Patterns in the predator–prey system with network connection and harvesting policy

Abstract

A diffusive predator-prey system with the network connection and harvesting policy is investigated in the present paper. The global existence and boundedness of the positive solutions to the parabolic equations are analyzed. Hereafter, a priori estimates and non-existence of the non-constant steady states are investigated for the corresponding elliptic equation. Next, we focus on the network connect model. The stability of the positive equilibrium, the Hopf bifurcation, and the Turing instability of networked system are explored. By using the multiple time scale (MTS), the direction of the Hopf bifurcation is determined. It is found that the networked system may admit a supercritical or subcritical Hopf bifurcation. For the Turing instability, the positive equilibrium will change its stability from an unstable state to a stable one with the change of the connecting probability. That is not the case in the model without network structure. Theoretical results also show that the model can create rich dynamical behaviors and numerical simulations well verify the validity of the theoretical analysis.