Numerical Study of Bifurcations Occurring at Fast Timescale in a Predator–Prey Model with Inertial Prey-Taxis

Bifurcations occurring in a system of partial differential equations (PDEs) describing spatiotemporal dynamics of predator and prey populations with prey-taxis have been studied numerically. The model of the local kinetics of the system assumes logistic reproduction of the prey and a simplest Lotka–Volterra functional response of the predator. Since the model ignores relatively slow and rare demographic processes of birth and death in the population of predator, the predator abundance is kept constant under the considered zero-flux boundary conditions. The abundance of predator populations together with the predator taxis coefficient were used as bifurcation parameters in the numerical study that have been made with help of two qualitatively different techniques of discretization: the Bubnov–Galerkin method and grid method of lines. It has been shown that the considered simple model of prey-taxis in predator–prey system demonstrates complex bifurcation transitions leading to periodic, quasi-periodic and chaotic spatiotemporal dynamics.