A minimal model of pursuit-evasion in a predator-prey system

A conceptual minimal model demonstrating spatially heterogeneous wave regimes interpreted as pursuit-evasion in predator-prey system is constructed and investigated. The model is based on the earlier proposed hypothesis that taxis accelerations of prey and predators are proportional to the density gradient of another population playing a role of taxis stimulus. Considering acceleration rather than immediate velocity allows obtaining realistic solutions even while ignoring variations of total abundances of both modelled populations. Linear analysis of the model shows that stationary homogeneous regime becomes oscillatory unstable with respect to small heterogeneous perturbations if either taxis activities or total population abundances are high enough. The ability for active directed movement of both prey and predators is the necessary condition for spatial self-organization. Numerical simulations illustrate analytical results. The relation between the proposed model and conventional two-component systems with cross-diffusion is discussed.