The stability of a predator-prey system with linear mass-action functional response perturbed by white noise.

The present paper deals with the problem of an ecoepidemiological model with linear mass-action functional response perturbed by white noise. The essential mathematical features are analyzed with the help of the stochastic stability, its long time behavior around the equilibrium of deterministic ecoepidemiological model, and the stochastic asymptotic stability by Lyapunov analysis methods. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.

Introduction

In an ecosystem, species does not exist alone while it spreads the disease: it competes with the other species for space or food or is predated by other species. Therefore, it is essential to consider the effect of interacting species when we study the dynamical behaviors of epidemiological models. Recently, epidemiological dynamics have been extensively applied in population biology. Some researchers have made some achievements (see [1-11]).

The authors in [2] proposed and analyzed a predator-prey system in which some of the susceptible phytoplankton cells were infected by viral contamination and formed a new group (infected). The role of viral disease in recurrent phytoplankton blooms was discussed. They considered that the contact rate follows the law of proportional mixing rate. They did not take into account in their model that the infected phytoplankton cells become susceptible again. The author in [6] studied an SI or SIS model with disease spread among the prey when there is logistic growth of the predator and prey populations and when the predators eat infected prey only. They have not regarded that infected populations contribute to the susceptible population toward its carrying capacity. The authors in [9] modified the model equations of [2] and also the model of [6]. They assumed that the contact rate follows the law of mass action rate. A portion of infected phytoplankter was being recovered and became susceptible. The authors in [10] assumed that pelicans feed not only on infected fish but on susceptible fish also. Feeding on infected fish enhances the death rate of pelicans and is considered to contribute negative growth, whereas feeding on susceptible fish enhances their growth rate and is considered to contribute positive growth. In their model they did not consider that the portion of infected fish recovered and became susceptible. On the basis of this model, the authors in [11] studied and compared the dynamics of the proposed ecoepidemiological model to explore the crucial system parameters and their ranges in order to obtain different theoretical behaviors predicted from the interactions between susceptible prey, infected prey, and their predators. For linear mass-action functional response function, the ecoepidemiological model takes the following form: where , , are the population densities of susceptible prey, infected prey, and predator, respectively, at time t, K is the carrying capacity, r is the growth rate of susceptible prey, λ is the force of infection, θ is the conversion efficiency, α and β are the attack rates on susceptible and infected prey, respectively, μ and δ are the death rates of the infected prey and predators, respectively.

The authors in [11] detail that system (1.1) has the following equilibria: , , , , and , where System (1.1) is unstable around for all parametric values, globally asymptotically stable around if and , globally asymptotically stable around if and , globally asymptotically stable around if and , and unstable around for all parametric values.

However, in this case, the effects due to environmental noise have been neglected. In fact, because of the existence of environmental noise, the parameters involved in system (1.1) are not absolute constants, and they fluctuate around some average values owing to continuous fluctuations in the environment. Therefore, the parameters in the model exhibit continuous oscillation around some average values but do not attain fixed values with the advancement of time. Consequently, the equilibrium population distribution fluctuates randomly around some average values. So many authors introduce stochastic perturbation into deterministic models to reveal the effect of environmental variability on the ecology and epidemiology system (see [12-16]). Keeping this in mind, we have modified the model (1.1) proposed by [11] and taken into account the effect of randomly fluctuating and stochastically perturbed force of infection λ in each equation of system (1.1): Consequently, , where is a standard Brownian motion, is the intensity of environmental white noise. Then system (1.1) becomes

In this paper, we study the dynamics of the ecoepidemiological model with linear mass-action functional response perturbed by white noise to explore the crucial system parameters and their ranges in order to obtain different theoretical behaviors predicted from the interactions between susceptible prey, infected prey, and their predators.

This paper is organized as follows. The existence and uniqueness of a positive solution are given in Section 2. In Section 3, we show that the equilibrium of system (1.2) is stochastically unstable. In Section 4, we discuss that the equilibrium of system (1.2) is stochastically asymptotically stable in the large under some conditions and investigate the convergence rate of the solution. In Section 5, we study the fluctuations of system (1.2) about its equilibrium under some conditions. In Section 6, we carry out an analysis of stochastically asymptotically stability around the equilibrium of system (1.2). Numerical results are obtained by varying the parameters of the ecoepidemiological model in Section 7.

Throughout this paper, we let be a complete probability space with filtration satisfying the usual conditions (i.e., it is increasing and right continuous with containing all -null sets), and we let be a scalar Brownian motion defined on the probability space.

Existence and uniqueness of a positive solution

In this section, we show that there is a unique globally positive solution of system (1.2).

Theorem 2.1

There is a unique positive solutionof system (1.2) a.s. for any initial value, and.

