Eur. Phys. J. Plus (2022) 137: 134
https://doi.org/10.1140/epjp/s13360-022-02358-7

Regular Article

Complex dynamics in a reaction-cross-diffusion model with refuge depending on predator–prey encounters

¹ School of Science, Zhejiang University of Science and Technology, 310023, Hangzhou, People’s Republic of China
² Department of Mathematics, Visva-Bharati, 731235, Santiniketan, India

^a rjhan@zust.edu.cn

Received: 27 September 2021
Accepted: 6 January 2022
Published online: 18 January 2022

Abstract

The contemporary study copes with a generalist predator–prey model with nonlinear cross-diffusion embracing prey refuge in proportion to both the species along with the ratio-dependent functional response. The investigation commences with the well-posedness keeping in view of the existence of all feasible non-negative equilibria together with their global dynamics for the corresponding temporal model. The refuge parameter plays a key role in the dynamics of the model in general and mediates the uniform persistence, the stability of the boundary and coexistent equilibria and even the Turing instability space, in particular. Another important observation is that refuge possessing a constant proportion predator–prey encounters leads to a weaker stabilizing effect than refuge possessing a constant proportion prey under certain system parameters. Subsequent exploration on three eminent classes of mechanism for Turing instability around the positive spatially homogeneous steady state reveals dynamical complexity stimulated by generalist predator of the model system. Some comparisons are given between refuge depending on prey and refuge depending on both the species. Finally, numerical simulations expose to view the growth of spatiotemporal patterns controlled by prey refuge together with both self- and cross-diffusion following the sequence of spots, stripe–spot mixtures, stripes, labyrinth, stripe–hole mixtures and holes as well. Thus the proposed model appears to be rich and complex from the dynamical point of view having novelty to contribute much to the system in the realm of ecology.