Break-out of dynamic balance of nonlinear ecosystems using first passage failure theory

Abstract

Normally, an existing nonlinear ecological or biological system is in dynamic balance. If it describes the evolution of two or more species, the dynamic balance is the state of species coexistence. Due to random evolution of the environments, such dynamic balance may be broken and possibly leading to a nearly extinction state of a certain species. The present paper tries to investigate such a phenomenon using the theories and methods of stochastic dynamics. Mathematically, the break-out of species coexistence can be modeled as the first passage problem. When the population of a species reaches a critical low limit or high limit, the coexistence of species is considered to be broken. For illustration, the theory of first passage failure is applied to a stochastic predator–prey system under random disturbances. Two important probabilistic measures, the probability density of the first passage time and the mean first passage time, are investigated. The former can be obtained by solving the backward Kolmogorov equation, while the latter can be calculated either by integrating from the probability density or directly from the Pontryagin equation. The accuracy of the analytical results is substantiated by those calculated from Monte Carlo simulations, and effects of several system parameters and the initial state are accessed. The method proposed may provide a useful tool for prediction and prevention of the break-out of dynamic balance of nonlinear systems, including those in engineering, ecology, biology, and other areas.