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Analysis of a stochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response
- Shuang Li^{1, 2}Email author and
- Xiaopan Wang^{3}
https://doi.org/10.1186/s13662-015-0448-0
© Li and Wang; licensee Springer. 2015
Received: 10 October 2014
Accepted: 19 March 2015
Published: 21 July 2015
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.
Keywords
- predator-prey
- Beddington-DeAngelis functional response
- stochastic
- stationary distribution
- stability
- persistence
- extinction
MSC
- 92B05
- 34F05
- 60H10
1 Introduction
Recently, epidemiological models have received much attention from scientists. Since the pioneering work of Kermack-Mckendrick, there have been many relevant papers [1–8], but only single-species is considered in these models. However, species does not exist alone; while species spreads the disease in the natural world, it also competes with other species for resource to exist, or is predated by their enemies. Therefore, it is more important to consider the effect of multi-species when we consider the dynamical behaviors of epidemiological models. There are not many papers [9–18] considering these two areas.
Let \(x(t)\) denote the population density of prey, \(y_{1}(t)\) and \(y_{2}(t)\) represent the population density of the susceptible predator and the infected predator, respectively.
Define the basic reproduction number \(R_{0}=\frac{a_{21}r}{(d_{2}+\frac {d_{3}a_{22}}{\beta})(a_{11}+mr+\frac{nd_{3}a_{11}}{\beta})}\), obviously, \(E^{*}\) exists when \(R_{0}>1\). In addition, we can compute a useful result: \(\hat{y}_{1}<\frac{d_{3}}{\beta}\) if \(R_{0}<1\).
The qualitative analysis of system (2) is in Additional file 1, here we mainly discuss the stochastic system. In the following section, we derive the existence of a positive solution of system (5), an estimate of the solution, and give the conditions of persistence in the mean or extinction for both populations. We also show that there exists a stationary distribution of the solution. As a result, conditions of global stability for model (2) are obtained.
2 Existence of the positive solution
In order for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and the local Lipschitz condition [37].
Theorem 1
For any initial value \(x_{0}>0\), \(y_{10}>0\) and \(y_{20}>0\), there is a unique solution \((x(t), y_{1}(t), y_{2}(t))\) of system (5) on \(t\geq0\), and the solution will remain in \(R_{+}^{3}\) with probability 1.
Proof
Define a \(C^{2}\)-function \(V: R_{+}^{3}\rightarrow R_{+}\) by \(V(x,y_{1},y_{2})=(x-1-\log x)+(y_{1}-1-\log y_{1})+(y_{2}-1-\log y_{2})\), by a similar way of the proof in Theorem 2.1 of [34], Theorem 2.1 of [35] and Lemma 2 of [36], we can have the required assertion. □
Though we cannot get an explicit solution for model (5), an estimate of positive solution of (5) can be derived, we firstly show a very useful lemma derived from [38].
Lemma 1
Theorem 2
3 Persistence in the mean and extinction
Lemma 2
Proof
Definition 1
- (1)
The population \(x(t)\) is said to be non-persistent in the mean if \(\langle x(t)\rangle^{*}=0\), where \(\langle f(t)\rangle =\frac{1}{t}\int_{0}^{t}f(s)\,ds\), \(f^{*}=\limsup_{t\rightarrow+\infty }f(t)\), \(f_{*}=\liminf_{t\rightarrow+\infty}f(t)\).
- (2)
The population \(x(t)\) is said to be weakly persistent in the mean if \(\langle x(t)\rangle^{*}>0\).
- (3)
The population \(x(t)\) is said to be strongly persistent in the mean if \(\langle x(t)\rangle_{*}>0\).
Lemma 3
[41]
- (I)If there are positive constants \(\lambda_{0}\), T and \(\lambda\geq0\) such thatfor \(t\geq T\), where \(\beta_{i}\) is a constant, \(1\leq i\leq n\), then \(\langle x\rangle^{*}\leq\lambda/\lambda_{0}\), a.s. (i.e., almost surely).$$\ln x(t)\leq\lambda t-\lambda_{0}\int_{0}^{t}x(s) \,ds+\sum_{i=1}^{n}\beta_{i}B_{i}(t) $$
- (II)If there are positive constants \(\lambda_{0}\), T and \(\lambda\geq 0\) such thatfor \(t\geq T\), where \(\beta_{i}\) is a constant, \(1\leq i\leq n\), then \(\langle x\rangle_{*}\geq\lambda/\lambda_{0}\), a.s.$$\ln x(t)\geq\lambda t-\lambda_{0}\int_{0}^{t}x(s) \,ds+\sum_{i=1}^{n}\beta_{i}B_{i}(t) $$
Theorem 3
- (i)
If \(r<\frac{1}{2}\sigma_{1}^{2}\), then the prey population \(x(t)\) will go to extinction a.s.
