Diffusion-reaction-advection equations for the predator-prey system in a heterogeneous environment

We analyze variants of considering the inhomogeneity of the environment in computer modeling of the dynamics of a predator and prey based on a system of reaction-diffusion-advection equations. The local interaction of species (reaction terms) is described by the logistic law for the prey and the Beddington–DeAngelis functional response, special cases of which are the Holling type II functional response and the Arditi–Ginzburg model. We consider a one-dimensional problem in space fora heterogeneous resource (carrying capacity) and three types of taxis (the prey to resource and from the predator, the predator to the prey). An analytical approach is used to study the stability of stationary solutions in the case of local interaction (diffusionless approach). We employ the method of lines to study diffusion and advective processes. A comparison of the critical values of the mortality parameter of predators is given. Analysis showed that at constant coefficients in the Beddington–DeAngelis model, critical values are variable along the spatial coordinate, while we do not observe this effect for the Arditi–Ginzburg model. We propose a modification of the reaction terms, which makes it possible to take into account the heterogeneity of the resource. Numerical results on the dynamics of species for large and small migration coefficients are presented, demonstrating a decrease in the influence of the species of local members on the emerging spatio-temporal distributions of populations. Bifurcation transitions are analyzed when changing the parameters of diffusion-advection and reaction terms.