Stochastic analysis of time-delayed ecosystems

The predator-prey-type ecosystem is investigated, taking into account the time-delay effect of the prey population on the predator population, as well as random variations in the birth rate of the preys and the death rate of the predators. The stochastic averaging procedure is applied to obtain the probability distributions of the predator and prey populations at the state of statistical stationarity. It is found that two system parameters, quantifying the effects of prey self-competition and the time delay, respectively, play the most important roles. Results are also obtained from Monte Carlo simulations to compare with the analytical results.