Dynamics of a non-autonomous stochastic mutualism model with jumps

In nature, we can find many examples where the interaction of two or more species is to the advantage of all. Mutualism occurs when one species provides some benefit in exchange for some benefit, for example, ants and aphids, in which the ants obtain honeydew food resources excreted by aphids while the aphids obtain increased survival by the non-trophic service of ant defense against natural enemies of the aphids. Mathematical deterministic mutualism models are widely used to study the dynamics of such population systems. In real life, there is a lot of randomness and stochasticity, such as environmental noise, so using stochastic models is more suitable. Suppose we want to take into account abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc., in the considered models. In that case, we must introduce Poisson noises into population models to describe such discontinuous systems. So, we take into account not only “small” jumps, corresponding to the centered Poisson measure, but also the “large” jumps, corresponding to the non-centered Poisson measure. The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a non-autonomous mutualism model disturbed by white noise, centered, and non-centered Poisson noises. We obtain sufficient conditions of stochastic ultimate boundedness, stochastic permanence, non-persistence in the mean, weak and strong persistence in the mean, and extinction of the solution to the considered system.