Single-species population models with stage structure and partial tolerance in polluted environments.

We propose stage-structured single-species population models with psychological effects and partial tolerance in polluted environments in this paper. First, the conditions of extinction and the time for extinction are investigated respectively. Especially, the time for extinction takes longer as the value of the psychological effects increases. Then the weak persistence in the mean around the pollution-free equilibrium and the stochastic permanence have been derived under some moderate conditions. Further, the existence of a periodic solution for the periodic single-species population has been determined. The corresponding numerical simulations verify the efficiency of the main theoretical results.