Qualitative analysis of a diffusive Crowley-Martin predator-prey model: the role of nonlinear predator harvesting

In this paper, we discuss a diffusive predator-prey system with mutually interfering predator and nonlinear harvesting in predator with Crowley-Martin functional response. The mathematical analysis of the system starts with the existence and uniqueness of solution of the system using C0 semigroup. The analysis reflects that the upper bound of rate of predator harvesting for the coexistence of the species can be guaranteed. In addition, we establish the existence and nonexistence of non-constant positive steady state. Explicit conditions on predator harvesting are obtained for local and global stability of interior equilibrium and also for the existence and nonexistence of non-constant steady-state solution. We also investigate conditions for Turing instabilities of the diffusive system analytically. Our results show that the effort of harvesting (g) provides a threshold value for existence of non-constant positive stationary solution. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the system exhibits interesting patterns. Some biological implications of obtained theoretical results have also been discussed.