Non-linear Waves of Interacting Populations with Density-Dependent Diffusion

In this study we consider a mathematical model based on diffusion-reaction type PDEs to describe the spatio–temporal behaviour of interacting populations. We extend the generalized reaction–diffusion model for spatio–temporal dynamics of interacting (competing) populations, proposed in [N. K. Vitanov, I. P. Jordanov, Z. I. Dimitrova, Applied Mathematics and Computation 215, 2950–2964 (2009)], by adding density–dependent diffusive terms to the equations therein. The new density–dependent terms describe the intra– and inter–specific spatial interactions among individuals leading to formation of aggregation structures or individual repulsion. We are interested in the particular case of one population migrating in one spatial direction. Then the generalized model is reduced to one PDE of diffusion–reaction kind. We extract a general analytical solution of the considered equation applying the modified method of simplest equation. The ordinary differential equation of Bernoulli is used as simplest equation. Numerical study of the obtained analytical solution is made. We show that the population density wave can vary in its profile depending on assumptions made about density–depending diffusion coefficients.