Stability analysis and optimal harvesting control of a cross-diffusion prey-predator system

In this paper, we consider a cross-diffusion prey-predator system with fear effect and prey refuge. The upper and lower bounds of the system are obtained by using priori estimates and Harnack Inequality. Then sufficient conditions for the local stability and global stability of the system are established. We obtain that the cross-diffusion coefficients can affect the stability of the original system, meanwhile the fear effect and prey refuge suppress the formation of Turing instability. By using the Leray-Schauder degree theory, we study the existence and nonexistence of the non-constant steady states. Moreover, we discuss the effects of the fear effect and prey refuge on the optimal harvesting. Finally, we obtain the optimal harvesting strategies under different fear effect values and prey refuge values, the different maximum sustainable yields (MSY) are correspondingly given. Numerical simulations are carried out to verify and illustrate these theoretical results.