Abstract
A predator-prey model with Beddington-DeAngelis functional response and disease in the predator population is proposed, corresponding to the deterministic system, a stochastic model is investigated with parameter perturbation. In Additional file 1, qualitative analysis of the deterministic system is considered. For the stochastic system, the existence of a global positive solution and an estimate of the solution are derived. Sufficient conditions of persistence in the mean or extinction for all the populations are obtained. In contrast to conditions of permanence for the deterministic system in Additional file 1, it shows that environmental stochastic perturbation can reduce the size of population to a certain extent. When the white noise is small, there is a stationary distribution. In addition, conditions of global stability for the deterministic system are also established from the above result. These results mean that the stochastic system has a similar property to the corresponding deterministic system when the white noise is small. Finally, numerical simulations are carried out to support our findings.
Highlights
Epidemiological models have received much attention from scientists
It is more important to consider the effect of multi-species when we consider the dynamical behaviors of epidemiological models
7 Conclusions A stochastic model corresponding to a predator-prey model with Beddington-DeAngelis functional response and disease in the predator population is investigated
Summary
Epidemiological models have received much attention from scientists. Since the pioneering work of Kermack-Mckendrick, there have been many relevant papers [ – ], but only single-species is considered in these models. We first introduce a deterministic predator-prey model with disease in the predator and Beddington-DeAngelis functional response. R is the intrinsic growth rate of preys, d and d are both death rates of susceptible predators and infected predators These parameters can be estimated by an average value plus an error term. Theorem Assume (x(t), y (t), y (t)) on t ≥ is the positive solution of system ( ) for initial value x > , y > and y > , there exist functions (t), φ(t), i(t), ψi(t) (i = , ), defined as above, such that φ(t) ≤ x(t) ≤ (t), ψi(t) ≤ yi(t) ≤ i(t) (i = , ), t ≥ , a.s
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