The influence of time-dependent delay on behavior of stochastic population model with the Allee effect

This paper presents the analysis of behavior of stochastic time-dependent delay population model with the Allee effect. We prove the existence-and-uniqueness of positive solution of considered model. Then, we find the sufficient conditions under which the population will become extinct. We also show that if the initial population size exceeds environmental carrying capacity and time delay is sufficiently long, considered population is non-persistent in mean. The sufficient conditions for asymptotical mean square stability and stability in probability of the positive equilibrium states of the model, in terms of Lyapunov functional method, are obtained. Finally, as an illustration, we apply our mathematical results and predict time which a population of the African wild dog Lycaon pictus needs to reach it is equilibrium states, and also confirm that population of brown tree snake Boiga irregularis is non-persistent in mean if the initial population size is greater than carrying capacity and time delay is long enough.