- (ii)
If \(r=\frac{1}{2}\sigma_{1}^{2}\), then the prey population \(x(t)\) is non-persistent in the mean a.s.
- (iii)
If \(r>\frac{1}{2}\sigma_{1}^{2}\), then the prey population \(x(t)\) is weakly persistent in the mean a.s.
- (iv)
If \(r>\frac{1}{2}\sigma_{1}^{2}+\frac{a_{12}}{n}\), then the prey population \(x(t)\) is strongly persistent in the mean a.s.
Proof
(iii) We need to show that there exists a constant \(\rho>0\) such that for any solution of system (5) with initial value \((x_{0}, y_{10},y_{20})\in R_{+}^{3}\) satisfying \(\langle x(t)\rangle^{*}\geq\rho >0\). Now we assume that the contrast is true, let \(\varepsilon_{1}>0\) sufficiently small such that \((-d_{2}-\frac{\sigma _{2}^{2}}{2})+a_{21}\varepsilon_{1}<0\), \((r-\frac{1}{2}\sigma _{1}^{2})-a_{11}\varepsilon_{1}>0\), then for \(\varepsilon_{1}>0\), there exists a solution \((\bar{x},\bar{y}_{1} ,\bar{y}_{2})\) with initial value \((x_{0}, y_{10},y_{20})\in R_{+}^{3}\) such that \(P\{\langle\bar{x}(t)\rangle^{*}<\varepsilon_{1}\}>0\).
Remark 1
By Theorem 3, we find that \(r-\frac {1}{2}\sigma_{1}^{2}\) is the threshold between weak persistence in the mean and extinction for the prey population. If \(\frac{1}{2}\sigma _{1}^{2}>r\), then the prey population will be extinct in the future, no matter whether the predator exists. It implies that environmental random perturbation plays a very important role in the biological system.
Theorem 4
- (i)
If \(a_{21}(r-\frac{1}{2}\sigma_{1}^{2})< a_{11}(d_{2}+\frac{\sigma _{2}^{2}}{2})\), then the susceptible predator population \(y_{1}(t)\) will go to extinction a.s.
- (ii)
If \(a_{21}(r-\frac{1}{2}\sigma_{1}^{2})=a_{11}(d_{2}+\frac{\sigma _{2}^{2}}{2})\), then the susceptible predator population \(y_{1}(t)\) is non-persistent in the mean a.s.
- (iii)
If \(\beta a_{21}(r-\frac{1}{2}\sigma_{1}^{2})<\beta a_{11}(d_{2}+\frac{\sigma_{2}^{2}}{2})+a_{11}a_{22}(d_{3}+\frac{\sigma _{3}^{2}}{2})\), then the infected predator population \(y_{2}(t)\) will go to extinction a.s.
- (iv)
If \(\beta a_{21}(r-\frac{1}{2}\sigma_{1}^{2})=\beta a_{11}(d_{2}+\frac{\sigma_{2}^{2}}{2})+a_{11}a_{22}(d_{3}+\frac{\sigma _{3}^{2}}{2})\), then the infected predator population \(y_{2}(t)\) is non-persistent in the mean a.s.
Proof
(iv) If \(\beta a_{21}(r-\frac{1}{2}\sigma_{1}^{2})=\beta a_{11}(d_{2}+\frac{\sigma_{2}^{2}}{2})+a_{11}a_{22}(d_{3}+\frac{\sigma _{3}^{2}}{2})\), then \(a_{21}(r-\frac{1}{2}\sigma_{1}^{2})>a_{11}(d_{2}+\frac{\sigma _{2}^{2}}{2})\) and \(r>\frac{1}{2}\sigma_{1}^{2}\). By the properties of superior limit, for sufficiently small \(\varepsilon_{4}>0\), there exists a constant \(T_{4}>0\) such that for all \(t>T_{4}\), \(\langle y_{1}(t)\rangle<\langle y_{1}(t)\rangle^{*}+\frac{\varepsilon _{4}}{\beta}\).
Remark 2
Observing conditions (i) and (iii) of Theorem 4, we can see that if condition (i) is true, (iii) must be verified. That is to say, if the susceptible predator population goes to extinction, the infected predator population will also die out, which is consistent with the reality. Though we have some difficulties to research persistence for the predator population now, we can consider it in another way in Section 5.
