A study of a spatiotemporal delayed predator–prey model with prey harvesting: Constant and periodic diffusion

This study investigates the emergence of Turing patterns in a predator–prey system with time delay at variable carrying capacity and prey harvesting in the presence of cross-diffusion. The impact of time delay and periodic diffusion on the stability of the equilibrium point and corresponding Turing pattern is primarily explored. Theoretical implications of Turing instability conditions caused by time delay and periodic diffusion are examined, followed by numerical simulations for ecologically meaningful control parameter values. The effects of carrying capacity specification parameters, time delay, and prey-harvesting effort on the emerging pattern are reported. Cross-diffusion results in a wide range of spatiotemporal pattern creation, including spot and stripe patterns, as well as both coexist under the zero flux border condition in unharvested or harvested dynamics. Furthermore, an intertwined pattern emerges under certain parameters, including time delay. The final comparison highlights the significant roles played by periodic diffusion, time delay, and harvesting effort in the evolution of the Turing pattern.