Evolution of Turing patterns of a predator–prey system with variable carrying capacity and harvesting

In the present study, we propose a modified form of the Rosenzweig–MacArthur model by incorporating prey harvesting and variable carrying capacity. These modifications are addressed both from a mathematical and ecological perspective. By considering harvesting effort as a bifurcating parameter, sufficient criteria are identified for the stability and occurrence of Hopf bifurcation. We also observe that the coexisting equilibrium stabilizes as harvesting effort is increased. On the contrary, increasing the influence parameter tends to destabilize the equilibrium point via limit cycles. As species live in both space and time, we extend our study by incorporating spatial dynamics into the system. The conditions for Turing instability are established, and it is demonstrated that the system cannot produce heterogeneous spatial patterns without cross-diffusion. Extensive numerical simulations are conducted, and various Turing patterns are observed as harvesting effort, influence parameter, and cross-diffusion coefficient are varied. Spatio-temporal chaotic patterns are also detected by choosing appropriate parametric values from the bifurcation plane. As a result, the population’s oscillatory behavior is desynchronized, potentially lowering the risk of extinction.