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- Open Access
Convergences of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes
- Yuhua Lin^{1},
- Xiangdong Xie^{2}Email author,
- Fengde Chen^{3} and
- Tingting Li^{3}
https://doi.org/10.1186/s13662-016-0887-2
© Lin et al. 2016
Received: 28 April 2016
Accepted: 2 June 2016
Published: 7 July 2016
Abstract
A stage-structured predator-prey model (stage structure for both predator and prey) with modified Leslie-Gower and Holling-II schemes is studied in this paper. Using the iterative technique method and the fluctuation lemma, sufficient conditions which guarantee the global stability of the positive equilibrium and boundary equilibrium are obtained. Our results indicate that for a stage-structured predator-prey community, both the stage structure and the death rate of the mature species are the important factors that lead to the permanence or extinction of the system.
Keywords
- global stability
- extinction
- stage-structure
- Leslie-Gower
- Holling-type II
- predator-prey
MSC
- 34D23
- 92B05
- 34D40
1 Introduction
In [5], the authors analyzed the dynamics of system (1.1), specially, by using the iterative technique, the authors obtained a set of sufficient conditions which guarantee the existence of a unique globally attractive positive equilibrium. Li et al. [6] found that some of the conditions in [5] are redundant, and they obtained the following result.
Theorem A
The organization of this paper is as follows: The main results are stated and proved in Sections 2 and 3, respectively. In Section 4, several examples together with their numerical simulations are presented to illustrate the feasibility of our main results. We end this paper by a brief discussion. For more work on the Leslie-Gower predator-prey system, one may refer to [36–39] and the references cited therein.
2 Main results
Consequently, we have the following theorem.
Theorem 2.1
Assume that inequality (2.1) holds, then system (1.6) admits a unique positive equilibrium point E.
Theorem 2.2
Remark 2.1
Remark 2.2
- (i)
If \(d_{12}= 0\), \(d_{21}\neq0\), in this case, \(\lambda _{3}>\lambda_{0}\), that is, introducting the mortality item of the predator species improving the coexistence rate of the two species.
- (ii)
If \(d_{12}\neq0\), \(d_{21}= 0\), in this case, \(\lambda _{0}>\lambda_{3}\), that is, introducting the mortality item of the prey species decreasing the chance of coexistence of both species.
Theorem 2.3
Theorem 2.4
Remark 2.3
From [5], we know that \(E_{0}(0,0,0)\) of the system (1.1) is unstable, which implies the extinction of both predator and prey species is impossible. However, if the death rates of the mature prey and predator species are large enough, (\(\mathrm{H}_{3}\)) in Theorem 2.3 would hold, and consequently both the prey and the predator species will be driven to extinction. By constructing a suitable Lyapunov function, Korobeinikov [27] showed that the unique positive equilibrium of the traditional Leslie-Gower predator-prey model is globally attractive, which means that it is impossible for the predator species to become extinct. However, Theorem 2.4 shows that if the death rate of the mature predator species is large enough, (\(\mathrm{H}_{4}\)) would hold and the predator species will be driven to extinction. Theorems 2.3 and 2.4 show that the death rates of the mature predator and prey species are two of the essential factors to determine the persistent property of the system.
Theorem 2.5
Corollary 2.1
If the parameters of system (1.2) satisfy the condition (2.1), then system (1.2) has a unique positive equilibrium point \(E'(x^{*}_{1},x^{*}_{2},y^{*}_{1},y^{*}_{2})\), where \(x^{*}_{1}= \frac{r_{1}x^{*}_{2}(1-e^{-d_{11}\tau_{1}})}{d_{11}}\), \(y^{*}_{1}=\frac{r_{2}y^{*}_{2}(1-e^{-d_{22}\tau_{2}})}{d_{22}} \).
Corollary 2.2
If the parameters of system (1.2) satisfy the conditions (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)), then \(E'\) is globally attractive.
Corollary 2.3
If the parameters of system (1.2) satisfy the condition (\(\mathrm{H}_{3}\)), then \(E'_{0}=(0,0,0,0)\) is globally attractive.
Corollary 2.4
If the parameters of system (1.2) satisfy the condition (\(\mathrm{H}_{4}\)), then \(E'_{1}=(x_{1*},x_{2*},0,0)\) is globally attractive, where \(x_{1*}=\frac{r_{1}x_{2*}(1-e^{-d_{11}\tau_{1}})}{d_{11}}\).
Corollary 2.5
If the parameters of system (1.2) satisfy the condition (\(\mathrm{H}_{5}\)), then \(E'_{2}=(0,0,y_{1*},y_{2*})\) is globally attractive, where \(y_{1*}=\frac{r_{2}y_{2*}(1-e^{-d_{22}\tau_{2}})}{d_{22}}\).
3 Proof of the main results
Now let us state several lemmas which will be useful in proving the main results.
Lemma 3.1
Assume that \(x_{2}(\theta)\geq0\), \(y_{2}(\theta)\geq0\) are continuous on \(\theta\in[-\tau,0]\), and \(x_{2}(0)>0\), \(y_{2}(0)>0\). Let \((x_{2}(t),y_{2}(t))^{T}\) be a any solution of system (1.6), then \(x_{2}(t)>0\), \(y_{2}(t)>0\) for all \(t>0\).
The proof of Lemma 3.1 is similar to the proof of Theorem 1 in [1], so we omit its proof.
Lemma 3.2
[2]
- (i)
if \(b\geq a_{1}\), then \(\lim_{t\rightarrow+\infty }x(t)= \frac{b-a_{1}}{a_{2}}\);
- (ii)
if \(b\leq a_{1}\), then \(\lim_{t\rightarrow+\infty}x(t)=0\).
Lemma 3.3
(Fluctuation lemma [23])
- (i)
\(x'(\gamma_{n})\rightarrow0 \) and \(x(\gamma_{n})\rightarrow\limsup_{t\rightarrow+\infty }x(t)=\overline{x}\) as \(n\rightarrow\infty\),
- (ii)
\(x'(\sigma_{n})\rightarrow0 \) and \(x(\sigma_{n})\rightarrow\liminf_{t\rightarrow+\infty }x(t)=\underline{x}\) as \(n\rightarrow\infty\).
Lemma 3.4
Proof
Now we start to prove the above results.
Proof of Theorem 2.2
Proof of Theorem 2.3
Proof of Theorem 2.4
Proof of Theorem 2.5
4 Numerical simulations
The following examples show the feasibility of our main results.
Example 4.1
Example 4.2
Example 4.3
Example 4.4
Example 4.5
5 Conclusion
Huo et al. [5] and Li et al. [6] studied the stability property of the positive equilibrium of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes. In those two papers, the authors only consider the stage structure of prey species and ignore that of predator species. Stimulated by [9–11], we consider a model with stage structure for both predator and prey species. By applying an iterative technique and the fluctuation lemma, sufficient conditions which guarantee the global attractivity of all the nonnegative equilibria are obtained. Our study indicates that both the stage structure of the species and the death rate of the mature predator and prey species are the important factors on the dynamic behaviors of the system. If the death rates of the mature prey and predator species are too large or the degree of the stage structure of the species is large enough, then at least one of the species will be driven to extinction. We would like to mention here that Example 4.5 shows that our result in Theorem 2.2 has room for improvement. We conjecture that condition (2.1) is enough to ensure the global attractivity of the positive equilibrium. We leave this for future work.
Declarations
Acknowledgements
The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. The research was supported by the Natural Science Foundation of Fujian Province (2015J01012, 2015J01019, 2015J05006) and the Scientific Research Foundation of Fuzhou University (XRC-1438).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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