A Stage-Structure Rosenzweig-MacArthur Model with Effect of Prey Refuge

Abstract

We proposed and analyzed the stage-structure Rosenzweig-MacArthur model incorporating a prey refuge. It is assumed that the prey is a stage-structure population consisting of two compartments known as immature prey and mature prey. The model incorporates the functional response Holling type-II. In this work, we investigate all the biologically feasible equilibrium points, and it is shown that the system has three equilibrium points. Sufficient conditions for the local stability of the non-negative equilibrium point of the model are also derived. All points are conditionally locally asymptotically stable. By constructing the Jacobian matrix and determining its eigenvalues, we analyzed the local stability of the trivial and non-predator points. Specially for the local stability of the coexistence point is analyzed by using the Routh-Hurwitz criterion. In addition, we investigated the effect of immature prey refuge. Our mathematical analysis exhibits that the immature prey refuge have played a crucial role in the behavioral system. When the effect of immature prey refuge (constant m) increases, it is can stabilize the non-predator point, where all the species can not exist together. And conversely, if contant m decreases, it is can stabilize the coexistence point then all the species can exist together. The work is completed with the numerical simulations to confirmed the analytical results.