Abstract and Applied Analysis

Volume 2014 (2014), Article ID 927217, 15 pages

http://dx.doi.org/10.1155/2014/927217

Research Article

Dynamics of a Stochastic Cooperative Predator-Prey System with Beddington-DeAngelis Functional Response

Dongwei Huang, Yu Li, and Yongfeng Guo

School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 21 March 2014; Revised 3 June 2014; Accepted 12 June 2014; Published 13 July 2014

Academic Editor: Imran Naeem

Copyright © 2014 Dongwei Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Stochastic cooperative predator-prey system with Beddington-DeAngelis functional response is studied. It presents an investigation of dynamic properties of the system. Our results show that there exists a unique positive solution to the system for any positive initial value, and the positive solution is stochastically bounded. Moreover, under some conditions, we analyze global asymptotic stability of the positive solutions. With small environmental noises, the stochastic system is getting more similar to the corresponding deterministic system. Neither of the species in the system will die out. Finally, simulations are carried out to conform to our result.

1. Introduction

As we all know, in mathematical biology, predator-prey system, competitive system, and cooperative system are the three rudimentary and important ecological systems. The dynamic relationship between species has long been and will continue to be a dominant theme in ecology due to its universal existence and importance. It is well-known that predator-prey systems are very important and extensive in the nature fields. One significant component of the predator-prey relationship is predator’s functional response, that is, the rate of prey consumption by an average predator. There are many significant functional responses in order to model various different situations. In fact, most of the functional responses are prey-dependent; however, some biologists have argued that in many cases, especially when predators have to search for food and therefore have to share or compete for food, the traditional predator-prey systems with prey-dependent functional response fail to model the interference among predators, the functional response should be predator-dependent. In [1, 2], Beddington and DeAngelis proposed the following predator-prey model with Beddington-DeAngelis functional response:

Skalski and Gilliam [3] compared statistical evidence from 19 predator-prey systems, and then they claimed that three predator-dependent functional responses (Hassell-Varley, Beddington-DeAngelis, and Growley-Martin) can provide better description of predator feeding over a range of predator-prey abundances. And the Beddington-DeAngelis type functional response was even suitable in some cases.

But most of this work is restricted to predator-prey systems, little has been done for cooperative systems [4, 5]. May [6] suggested the following set of equations:

to describe a pair of mutualists.

However, there is often the interaction among multiple species in nature, whose relationship is more complex than those in two species. Therefore, it is more realistic to consider the multiple-species predator-prey systems. In order to continue studying such models, in this paper, we consider a cooperative predator-prey system with Beddington-DeAngelis functional responses at first:

where species is the prey of and , , and are cooperative species. All the parameters in system (3) are positive constants.

In fact, population dynamics is inevitably affected by environmental white noise which is an important component in an ecosystem. But the model (3) is deterministic and does not incorporate the effect of environmental noise. May [6] also pointed out the fact that due to environmental fluctuation, the birth rates, carrying capacity, competition coefficients, and other parameters involved in system exhibit random fluctuation to a greater or a lesser extent. Therefore many scholars rewrote the deterministic models as stochastic ones subjected to stochastic noises, for studying the effect of environmental variability on the population dynamics [7–9].

The parameters in the real ecosystems are often subject to lots of environmental noises, since they relate to climate, geographical distribution, geological features, human disaster, human intervention, and other environmental factors. Therefore, the logistics and energy flow, in which they are determined by groups, are fluctuating. The oscillation in population biomass is directly manifested as birth and death rates of random perturbation. Currently, one of the main ways considered in the literature to model the effect of the environmental fluctuations in population dynamics is to assume that the most sensitive parameter is the intrinsic growth rate. Thus, in this paper we introduce some stochastic perturbation into the intrinsic growth rate. Therefore, the intrinsic growth rate can be written as an average growth rate adding some small random perturbed terms. In general, by the well-known central limit theorem, the small terms follow some normal distributions, so we can use standard Brownian motions to represent the environmental fluctuations.

In this paper, taking into account the effect of randomly fluctuating environment, we introduce stochastic perturbation into growth rates , , and to become , , and in system, where represents the intensity of the noise and is a standard white noise, namely, is a standard Brownian motion defined on a complete probability space . Then the stochastic system takes the following form:

Considering system (4), the initial conditions , , and will be referred to.

2. Global Positive Solutions

Lemma 1. For any initial value , where , system (4) has a unique positive local solution for almost surely, where is the explosion time.

