Stochastic Lotka--Volterra Population Dynamics with Infinite Delay

We discover in this paper that when environmental noise is strongly dependent on the population size, this noise may suppress the population explosion in a finite time and guarantee the global positive solution. When the environmental noise is weakly dependent on the population size, the conditions that guarantee the global positive solution are independent of this environmental noise. We also discuss the pth moment boundedness, stochastic ultimate boundedness, and moment average boundedness in time under two classes of conditions. These properties are natural requirements from the biological point of view.

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