Proof

Obviously, the coefficients of equation (1.2) satisfy the local Lipschitz condition. Therefore, there is a unique local solution on , where is the explosion time. Moreover, if , then for a.s. In fact, note that Therefore,

Let . Then by choosing such that and . Hence, by the comparison theorem we get and which is independent of the initial values.

Now, we are going to show that this solution is global, that is, that a.s. Let be sufficiently large so that , , and all lie within the interval . For each integer , define the stopping time where we set (as usual, ∅ denotes the empty set). Clearly, is increasing as . Set , whence a.s. If we can show that a.s., then and a.s. for all . In other words, to complete the proof, all we need to show is that a.s. If this statement is false, then there is a pair of constants and such that Hence, there is an integer such that Define the -function by The nonnegativity of this function can be seen from the inequality () for all . Using Itô’s formula, we get where By a similar proof as in Li and Mao [16], Theorem 2.1, we can obtain the desired assertion; see Appendix 2. □

Remark 2.1

From this theorem we know that the region is a positively invariant set of system (1.2), where B and η are determined in the proof of Theorem 2.1. From now on we always assume that the initial value .

Stochastic instability around the equilibrium

System (1.1) is unstable around for all parametric values. It is obvious that is still an equilibrium of system (1.2). In this section, we show that the equilibrium of system (1.2) is stochastically unstable.

Theorem 3.1

Letbe the solution of system (1.2) with initial value. Then the equilibriumof system (1.2) is stochastically unstable.

If not, there must be and such that and , , and for , . Hence, Then Let , which is a real-valued continuous local martingale, , and Then by the strong law of large numbers we have Therefore, which is a contradiction, and the proof of this theorem is completed. □

Global asymptotic stability around the equilibrium

System (1.1) is globally asymptotically stable around if and . It is obvious that is still an equilibrium of system (1.2). In this section, we first show that it is stochastically asymptotically stable in the large under some conditions. Then we investigate the convergence rate of the solution.

Theorem 4.1

Letbe the solution of system (1.2) with initial value. Ifand, then the equilibriumof system (1.2) is stochastically asymptotically stable in the large.

Define the function by Let L be the generating operator of system (1.2). Then which is negative-definite according to and , that is, and . Therefore, by Lemma A.2 (Mao [17]) the equilibrium of system (1.2) is stochastically asymptotically stable in the large. □

In the remainder of this section, we compute the convergence rate of , , and .

Theorem 4.2

Letbe the solution of system (1.2) with initial value. Assume thatThenMoreover,

, or

, or

.

By Itô’s formula we have Let We will analyze the following two cases.

(i) . Then we have Therefore, and Let , which is a real-valued continuous local martingale, , and Then by the strong law of large numbers we have which by (4.1) implies that By condition (a) it is easy to see that that is, tends to zero exponentially almost surely. In other words, the infected prey population dies out with probability one.

(ii) . Then we have Therefore, Similarly, as in (i), we get Using condition (b), we then obtain that that is, tends to zero exponentially almost surely. In other words, the infected prey population dies out with probability one.

In the same way, by Itô’s formula we have Therefore, Condition (c) implies thats is, tends to zero exponentially almost surely. In other words, the predator population dies out with probability one.

By Itô’s formula we have Therefore, and This, together with (4.2), (4.3), and (4.4), implies that Due to we obtain  □

Asymptotic behavior around the equilibrium of system (1.1)

The equilibrium of system (1.1) exists if , but it is not an equilibrium of system (1.2). In this section, we first compute the convergence rate of . Then we study the fluctuations of system (1.2) about its equilibrium under some conditions.

Theorem 5.1

Letbe the solution of system (1.2) with initial value. Ifand, thenandwhereis the boundary equilibrium of system (1.1), , and.

By Itô’s formula, we can easily show that, for , Therefore, It then follows from the condition that that is, tends to zero exponentially almost surely. In other words, the predator population dies out with probability one. That is to say, we can see that .

Since is the boundary equilibrium of system (1.1), we have Define where is a positive constant, which is determined later. By Itô’s formula and (1.2) we compute where Let . Then where By the Cauchy inequality we can easily show that By choosing such that , that is, , we see that Notice that Integrating both sides of from 0 to t yields Let By the boundedness of and and by (5.1) we can show that Let which is a real-valued continuous local martingale, , and Then by the strong law of large numbers we have It then follows from (5.2) that Then we obtain where .

Hence, the proof of this theorem is completed. □

Stochastic asymptotic stability around the equilibrium

Since is the boundary equilibrium of system (1.1), we have

The stochastic system (1.2) can be centered at its equilibrium by the change of variables

We obtain the following system:

It is easy to see that the stability of the equilibrium of system (1.2) is equivalent to the stability of the zero solution of system (6.1).

