A delayed Holling type III functional response predator-prey system with impulsive perturbation on the prey
- Shunyi Li^{1} and
- Wenwu Liu^{1}Email author
https://doi.org/10.1186/s13662-016-0768-8
© Li and Liu 2016
Received: 19 October 2015
Accepted: 21 January 2016
Published: 5 February 2016
Abstract
A Holling type III functional response predator-prey system with constant gestation time delay and impulsive perturbation on the prey is investigated. The sufficient conditions for the global attractivity of a predator-extinction periodic solution are obtained by the theory of impulsive differential equations, i.e. the impulsive period is less than the critical value \(T_{1}^{*}\). The conditions for the permanence of the system are investigated, i.e. the impulsive period is larger than the critical value \(T_{2}^{*}\). Numerical examples show that the system has very complex dynamic behaviors, including (1) high-order periodic and quasi-periodic oscillations, (2) period-doubling and -halving bifurcations, and (3) chaos and attractor crises. Further, the importance of the impulsive period, the gestation time delay, and the impulsive perturbation proportionality constant are discussed. Finally, the impulsive control strategy and the biological implications of the results are discussed.
Keywords
predator-prey system impulsive perturbation time delay extinction permanence chaos1 Introduction
The time delay population dynamics system describes the current state of the population not only related to the current state but also related to the state of the population in the past. That is to say, the time delay effect is very important in population dynamics, which tends to destabilize the positive equilibria and cause a loss of stability, bifurcate into various periodic solutions, even make chaotic oscillations. Recently, there has been much work dealing with time delayed population systems (see [1–12]). For example, a stage-structured prey-predator system with time delay and Holling type-III functional response is considered by Wang et al. [4], the existence and properties of the Hopf bifurcations are established. A delayed eco-epidemiology model with Holling-III functional response was instigated by Zou et al. [5], and one found that time delay may lead to Hopf bifurcation under certain conditions. The Hopf bifurcation of a delayed predator-prey system with Holling type-III functional response also has been considered in [6–8]. The existence of positive periodic solutions of a delayed nonautonomous prey-predator system with Holling type-III functional response was considered in [9–11], by using the continuation theorem of coincidence degree theory. Therefore, time delays would make the prey-predator system subject to periodic oscillations via a local Hopf bifurcation, and destroy the stability of the system.
Recently, more and more authors have discussed the impulsive perturbation on prey-predator systems, since the system would be stabilized by impulsive effects, and would make the system subject to complex dynamical behaviors [13–21]. For example, a predator-prey model with impulsive effect and generalized Holling type III functional response was studied by Su et al. [15], and the sufficient conditions for the existence of a pest-eradication periodic solution and permanence of the system are obtained. The existence of positive periodic solutions of the nonautonomous prey-predator system with Holling type III functional response and impulsive perturbation is considered in [17, 18]. By using a continuation theorem of coincidence degree theory, the sufficient conditions for the existence of a positive periodic solution are obtained for the system.
Note that harmful delays would destroy the stability of the system via bifurcations and even lead the system to extinction. At this point, the impulsive control strategies can be considered, which can both improve the stability of the system and control the amplitude of the bifurcated periodic solution effectively. For example, a time delayed Holling type II functional response prey-predator system with impulsive perturbations is investigated by Jia et al. [22], and the problems of the predator-extinction periodic solution and the permanence of the system are investigated. So, how does the dynamical behavior go when the delayed system with impulsive effect? Especially, what would happen for the delayed predator-prey system with Holling type III functional response under impulsive perturbation?
The paper is arranged as follows. Some notations and lemmas are given in the next section, and we consider the existence and global attraction of the predator-extinction periodic solutions of the system. The sufficient conditions for the permanence of the system are given by using the theory on impulsive and delay differential equation. Numerical examples are given to support the theoretical research, and some complex dynamic behaviors are shown. For example, we see period-halving and period-doubling bifurcations, periodic and high-order quasi-periodic oscillations, even chaotic oscillation. The importance of the impulsive period T, the gestation time delay τ, and the impulsive perturbation proportionality constant p are discussed. Finally, the impulsive control strategy and biological implications of the results are discussed.
2 Preliminaries
- (1)
V is continuous in \((t,x) \in(nT,(n + 1)T] \times\mathbb{R}_{+} ^{2} \) and for each \(x\in\mathbb{R}_{+}^{2}\), \(n\in\mathbb{N}\), \(\lim_{(t,y) \to(nT^{+} ,x)}V(t,y) = V(nT^{+} ,x)\) exists.
- (2)
V is locally Lipschitzian in x.
Definition 2.1
Definition 2.2
System (1) is said to be permanent if there exist two positive constants m, M, and \(T_{0}\) such that each positive solution \(X(t)=(x(t),y(t))\) of the system (1) satisfies \(m \le x(t) \le M\), \(m \le y(t) \le M\) for all \(t>T_{0}\).
