Pattern dynamics in a predator–prey model with Smith growth function and prey refuge in predator poisoned environment

In this article, we investigate temporal as well as spatiotemporal dynamics of a predator–prey system where prey grows according to Smith growth function and prey refuges in predator poisoned environment. Self as well as cross diffusion are taken into consideration to describe spatial moment of the species which make the system more realistic. This study aims to investigate the role of prey refuge, toxin-carrying carcass and the impact of cross-diffusion on the spatial distribution of species densities and associated patterns. The existence of equilibria and their stability conditions for non-spatial system are derived. We discuss the occurrence of Hopf bifurcation and detect the stability of limit cycle by computing first Lyapunov number. Temporal dynamics of the system is studied by varying the significant parameters. Study reveals that both prey refuge and fatality rate of predator due to toxin-carrying carcass, initially makes the system stable for low refuge and fatality rate but predator population goes to extinction as those effects increase. Turing instability conditions are listed and different instability regions so formed are discussed. In Turing region, pattern selection with the help of amplitude equation via weakly nonlinear analysis are performed which also numerically supported. Intensity of refuge and fatality rate due to toxin carrying carcass result paradox with real ecological world due to the presence of cross diffusion. Various kind of patterns such as mixture of stripe and spot, spot, irregular chaotic patterns emerge due to Hopf and Turing instability. Particularly, spatiotemporal chaos is a controversial topic because of its significant consequences for population dynamics. The results of this work are expected to contribute to a better understanding the level of prey refuge, use of toxin-carrying carcass considering movement in population and effect of cross-diffusion.