Theorem 6.1

Letbe the solution of system (1.2) with initial value. Ifthenis stochastically asymptotically stable.

It is easy to see that we only need to prove that the zero solution of (6.1) is stochastically asymptotically stable.

Let . Define the Lyapunov function as follows: where , , are positive constants, which are determined later. By Itô’s formula we compute where Since is the boundary equilibrium of system (1.1), we get Moreover, using the Cauchy inequality, we obtain Further, where where By the condition we get that .

We further have where Then we obtain where In (6.2), we choose such that .

Moreover, using the Cauchy inequality, we obtain Substituting (6.3) into (6.2) yields and so we have Let so that Then we get Finally, we obtain Put so that Then we get

Let . Then Hence, is negative-definite in a sufficiently small neighborhood of for . From Lemma A.3 of Mao [17] we therefore conclude that the zero solution of (6.1) is stochastically asymptotically stable. □

Numerical simulations

In this section, we make numerical simulations to illustrate our results by using Milstein’s higher-order method [18]. Variables and parameters used in the models of susceptible prey-infected prey-predator population interaction are given by Chattopadhyay et al. [11], Table 2, where

First, we take , , . In this case, We can therefore conclude by Theorem 4.1 that the equilibrium of system (1.2) is stochastically asymptotically stable in the large. The numerical simulations in Figure 1 support these results clearly.

Figure 1

Numerical simulation of the solution of system ( ) and its corresponding deterministic system ( ) with , , and with the initial values , , .

Noting that and we see that conditions (a) and (c) of Theorem 4.2 are satisfied. Therefore, by Theorem 4.2, for the initial values , , and , the solution of system (1.2) obeys The numerical simulations in Figure 1 support these results clearly, illustrating extinction of the infected prey and the predator.

Next, we choose and . Then and the conditions are satisfied. Therefore, by Theorem 5.1, tends to zero exponentially with probability one. We see that the difference between the solution of system (1.2) and in time average is related to the intensity of the white noise. The weaker the white noise, the smaller the difference. The numerical simulations in Figure 2 support these results clearly, illustrating that the solution of system (1.2) is surrounding randomly oscillating, and the extent vibrating enhances gradually with gradual increase of σ.

Figure 2

Numerical simulation of the solution of system ( ) and its corresponding deterministic system ( ) with initial value , , : (a) is with , ; (b) with , , ; (c) with , , ; (d) with , , .

Finally, we take , , . In this case, we compute We can therefore conclude, by Theorem 6.1, that the equilibrium of system (1.2) is stochastically asymptotically stable. The numerical simulations in Figure 3 support these results clearly.

Figure 3

Numerical simulation of the solution of system ( ) and its corresponding deterministic system ( ) with , , and with initial values , , .

Conclusion

In this paper, we have proposed and analyzed an ecoepidemiological model with linear mass-action functional response perturbed by white noise. Based on this model, we mainly have showed that system (1.2) has a unique positive global solution and investigated how the four equilibria , , , and of system (1.1) will be under stochastic perturbation. The key parameters are one ecological parameter α, predators’ attack rate on susceptible prey, and one epidemiological parameter λ, the rate of infection.

(i) System (1.1) is unstable around for all parametric values. We show that the equilibrium of system (1.2) is stochastically unstable in Theorem 3.1.

(ii) If and , then system (1.1) is globally asymptotically stable around the equilibrium . Theorem 4.1 shows that if and , then the equilibrium of system (1.2) is stochastically asymptotically stable in the large. Theorem 4.2 shows that, under some conditions, the disease will die out, the predator population will go into extinction, and the prey population will approach the carrying capacity K. Biologically, it implies that if both the infection rate and the search rate of susceptible prey are low, then the infected prey and predator population cannot survive, and the system converges to the equilibrium where only healthy prey exists.

(iii) If , then the equilibrium of system (1.1) exists. Theorem 5.1 shows that the difference between the solution of system (1.2) and in time average is only relation with the intensity of the white noise. The weaker the white noise, the smaller the difference. So there is approximate stability, provided that σ is sufficiently small. Biologically, this implies that if the infection rate is too high and the search rate of susceptible population is moderate, then the predator population cannot survive, and the system converges to the equilibrium where susceptible prey and infected prey coexist in the form of a stable equilibrium.

(iv) If and , then system (1.1) is globally asymptotically stable around the equilibrium . We also show (Theorem 6.1) that if and , then, under certain conditions, the equilibrium of system (1.2) is stochastically asymptotically stable. Biologically, it implies that in case of higher infection rate and higher predation rate, all trajectories with the default values converge to the disease-free equilibrium , where susceptible prey and predator population coexist in the form of a stable equilibrium.