Abstract
In this paper, a reaction-diffusion prey-predator system including the fear effect of predator on prey population and group defense has been considered. The conditions for the onset of cross-diffusion-driven instability are obtained by linear stability analysis. The technique of multiple time scales is employed to deduce the amplitude equation near Turing bifurcation threshold by choosing the cross-diffusion coefficient as a bifurcation parameter. The stability analysis of these amplitude equations leads to the identification of various Turing patterns driven by the cross-diffusion, which are also investigated through numerical simulations.
Highlights
The dynamics of interacting predator–prey models have been extensively studied by several researchers interested in characterizing the long-term behavior of the species
A more realistic formulation of predator–prey interactions cannot be reduced to the simple description of direct predation effects; it requires the modelling of non-consumptive effects of predators
While self-diffusion terms are used to model the random movement of prey and predator individuals, cross-diffusion terms are included in addition to population models to account for the influence of species interactions on the individuals’ movement
Summary
The dynamics of interacting predator–prey models have been extensively studied by several researchers interested in characterizing the long-term behavior of the species. Cross-diffusion-driven spatial patterns are studied by deriving through multiple scale analysis the amplitude equation This is the tool of choice to understand the spatial dynamics of a reaction-diffusion system for parameter values in the vicinity of a Turing bifurcation point. This approach can be extended to other interacting models with different functional responses, and in other fields of applied mathematics where nonlinear mathematical models having a similar structure are considered ([35,36]) and a comparative study of the pattern formation scenario can be explored. Details of the derivation of amplitude equations are given in Appendix A
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