Stochastic prey-predator model with small random immigration

In this paper, we introduce a novel stochastic prey-predator model under random small immigration. Mainly, we prove boundedness for the solution of the model using probabilistic and analytic types of inequalities. Furthermore, possible conditions on the immigration for achieving stochastic square stability are obtained. The immigration of both prey and predator is assumed to be either constant and stochastically perturbed or proportional to the population and stochastically perturbed. In all cases, we arrived at the fact that stability can only be achieved if the immigration is small enough. We also show that as random immigration increases, the dynamic becomes destabilized and could lead to chaos. Lastly, we perform a computational analysis in order to verify the obtained theoretical results.