Long-time behavior of a stochastic logistic equation with distributed delay and nonlinear perturbation

In this paper, we study a stochastic logistic model with distributed delay and nonlinear perturbation. We first use the linear chain technique to transfer the one-dimensional stochastic model with strong kernel or weak kernel into an equivalent system with degenerate diffusion. Then we establish sufficient and necessary criteria for extinction of the population with probability one. Moreover, since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We show that the densities of the distributions of the solutions can converge in L1 to an invariant density. The results show that the larger white noise can lead to the extinction of the population while the smaller white noise can guarantee the existence of a stable stationary distribution which leads to the stochastic persistence of the population. Numerical simulations are provided to demonstrate the theoretical results.