Remark 3
It follows from Theorems 3, 4 and Theorem A.4 in Additional file 1 that the solution of stochastic system (5) satisfies \(\langle x(t)\rangle^{*}\leq\frac{r-\frac{1}{2}\sigma_{1}^{2}}{a_{11}}\) when \(r>\frac{1}{2}\sigma_{1}^{2}\). If \(r>\frac{1}{2}\sigma_{1}^{2}+\frac {a_{12}}{n}\), then \(\langle x(t)\rangle_{*}\geq\frac{r-\frac{1}{2}\sigma_{1}^{2}-\frac {a_{12}}{n}}{a_{11}}\). When \(\sigma_{1}^{2}=0\), the upper bound and lower bound are the same with \(\overline{K}\) and \(\underline{K}\) in Theorem A.4, this is consistent with our expectations. Besides, if \(a_{21}(r-\frac{1}{2}\sigma_{1}^{2})>a_{11}(d_{2}+\frac{\sigma _{2}^{2}}{2})\), \(\langle y_{1}(t)\rangle^{*}\leq\frac{a_{21}(r-\frac {1}{2}\sigma_{1}^{2})-a_{11}(d_{2}+\frac{\sigma_{2}^{2}}{2})}{a_{11}a_{22}}\). \(\langle y_{2}(t)\rangle^{*}\leq\frac{\beta a_{21}(r-\frac{1}{2}\sigma _{1}^{2})-\beta a_{11}(d_{2}+\frac{\sigma _{2}^{2}}{2})-a_{11}a_{22}(d_{3}+\frac{\sigma _{3}^{2}}{2})}{a_{11}a_{22}a_{33}}\) if \(\beta a_{21}(r-\frac{1}{2}\sigma _{1}^{2})>\beta a_{11}(d_{2}+\frac{\sigma _{2}^{2}}{2})+a_{11}a_{22}(d_{3}+\frac{\sigma_{3}^{2}}{2})\). These results are all the same conclusions with \(\overline{K}_{1}\) and \(\overline{K}_{2}\) in Theorem A.4 when \(\sigma_{i}^{2}=0\) (\(i=1,2,3\)). Furthermore, we find that the upper bound and lower bound of the solution for the stochastic system are smaller than those for the deterministic system. It means environmental random perturbation can reduce the size of the population to a certain extent.
4 The long time behavior of solution
For system (2), by analyzing the characteristic equation of four equilibria, we can easily get sufficient conditions of local stability for these equilibria. At this time, we know \(O(0,0,0)\) is saddle, not stable. If \(a_{21}r< d_{2}(a_{11}+mr)\), then \(E_{1}\) is locally asymptotically stable, and the disease-free equilibrium \(E_{2}\) does not exist. When \(r>\frac{a_{12}}{n}\), \(a_{21}r>d_{2}(a_{11}+mr)\) and \(R_{0}<1\) are verified, the disease-free equilibrium \(E_{2}(\hat{x},\hat{y}_{1},0)\) is locally asymptotically stable, \(E^{*}\) does not exist; when \(R_{0}>1\) and \(r>\frac{a_{12}}{n}\), the positive equilibrium \(E^{*}\) is locally asymptotically stable.
For stochastic system (5), \(E_{1}\) and \(E_{2}\) are no longer equilibria, but in this section we can study the asymptotic behavior of solution around them. Meanwhile, the conditions of global asymptotic behavior for \(E_{1}\) and \(E_{2}\) are derived.
Theorem 5
Proof
When \(\sigma_{i}=0\) (\(i=1,2,3\)), system (5) becomes system (2). By Theorem 5, we know the stability of equilibrium \(E_{1}\).
Corollary 1
If \(a_{21}r< a_{11}d_{2}\), the equilibrium \(E_{1} (\frac{r}{a_{11}},0,0)\) of system (2) is globally asymptotically stable.
Remark 4
It is not difficult to find that the solution of stochastic system (5) fluctuates around equilibrium \(E_{1} (\frac{r}{a_{11}},0,0)\) of system (2) when \(E_{1}\) is globally asymptotically stable. The intensity of fluctuation is relevant to \(\sigma_{1}^{2}\). The smaller \(\sigma_{1}^{2}\) is, the weaker the fluctuation is.
Theorem 6
Proof
When \(\sigma_{i}=0\) (\(i=1,2,3\)), it is easy to get the following.
Corollary 2
Assume \(a_{21}r>d_{2}(a_{11}+mr)\), the disease-free equilibrium \(E_{2} (\hat {x},\hat{y}_{1},0)\) of system (2) is globally asymptotically stable when \(a_{11}>m(r-a_{11}\hat{x})\) and \(R_{0}<1\).
