling wave solution of the model system.We assume that nonlinearity in growth rates and interaction coefficients in the generalized model exist according to high density of their individuals. An analytical solution of the extended model is derived. Traveling wave solutions of these equations are of special interest as they describe the motion of wave fronts or the motion of boundary between two different states existing in this system. Numerical simulations of this solution demonstrate propagation of nonlinear waves in the considered extended model. The characteristics of the the obtained traveling wave solution are visualized and discussed.