Analysis of Prey – Predator System with Prey Population Experiencing Critical Depensation Growth Function

Abstract

Abstract:In this paper we have presented a pair of coupled differential equations to represent a prey -predator system. It is assumed that the growth of the prey population follows critical depensation function and that of the predator population is negative in absence of the prey population. The critical depensation function is special since the growth rate is negative initially but positive later on. This function is stable both at the origin and at the carrying capacity while unstable at the critical mass quantity. The maximum and minimum rates of the critical depensation model are verified. It can be interpreted here that the prey represent fish and the predator represent a kind of birds that mostly feeding on fish to live. We showed the solution of the model is positive and bounded. The mathematical model of the system consisting of 7 parameters is constructed and shown that the non -dimensionalization decreases the number of model parameters to 4. The deterministic behavior of the model around feasible equilibrium points and criteria of the interior positive equilibrium points and their stability are explained. The trivial equilibrium point is always stable while the two axial equilibrium points and the lone interior equilibrium point are either stable or unstable depending on the conditions imposed on the parameters. The criterion for the existence of the limit cycle and the region of existence of interior equilibrium point are identified. Global stability of interior equilibrium points is also studied. For the interior equilibrium point of the model (i) the region of existence is identified (ii) Dulac's criteria is applied to find the limit cycle and (iii) Lyaponov function is used to analyze the global stability. Simulation study of the model is conducted in support of the analytical analysis. To solidify the analytical results numerical simulations are provided for hypothetical set of parametric values.