Proof. Consider the following equations:

on with initial value , , and . The coefficients of (5) satisfy the local Lipschitz condition, thus there is a unique local solution on . Then , , and are the unique positive local solutions with initial value , , and by Itô’s formula.

Theorem 2. For any initial value , there is a unique solution of system (4) on , and the solution will remain in with probability 1.

Proof. According to Lemma 1, we only need to show that . Let be sufficiently large for , , and lying within the interval . For each integer , we define the stopping times

Obviously, is increasing as . Set ; hence, almost surely. Now, we only need to show that . If this statement is false, there is a pair of constants and such that . Thus there exists an integer such that

Define a function : by

The nonnegativity of this function can be seen from and . If , we have

then

where is a positive number. Substituting this inequality into (9), we see that

Integrating both sides of the above inequality from 0 to and then taking the expectations leads to

Set , then we get by inequality (7). Obviously, for every , there are at least , , and which equal either or , then is no less than

It then follows from (9) that

where is the indicator function of , letting , we have that

This completes the proof.

3. Stochastic Boundedness

Definition 3. The solution of system (4) is said to be stochastically ultimately bounded, if for any , there is a positive constant , such that for any initial value , the solution of system (4) has the property that

Assumption A. For any initial value , there exists such that

Lemma 4. Assume that Assumption A holds. Let be a positive solution of (4) with any initial value , for all , then

where

Proof. Define the function , for and . By Itô’s formula we get

Integrating from 0 to and taking expectations yields

So,

Let , then we have

From (17), we know

which by the standard comparison argument shows that

that is,

Define the function , for and . By Itô’s formula we get

Integrating from 0 to and taking expectations yields

So,

Let , then we have

From (17), we know that

which by the standard comparison argument shows that

that is,

Similarly, we can show that

This completes the proof.

Theorem 5. Assume that Assumption A holds, the solutions of system (4) with initial value are stochastically ultimately bounded.

Proof. If , its norm here is denoted by , then

by Lemma 4, , . is dependent on and is defined by . By virtue of Chebyshev inequality, we can easily obtain that the solution of system (4) is stochastically ultimately bounded.

4. Stochastic Permanence

Definition 6 (see [10]). The solution of system (4) is said to be stochastically permanent, if for any , there exist a pair of positive constants and such that for any initial value , the solution of system (4) has the properties that

Assumption B. One has

.

Theorem 7. Under Assumption B, for any initial value , the solution of system (4) satisfies that

where is an arbitrary positive constant satisfying

and is an arbitrary positive constant satisfying

Proof. Define for , then

Define for , by Itô’s formula, we get

where

Under Assumption B, choosing a positive constant such that it satisfies (38). By Itô’s formula, we get

where

then choosing a positive constant such that it obeys (39), by Itô’s formula,

where

where

Hence, it implies that there exists a positive constant such that

Then we have

Integrating both sides of the above inequality from 0 to and then taking the expectations leads to

where . So

Since that , where , obviously

as required.

Theorem 8. Under Assumption B, system (4) is stochastically permanent.

Proof. By Theorem 5, we know that

Now, for any , let . Then by Chebyshev’s inequality, we can obtain the conclusion easily.

5. Global Asymptotic Stability

Definition 9. Let be a positive solution of system (4). If we say that is globally asymptotically stable in expectation, it means that any other solution of system (4) has and that we have initial value . That is

Lemma 10 (see [11]). Suppose that an n-dimensional stochastic process on satisfies the condition

for some positive constants , , and . Then there exists a continuous modification of which has the property that for every there is a positive random variable such that

In other words, almost every sample path of is locally but uniformly Hölder continuous with exponent .

Lemma 11. Let be a solution of system (4) on with initial value , then almost every sample path of is uniformly continuous on .

Proof. From system (4), we have the following stochastic integral equation

Let   − , = , notice that

On the other hand, by the moment inequality (see [12]) for stochastic integrals, we have that for and ,

where . Let , , and , we obtain

where . Then, we have that almost every sample path of is locally but uniformly Hölder continuous with exponent for every and therefore almost every sample path of is uniformly continuous on . Similarly, we can show that almost every sample path of and is uniformly continuous on .

Lemma 12 (see [13]). Let be a nonnegative function defined on such that is integrable on and is uniformly continuous on . Then .

Theorem 13. If

then system (4) is globally asymptotically stable.

