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Stability and Hopf bifurcation for a ratio-dependent predator-prey system with stage structure and time delay
- Lingshu Wang^{1}Email author and
- Guanghui Feng^{2}
https://doi.org/10.1186/s13662-015-0548-x
© Wang and Feng 2015
- Received: 7 April 2015
- Accepted: 17 June 2015
- Published: 19 August 2015
Abstract
A ratio-dependent predator-prey system with time delay due to the gestation of the predator and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of the predator-extinction equilibrium and the coexistence equilibrium of the system are discussed, respectively. Further, the existence of Hopf bifurcation at the coexistence equilibrium is also studied. By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the analytical results.
Keywords
- predator-prey system
- stage structure
- time delay
- stability
- Hopf bifurcation
MSC
- 34K18
- 34K20
- 34K60
- 92D25
1 Introduction
It is well known by the fundamental theory of functional differential equations [11] that system (1.2) has a unique solution \((x_{1}(t), x_{2}(t), y_{1}(t), y_{2}(t))\) satisfying initial conditions (1.3).
The organization of this paper is as follows. In the next section, we investigate the local stability of the predator-extinction equilibrium and the coexistence equilibrium of system (1.2). Further, we study the existence of a Hopf bifurcation for system (1.2) at the coexistence equilibrium. In Section 3, by means of an iterative technique, sufficient conditions are derived for the global stability of the coexistence equilibrium of system (1.2). By comparison arguments, we discuss the global stability of the predator-extinction of system (1.2). Numerical simulations are carried out to illustrate the main results. A brief discussion is given in Section 4 to conclude this work.
2 Local stability and Hopf bifurcation
In this section, we discuss the local stability of equilibria and the existence of a Hopf bifurcation at the coexistence equilibrium of system (1.2).
- (H1)
\({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac{a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m}>0}\),
- (H2)
\({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m} (1+\frac{d_{4}(r_{2}+d_{3})}{a_{2}r_{2}} )>0}\),
- (H3)
\({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m} (1+2\frac {d_{4}(r_{2}+d_{3})}{a_{2}r_{2}+d_{4}(r_{2}+d_{3})} )>0}\),
- (H4)
\({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}<\frac {a_{2}r_{2}-d_{4}(r_{2}+d_{3})}{a_{2}r_{2}m} (1+2\frac {d_{4}(r_{2}+d_{3})}{a_{2}r_{2}+d_{4}(r_{2}+d_{3})} )}\),
We therefore obtain the following results.
Theorem 2.1
- (i)
Let \(rr_{1}>d_{2}(r_{1}+d_{1})\), if \(a_{2}r_{2}< d_{4}(r_{2}+d_{3})\), then the predator-extinction equilibrium \(E_{1}(x_{1}^{+}, x_{2}^{+}, 0, 0)\) is locally asymptotically stable; if \(a_{2}r_{2}>d_{4}(r_{2}+d_{3})\), then the equilibrium \(E_{1}\) is unstable.
- (ii)
If (H3) holds, then the positive equilibrium \(E^{*}\) is locally asymptotically stable for all \(\tau\geq0\).
- (iii)
If (H2) and (H4) hold, then there exists a positive number \(\tau_{0}\) such that the coexistence equilibrium \(E^{*}\) is locally asymptotically stable if \(0<\tau<\tau_{0}\) and unstable if \(\tau>\tau_{0}\). Further, system (1.2) undergoes a Hopf bifurcation at \(E^{*}\) when \(\tau=\tau_{0}\).
3 Global stability
In this section, we are concerned with the global stability of the coexistence equilibrium \(E^{*}\) and the predator-extinction equilibrium \(E_{1}\) of system (1.2), respectively.
Theorem 3.1
- (H5)
\(a_{2}r_{2}<2d_{4}(r_{2}+d_{3})\), \({\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac{1}{m}}\).
Proof
Theorem 3.2
- (H6)
\(a_{2}r_{2}< d_{4}(r_{2}+d_{3})\), \(\frac{rr_{1}-d_{2}(r_{1}+d_{1})}{a_{1}d_{2}}>\frac{1}{m}\).
Proof
We now give some examples to illustrate the main results above.
Example 1
Example 2
Example 3
4 Discussion
In this paper, we have incorporated stage structure for both the predator and the prey into a predator-prey model with time delay due to the gestation of the predator. By using the iteration technique and comparison arguments, we have established sufficient conditions for the global stability of the coexistence equilibrium and the predator-extinction equilibrium. As a result, we have shown the threshold for the permanence and extinction of the system. By Theorems 3.1 and 3.2, we see that: (i) if (H6) holds, the predator population will go to extinction; (ii) if (H3) and (H5) hold, then both the prey and the predator species of system (1.2) are permanent.
Declarations
Acknowledgements
We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research was funded by the National Natural Science Foundation of China (11101117), the Scientific Research Foundation of Hebei Education Department (No. QN2014040) and the Foundation of Hebei University of Economics and Business (No. 2015KYQ01).
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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