Spatial patterns through Turing instability in a reaction–diffusion predator–prey model

Pattern formation in nonlinear complex systems is one of the central problems of the natural, social and technological sciences. In this paper, we consider a mathematical model of predator–prey interaction subject to self as well as cross-diffusion, arising in processes described by a system of reaction–diffusion equations (coupled to a system of ordinary differential equations) exhibiting diffusion-driven instability. Spatial patterns through Turing instability in a reaction–diffusion predator–prey model around the unique positive interior equilibrium of the model are discussed. Furthermore, we present numerical simulations of time evolution of patterns subject to self as well as cross-diffusion in the proposed spatial model and find that the model dynamics exhibits complex pattern replication in the two-dimensional space. The obtained results unveil that the effect of self as well as cross-diffusion plays an important role on the stationary pattern formation of the predator–prey model which concerns the influence of intra-species competition among predators.