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

Abstract

We have studied properties of nonlinear waves in a mathematical model of a predator-prey system with pursuit and evasion. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the "taxis", represented by nonlinear "cross-diffusion" terms in the mathematical formulation. We have shown that the dependence of the velocity of wave propagation on the taxis has two distinct forms, "parabolic" and "linear". Transition from one form to the other correlates with changes in the shape of the wave profile. Dependence of the propagation velocity on diffusion in this system differs from the square-root dependence typical of reaction-diffusion waves. We demonstrate also that, for systems with negative and positive taxis, for example, pursuit and evasion, there typically exists a large region in the parameter space, where the waves demonstrate quasisoliton interaction: colliding waves can penetrate through each other, and waves can also reflect from impermeable boundaries.

Cross-diffusion and pattern formation in reaction–diffusion systems

“…Examples of systems that exhibit cross-diffusion include strong electrolytes, [23][24][25] micelles, 26,27 or microemulsions, 28 and systems containing molecules of significantly different sizes, for example protein-salt. [29][30][31] Similar phenomena also occur in biological systems, but since the transport process is driven by an input of energy, as, for example, in bacterial chemotaxis [32][33][34] or in predator-prey systems, [35][36][37][38] these are not true diffusion processes. Nonetheless, the mathematical description of biological or ecological cross-diffusion is the same as in physicochemical systems, as we discuss below.…”

Families of traveling impulses and fronts in some models with cross-diffusion

“…The approximation of the taxis term is an "upwind" explicit scheme [25] which is frequently used for cross-diffusion systems (e.g., [26]). More precisely, …”

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

“…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].…”

“…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.…”