Stochastic dynamics of populations with refuge in polluted turbidostat

Environmental pollution and random perturbation from the environment influence populations’ dynamics by alternating the habitat structure. To avoid disasters above or threats from predators, populations have self-defensive strategies such as camouflage or using a refuge. A turbidostat model with toxicant and prey refuge under the deterministic and stochastic environments is investigated in this work. We first analyze the positive solution's existence and uniqueness under the effect of shelter and stochastic perturbation. We further determine the sufficient conditions of extinction and permanence for each population with the impact of toxicant, prey refuge under the stochastic environment. Our results reveal the dynamics of populations under the influence of the factors above. Several numerical examples are provided to verify the theoretical analysis and simulate the effect of the phenomena above in the population dynamics for both deterministic and stochastic cases.