Nonlinear Stochastic Ecosystem of Two Competing Species

Abstract

A nonlinear stochastic model describing two competing species of grass and woody vegetation is proposed. Two types of stochastic processes are proposed to model the variation part of the stocking rate. One is Gaussian white-noise, and another is randomized sinusoidal process. The system nonlinear behaviors are investigated for the deterministic case and the cases with two different stochastic variations. Some key characteristics of the system are found to be different for cases with and without stochastic variations. With the stochastic variation, a single stable state in the deterministic system is diffused into a region of stable states, and a separatrix dividing the two attraction zones no longer exists. The system may follow different trajectories and lead to different outcomes, beginning from the same initial state. It is also found that the stationary probability of the system response depends on the initial conditions, a special phenomenon for the investigated nonlinear system. Although the white-noise process and the randomized sinusoidal process are quite different in nature, the qualitative behaviors of the system are similar. Furthermore, effects of the initial state and the intensity of the stochastic variations on the system behaviors are investigated.