Proof. Define

then is a continuous positive function on . A direct calculation of the right differential of , and then applying Itô’s formula, we have

Integrating from 0 to and taking expectations yields

So

By Lemma 4

Thus

Integrating both sides leads to

Therefore

Then we get

Therefore from Lemmas 11 and 12, we have

This completes the proof of Theorem 13.

6. Extinction

Theorem 14. For any initial value , the solution of system (4) obeys

Proof. Define Lyapunov functions , , and , respectively, hence by Itô’s formula, we get

Then we have

Dividing on the both sides and letting , we can derive

as required.

So we can obtain that if , , and hold, then for any initial value , the solution of system (4) will be extinct exponentially with probability one.

7. Numerical Simulations

At last, we numerically simulate the solution of system (4) to substantiate the analytical findings. By the method mentioned in [14], we consider the discretization equation:

where , , and are Gaussian random variables that follow .

For example, in Figure 1, we choose initial value and parameters satisfying conditions of Theorem 7; the system is stochastically permanent. In Figure 2, (a) is the solution to the deterministic system (4), (b) is the solution to the stochastic system (4), and (c) is the phase diagram of the stochastic system (4). Comparing (a) with (b), we find that with small environmental noise, the stochastic system is getting more similar to the deterministic. In Figure 3, (a) is the solution to the deterministic system (4) and (b) is the solution to the stochastic system (4). Comparing (a) with (b), we find that the sufficiently large environmental noise make the stochastic system extinct. The phase diagram of is displayed in (c).

Figure 1: Solutions of system (4) for = = = = = = = .

(a)

(b)

Figure 2: Solutions of system (4) for = = = = = = .

(a)

(b)

(c)

Figure 3: Solutions of system (4) for = = = = .

8. Discussion

This paper has been devoted to dynamics of a stochastic cooperative predator-prey system with Beddington-DeAngelis functional. Firstly we show that, although the coefficients in the model neither satisfy the linear growth condition nor local Lipschitz continuity, the stochastic model has a globally positive solution. Then we know that the positive solution is stochastically bounded. Moreover, under some conditions, we analyze global asymptotic stability of the positive solutions. We can find that the stochastic model will preserve this nice property provided that the noise is sufficiently small. Some meaningful questions deserve further investigation. One way we can consider colored noise in the models owing to sudden environmental changes caused by seasons or other reasons. Moreover, it is worth to rebuild our model with some parameters, not the ones studied above, which are also subject to stochastic excitation. The ecosystem we considered has some limitations, for example, both predators and with the common prey and we only established a model with a mutual cooperation between both predators. It is worth to discuss mechanics of the population contained predators with a mutual competition in other ecosystems.

Conflict of Interests

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

Acknowledgment

This research was supported by the National Natural Science Fund of China (NNSF: 11102132).

References

J. R. Beddington, “Mutual interference between parasites or predat ors and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, pp. 331–340, 1975.

D. L. DeAngelis, R. A. Goldstein, and R. V. O’Neill, “A model for trophic interaction,” Ecology, vol. 56, pp. 881–892, 1975.

G. T. Skalski and J. F. Gilliam, “Functional responses with predator interference: viable alternatives to the Holling type II model,” Ecology, vol. 82, no. 11, pp. 3083–3092, 2001.

B. S. Goh, “Stability in models of mutualism,” The American Naturalist, vol. 113, no. 2, pp. 261–275, 1979.

P. H. Yang and R. Xu, “Global asymptotic stability of periodic solution in n-species cooperative system with time delays,” Journal of Biomathematics, vol. 13, pp. 841–862, 1998.

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001.

X. Li and X. Mao, “Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 2, pp. 523–545, 2009.

M. Liu, K. Wang, and Q. Wu, “Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle,” Bulletin of Mathematical Biology, vol. 73, no. 9, pp. 1969–2012, 2011.

Ch. Ji, D. Jiang, and X. Li, “Qualitative analysis, of a stochastic ratio-dependent predator-prey system,” Journal of Mathematical Analysis and Applications, vol. 235, pp. 1326–1341, 2011.

D. Jiang, N. Shi, and X. Li, “Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 588–597, 2008.

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, NY, USA, 2nd edition, 1991.

X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 2nd edition, 1997.

I. Barbalat, “Systems dequations differentielles d'isci d'oscillations nonlinearires,” Revue Roumaine de Mathematiques Pures et Appliquees, vol. 4, pp. 267–270, 1959.

D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001.