Pattern dynamics of a harvested predator–prey model

Abstract

This paper investigates the pattern dynamics of a harvested predator–prey model with no-flux boundary conditions. Firstly, we analyze the positive equilibrium types of the local temporal model. We find that they can be classified as nodes, foci, or centers depending on the harvesting coefficient within a certain parameter range. Furthermore, the direction of the Hopf bifurcation is determined by employing the first Lyapunov coefficient. In the subsequent analysis, we present the conditions for the existence of Turing instability and classify the different pattern selections using amplitude equations with the assistance of weakly nonlinear analysis by treating the harvesting coefficient as a critical parameter. Finally, the spot patterns and mixed patterns are respectively displayed in 2D space, on spherical and torus surfaces with various harvesting coefficient values. Especially, we can numerically demonstrate that the diffusion rate of the prey population will strongly affect the pattern structures of the model. These results can provide a reference for understanding the interaction dynamics of the model.