Bifurcations in a diffusive predator-prey system with linear harvesting

Complex spatiotemporal dynamical behaviors of a diffusive predator-prey system with Michaelis-Menten type functional response and linear harvesting are investigated. Firstly, the critical conditions for the occurrence of Turing instability, which are necessary and sufficient, are derived in a novel way. Then, the existence conditions of codimension-1 Turing bifurcation, Hopf bifurcation, and codimension-2 Turing-Turing bifurcation, Turing- Hopf bifurcation are established. Furthermore, the detailed bifurcation set is given by calculating the amplitude equation with the method of the multiple time scale near the Turing-Hopf bifurcation. We find that the system may exhibit nonconstant steady-state solutions, spatially homogeneous periodic solutions, and spatially inhomogeneous periodic solutions, which can be verified by a series of numerical simulations. These investigations not only explain the effect of diffusion and harvesting on the dynamic behavior of the system, but also reveal the mechanism of spatiotemporal complexity in the diffusive predator-prey system.

Automated summary

This paper investigates the complex spatiotemporal dynamical behaviors of a diffusive predator-prey system with Michaelis-Menten type functional response and linear harvesting. Firstly, the necessary and sufficient critical conditions for Turing instability are derived in a novel way. Then, the existence conditions of codimension-1 Turing bifurcation, Hopf bifurcation, codimension-2 Turing-Turing bifurcation, and Turing-Hopf bifurcation are established. The detailed bifurcation set is obtained by calculating the amplitude equation with the method of the multiple time scale near the Turing-Hopf bifurcation. Numerical simulations verify the existence of nonconstant steady-state solutions, spatially homogeneous periodic solutions, and spatially inhomogeneous periodic solutions. These investigations provide insights into the effect of diffusion and harvesting on the dynamic behavior of the system and reveal the mechanism of spatiotemporal complexity in the diffusive predator-prey system.