Stationary distribution and extinction of a stochastic two-stage model of social insects with egg cannibalism

Considering the cannibalism behavior of many groups of animals in nature, a two-stage model of social insects with egg cannibalism and nonlinear perturbations is constructed and investigated in this paper. First, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution ψ(⋅) of the stochastic model. In a biological implication, the existence of the distribution ψ(⋅) implies the long-term persistence of social insects. Moreover, sufficient conditions for exponential extinction of social insects are obtained. Finally, our theoretical results are further discussed by several numerical simulations.