Remark 5
The solution of stochastic system (5) fluctuates around the disease-free equilibrium \(E_{2} (\hat{x},\hat {y}_{1},0)\) of deterministic system (2) when \(E_{2}\) is globally asymptotically stable. The values of \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\) determine the extent of fluctuations.
Remark 6
According to \(\hat{y}_{1}=\frac{(1+m\hat{x})(r-a_{11}\hat {x})}{a_{21}-nr+na_{11}\hat{x}}\), we have \(\hat{x}>\frac {r}{a_{11}}-\frac{a_{12}}{na_{11}}\) when \(r>\frac{a_{12}}{n}\). The condition \(a_{11}>m(r-a_{11}\hat{x})\) in Theorem 6 is equivalent to \(\hat{x}>\frac{r}{a_{11}}-\frac{1}{m}\), thus, if \(ma_{12}< na_{11}\), the condition \(a_{11}>m(r-a_{11}\hat{x})\) must be verified. Therefore, the condition \(a_{11}>m(r-a_{11}\hat{x})\) in Theorem 6 can be replaced by the condition \(a_{12}<\min\{\frac{na_{11}}{m}, nr\}\) which is easier to verify.
5 Stationary distribution
Before giving the main theorems, we first give a lemma [43].
Assumption B
- (B.1)
In the domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix \(A(x)\) is bounded away from zero.
- (B.2)
If \(x\in E_{l}\backslash U\), the mean time τ at which a path issuing from x reaches the set U is finite, and \(\sup_{x\in K}E_{x}\tau <\infty\) for every compact subset \(K \subset E_{l}\).
Lemma 4
[43]
Remark 7
To validate (B.1), it suffices to prove that F is uniformly elliptical in U, where \(F_{u}=b(x)\cdot u_{x}+[tr(A(x)u_{xx})]/2\), that is, there is a positive number M such that \(\sum_{i,j=1}^{k}a_{ij}(x)\xi_{i}\xi_{j}\geq M|\xi|^{2} \), \(x\in U\), \(\xi\in R^{k}\) (see p.103 of [44]). To verify (B.2), it is sufficient to show that there exists some neighborhood U and a non-negative \(C^{2}\)-function V such that for any \(x\in E_{l}\backslash U\), LV is negative. (For details, we refer to p.1163 of [45].)
Theorem 7
Proof
Without considering the random fluctuations of environment, that is say, \(\sigma_{i}\ (i=1,2,3)=0\), according to the proof of Theorem 7, it is easy to get the following conclusion.
Corollary 3
Assume \(R_{0}>1\), when \(a_{11}>m(r-a_{11}x^{*})\), the positive equilibrium \(E^{*}(x^{*},y_{1}^{*},y_{2}^{*})\) of system (2) is globally asymptotically stable.
6 Numerical simulation
7 Conclusions
A stochastic model corresponding to a predator-prey model with Beddington-DeAngelis functional response and disease in the predator population is investigated. We show that system (5) has a unique global positive solution as this is essential in any population dynamics model. We also give an estimate of the solution by the comparison theorem. The threshold between persistence in the mean and extinction for prey population is given. Sufficient conditions of extinction for both the susceptible predator and the infected predator are obtained. Furthermore, sufficient conditions of permanence for deterministic system (2) are derived in Additional file 1, which can give us a contrast between stochastic system (5) and its corresponding deterministic system (2). This shows that environmental perturbation will make the reduced population size. There is a stationary distribution for system (5) when the environmental noise is very small, we can consider it as stability in stochastic sense. By the way, conditions of global stability of system (2) can be established. Numerical simulations illustrate that if the positive equilibrium of the deterministic system is globally stable, then the stochastic model will preserve this nice property provided the noise is sufficiently small, but it is not true when the noise is large. All these consequences imply that the environmental white noise has an important effect on biological systems; therefore, it is more realistic and suitable to include random effects in the models.
Some interesting questions deserve further investigation. Here, we cannot get the condition of persistence for the predator population at present. In fact, there are some difficulties that cannot be overcome at present, we leave them for future research. Moreover, it is interesting to study other parameters perturbed by the environmental noise.
Declarations
Acknowledgements
Shuang Li would like to thank Professor Xinan Zhang, her teacher, for his helpful comments which improved this work. This work is supported by Science and Technology Search Key Project of Education Department of Henan Province in China (No. 14A110019) and Foundation for Ph.D. of Henan Normal University (No. qd13043).
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Authors’ Affiliations
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