Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

Abstract

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.