Effects of density dependent cross-diffusion on the chaotic patterns in a ratio-dependent prey-predator model

Abstract

Spatio-temporal chaos is an intriguing part of the spatio-temporal pattern formation, observed in many interacting population models when their heterogeneous distributions within their habitats and movement from one location to the other are taken care of within the modeling approach. When the homogeneous steady-states become unstable, the solutions of the corresponding reaction–diffusion systems never approach a stationary state rather exhibit an irregular nature with respect to both space and/or time. Cross-diffusion terms are incorporated in a system of reaction–diffusion equations to model the situation where presence, absence, abundance of one species influence the movement of another species and vice-versa. In this work, cross-diffusion is considered in a prey–predator model with ratio-dependent functional response along with the self-diffusion terms. After deriving the Turing instability conditions in terms of cross-diffusion parameters, extensive numerical simulations are carried out to study the effect of cross-diffusion on the chaotic dynamics and stationary Turing patterns generated in the system containing self-diffusion terms only. Appropriate numerical tools are used to characterize the spatio-temporal chaos. Route to spatio-temporal chaos and its disappearance are discussed in detail. The chaotic dynamics of the self-diffusion model may be suppressed leading to a stationary state or preserved depending on the cross-diffusion coefficients. The stationary patches of both the species generated in the Turing domain remain stationary but their configurations may change due to the effect of density-dependent cross-diffusion. On the other hand, stationary patches generated in the Turing-Hopf domain change to spatio-temporal chaos for higher dispersal rates of predator avoidance by prey.