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.

It is well known that predator-prey interaction is one of basic interspecies relations for ecosystems, and it is also the basic block of more complicated food chain, food web, and biophysical network structure [

The classical predator-prey model has received extensive attentions from mathematicians as well as ecologists [

In this paper, we consider the usual logistic form of the growth function for prey in the absence of predator as

For more biological motivation in population dynamics, we take into account the density-dependence of predator population. And the corresponding Hassell-Varley type predator-prey model is described by the following form:

On the other hand, most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average. So that the population density never attains a fixed value with the advancement of time but rather exhibits continuous oscillation around some average values [

The organization of this paper is as follows. In Section

In this section, we investigate the effects of fluctuating environments on the dynamical behaviors of model (

For model (

There is a unique local solution

The proof of this lemma is rather standard and hence is omitted.

Lemma

In particular, let us consider the one-dimensional stochastic population model

Besides, for the following equation

Consequently, we obtain

On the other hand, the equation

From the representation of solutions

There is a unique positive solution

In this subsection, we show that the solution

The solution

From (

Set

It is well known that the property of permanence is more desirable since it means the long time survival in a population dynamics. Now, the definition of stochastic permanence will be given below [

The solution

For any initial value

Set a function

Let

In addition, we know that

Based on the results of Theorem

Assume that

In a view of ecology, the coexistence of species may be a good situation. In the following, we consider the stochastic persistence (i.e., stochastic persistence in mean) of the species.

Assume that

Denoting

Letting

On the other hand,

Assume that

Denoting

Furthermore,

From (

From Theorem

Combining the above arguments, we can get the theorem as follows.

Let

Furthermore, set

Similarly, we obtain

Based on the above, we obtain the following theorem which means that if the noise satisfies some conditions, then both species

Let

In this section, we perform some numerical simulations for model (

When choosing the values of parameters

Phase portrait of model (

Figure

Solutions of model (

In Figure

Solutions of model (

In this paper, we consider a stochastic Hassell-Varley type predator-prey model. The value of this study lies in twofolds. First, it verifies some relevant properties of the corresponding stochastic model (

In order to study the stochastic model (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the editor and the anonymous referees for the very helpful suggestions and comments which led to improvement of their original paper. This research was supported by the National Science Foundation of China (no. 11301263).