Abstract
We investigate a Hassell-Varley type predator-prey model with stochastic perturbations. By perturbing the growth rate of prey population and death rate of predator population with white noise terms, we construct a stochastic differential equation model to discuss the effects of the environmental noise on the dynamical behaviors. Applying the comparison theorem of stochastic equations and Itô’s formula, the unique positive global solution to the model for any positive initial value is obtained. We find out some sufficient conditions for stochastically asymptotically boundedness, permanence, persistence in mean and extinction of the solution. Furthermore, a series of numerical simulations to illustrate our mathematical findings are presented. The results indicate that the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model when the noise intensity is small under some conditions, but while the noise intensity is sufficiently large, the species may die out, which does not happen in the deterministic model.
Highlights
It is well known that predator-prey interaction is one of basic interspecies relations for ecosystems, and it is the basic block of more complicated food chain, food web, and biophysical network structure [1]
When choosing σ1 = 0.15 and σ2 = 0.9 (Figure 3(a)), the conditions of Theorem 9 are satisfied; we can find that prey population N(t) of model (6) is permanent and predator population P(t) will die out
The value of this study lies in twofolds. It verifies some relevant properties of the corresponding stochastic model (6), which shows the global existence, boundedness and stochastic permanence, persistence in mean, and extinction of the positive solution. It illustrates the dynamics of the model via numerical simulations, which shows that if the noise is not large and satisfies some conditions, the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model (4), while if the noise is sufficiently large and satisfies some conditions, it will force two species in the model to die out
Summary
It is well known that predator-prey interaction is one of basic interspecies relations for ecosystems, and it is the basic block of more complicated food chain, food web, and biophysical network structure [1]. The function f(N) represents the density-dependent specific growth rate of prey in absence of predator. The predatorprey model with Hassell-Varley type functional response has been studied in the ecological literature [6, 9,10,11]. The corresponding Hassell-Varley type predator-prey model is described by the following form: rN As our knowledge is concerned, the work of a modified Hassell-Varley type predator-prey model with stochastic perturbations seems rare. We attempt to study the stochastic behaviors of the modified Hassell-Varley type predation model in a random fluctuating environment.
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