Stability and Numerical Simulation of Prey-predator System with Holling Type-II Functional Responses for Adult Prey

Abstract

In this paper, we analyze the local stability of the prey-predator model. This model was constructed from two prey involving stage-structure and one predator. The population of prey is divided into two subpopulations, adult prey and immature. The ratio of intrinsic growth of predator population density divided by adult prey conversion factors into a predator. Reduction of predator populations has a reciprocal relationship with the availability of favorite foods determined by the environmental resources. We also consider that Holling type II the functional response for the predator. We have investigated the local stability and the solution of the system. We have studied the existence and feasibility of various equilibrium points. The system has three positive equilibria, namely the original, the extinction of the predator and the interior point. We explore the stabilities of all nonnegative equilibrium point by the real parts of the eigenvalues of the Jacobian matrix at each equilibrium point must be negative. We also use the Routh-Hurwitz criterion to investigate the stability at the interior equilibrium point. This study aims to detect the limit cycle with its phase portrait and its time-series graphs of the prey-predator system. Numerical simulated are represented as phase portraits to draw the stability of the equilibrium point. In our Numerical outcomes confirmed the analytical results and the local behavior prey-predator model which start from their positive initial condition. The parameter value is taken mainly from the literature and assumption. By a computational Python using the fourth-order Runge-Kutta method, we solved the prey-predator system and showed some new phase portraits such as the existence of the stable or unstable equilibrium point under a suitable value of the parameter. The next research, we will explore the global stability analysis behavior of the interior equilibrium point.