Dynamics of Prey Predator Model with Crowley-Martin Functional Response, Prey Refuge and Immigration of Both Species

Abstract

In this paper, we introduce a novel prey-predator model with Crowley-Martin functional response, prey refuge, and immigration of both species. The existence, positivity, and boundedness of solutions are proved. The equilibrium points and stability are discussed. Theoretical analysis shows that if the model includes prey refuges but does not involve immigration, one or both species will go extinct. However, if the model includes both prey refuges and immigration, then no species will go extinct. In addition, the conditions for the stability of the system locally and globally are obtained, which show that the refuge of prey and the immigration of both species play important roles in the uniqueness of the coexistence equilibrium point, the stability of the system, and the existence of a limit cycle. Numerical simulations are performed to verify and explain the analytical results. It has been found that increasing the rate of refuges to a sufficient extent leads to the extinction of predators, but with the presence of constant immigration it leads to the survival of the system. The numerical results also show that prey refuges and constant immigration lead to system stability and increased species density. The results of this study are considered as conservation strategies for species survival and biodiversity continuity. Some ecological explanations are offered in this study.