Stochastic Nonlinear Aspect of Noise in a Two Predators – One Prey Mathematical Model Induced Cyclic Oscillations

Abstract

This paper deals with stochastic stability of a three species ecosystem consisting of a prey and two competing predators. The prey and predators are being continuously harvested. The mathematical model is defined by the system of three first order nonlinear ordinary differential equations. The local stability by using Routh–Hurwitz criteria and the global stability by using Lyapunov’s function are discussed. The condition for the existence of bionomic equilibrium of the system is identified. The optimal harvesting scheme is designed using Pontryagin’s maximum principle. Also the population intensities of fluctuations (variances) around the positive equilibrium due to Gaussian additive white noise are computed using Fourier transform methods. The stability of the deterministic system and chaotic nature of stochastic system are portrayed for the chosen set of parameters in MATLAB simulations.