Cross-diffusion-driven instability in an interacting species model with prey refuge

•The cross-diffusive predator-prey model with the inclusion of prey refuge is formulated.•A brief re-examination of different dynamical properties corresponding to its temporal counterpart is performed.•The stability analyses of the cross-diffusive system along with Turing space are carried out successfully.•Special emphasis is paid on the impact of prey refuge in the cross-diffusive system to observe how the growth of the interacting species gets substantially perturbed.•The emergence of spatial patterns based on numerical simulations is pointed out.•Conclusions are drawn from several numerical outcomes of the proposed model. The present study appertains to a reaction-diffusion system embracing two-dimensional continuous Beddington-DeAngelis predator-prey model incorporating intra-specific competition among predators and prey refuge in proportion to both the species as well. The existence of all conceivable ecologically significant equilibria is explored and consequently the diffusion-driven instability around the coexistence equilibrium position is reviewed. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of diffusion-driven instability. The dynamical system in Turing space emerges further to get influenced by prey refuge while it unveils diffusion controlled spatiotemporal pattern formation namely, growth of spots, stripe-spot mixtures, stripes, labyrinthine, stripe-hole mixtures and holes reproduction. The quantitative analysis reveals that the interaction of both self- and cross-diffusion plays a significant role on the pattern formation of the present system in a way to enrich the pattern at a greater height.