Quasisoliton Interaction of Pursuit-Evasion Waves in a Predator-Prey System

Abstract

We consider a system of partial differential equations describing two spatially distributed populations in a "predator-prey" interaction with each other. The spatial evolution is governed by three processes: positive taxis of predators up the gradient of prey (pursuit), negative taxis of prey down the gradient of predators (evasion), and diffusion resulting from random motion of both species. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the taxis and is entirely different from waves in a reaction-diffusion system. Unlike typical reaction-diffusion waves, which annihilate on collision, these "taxis" waves can often penetrate through each other and reflect from impermeable boundaries.

Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator–prey pursuit and evasion example

“…To explain specific phenomena characteristic of various population communities, we have performed studies of pursuit-evasion waves in a mathematical model of a predator-prey system. Our results presented in [1] and here demonstrate that population taxis waves exhibit special properties, which provide insight not only into specific mechanisms of population dynamics, but also broaden the class of nonlinear waves to include quasi-solitons in reaction-diffusion-taxis systems.…”

A Minimal Model of Pursuit-Evasion in a Predator-Prey System

“…We base our model on the hypothesis that taxis acceleration is proportional to the density gradient of another population playing a role of taxis stimulus. This hypothesis allows overcoming known limitations of conventional spatial predator-prey models based on crossdiffusion equations that assume the direct dependence of taxis velocity on density gradient of population-stimulus [25,7,4,40].…”

“…However, in the pursuit-evasion model of [40,41,39,38] (as well as in other models of taxis and diffusive movements of populations, that exhibit heterogeneous solutions under boundary conditions (2.3)) local kinetics (i.e., the birth/death processes of both populations) being coupled with spatial behaviour of species plays a critical role in emergence of spatial structures. Moreover, existence of stable spatially heterogeneous solutions requires essential nonlinearity of the functions describing local dynamics of system, in interplay with spatial behaviour of animals.…”

“…Citing from a recent instance, we would like to consider here a pursuit-evasion model investigated by Tsyganov and co-authors [40,41,39,38]:…”

“…Thus, the local kinetics of model (2.2) consists of logistic reproduction of prey, Holling type-III trophic function, and constant rate of predator mortality. The authors have demonstrated that the model has very interesting spatially heterogeneous regimes including solitonic [40,41] and spiral waves [38]. Basing on the mathematical models of type (2.2) and experimental work with bacterial populations, such waves in excitable cross-diffusion systems have been identified as a new class of nonlinear waves [39].…”