The solution of system (1) is a piecewise continuous function \(x:\mathbb{R}_{+} \mapsto\mathbb{R}_{+} ^{2} \), \(x(t)\) is continuous on \((nT,(n + 1)T]\), \(n\in\mathbb{N}\), and \(x(nT^{+} ) = \lim_{t \to nT^{+} } x(t)\) exists, the smoothness properties of f guarantee the global existence and uniqueness of solutions of system (1), for details see [23, 24].
Lemma 2.1
Let \(X(t)\) be a solution of system (1) with \(X(0^{+})\geq0\), then \(X(t)\geq0\) for all \(t\geq0\) and further \(X(t)>0\) for all \(t\geq0\) if \(X(0^{+})>0\).
Lemma 2.2
[23]
Lemma 2.3
If \((1 - p){e^{rT}} > 1\), system (1) has a predator-extinction periodic solution \(X(t)=({x^{*}}(t),0)\) for \(t \in (nT,(n + 1)T]\), and for any solution \(X(t)=(x(t),y(t))\) of system (1), we have \(x(t) \to{x^{*}}(t)\) as \(t \to + \infty\).
Lemma 2.4
- (i)
If \(a< b\), then \(\lim_{t\to+\infty} x(t)=0\).
- (ii)
If \(a>b\), then \(\lim_{t\to+\infty} x(t)=+\infty\).
3 Extinction and permanence
Theorem 3.1
If \({\Re_{1}} < 1\) and \((1 - p){e^{rT}} > 1\), then the predator-extinction periodic solution \(X(t) = ({x^{*}}(t),0)\) of system (1) is globally attractive.
Proof
Theorem 3.2
There exists a constant \(Y_{0}=M_{1}/d-kK>0\), such that \(x(t) \le K\) and \(y(t) \le Y_{0}\) for any solution \(X(t) = (x(t),y(t))\) of system (1) with all t large enough.
Proof
Theorem 3.3
If \({{\Re}_{2}}>1\) and \(r>\alpha KY_{0}\), then system (1) is uniformly persistent.
Proof
On the one hand, if \(y(t)\ge m_{2}^{*}\) holds true for all sufficiently large t, then our aim is reached. On the other hand, suppose \(y(t)\) is oscillatory about \(m_{2}^{*}\). Let \({{m}_{2}}=\min\{m_{2}^{*}/2, m_{2}^{*}\exp(-d\tau)\}\) and we will prove that \(y(t)\ge{{m}_{2}}\). There exist two positive constants t̄ and ω such that \(y(\bar{t})=y(\bar{t}+\omega )=m_{2}^{*}\) and \(y(t)< m_{2}^{*}\) for \(t\in(\bar{t},\bar{t}+\omega)\). The inequality \(x(t)>\rho\) holds true for \(t\in(\bar{t},\bar {t}+\omega)\) when t̄ is large enough.
Since there is no impulsive effect on \(y(t)\), \(y(t)\) is uniformly continuous. Then, there exists a constant \(T_{3}\) (with \(0<{{T}_{3}}<\tau \) and \({{T}_{3}}\) is dependent of the choice of t̄) such that \(y(t)>m_{2}^{*}/2\) for all \(t\in[\bar{t},\bar{t}+{{T}_{3}}]\).
If \(\omega\le{{T}_{3}}\), our aim is reached. If \({{T}_{3}}<\omega\le \tau\), by the second equation of system (1) we get \({y}'(t)\ge -dy(t)\) for \(t\in(\bar{t},\bar{t}+\omega]\). Then we get \(y(t)\ge m_{2}^{*}\exp(-d\tau)\) for \(\bar{t}< t\le\bar{t}+\omega\le\bar{t}+\tau\) since \(y(t)=m_{2}^{*}\). Therefore, \(y(t)\ge{{m}_{2}}\) for \(t\in(\bar{t},\bar {t}+\omega]\).
If \(\omega\ge\tau\), from the second equation of system (1), then we get \(y(t)\ge{{m}_{2}}\) for \(t\in(\bar{t},\bar{t}+\tau]\). Thus, we have \(y(t)\ge{{m}_{2}}\) for \(t\in[\bar{t}+\tau,\bar{t}+\omega]\). According to the above proof. Since the interval \([\bar{t},\bar {t}+\omega]\) is arbitrarily chosen, we have \(y(t)\ge{{m}_{2}}\) for sufficiently large t. From our arguments above, the choice of \({{m}_{2}}\) is independent of the positive solution of system (1) which shows that \(y(t)\ge{{m}_{2}}\) holds for sufficiently large t.
Remark 1
Remark 2
Note that \({{f}_{1}}(p,T)>{{f}_{2}}(p,T)\), so \(T_{1}^{*}< T_{2}^{*}\) for \({{F}_{1}}({{f}_{1}}(p,T_{1}^{*}))=0\) and \({{F}_{2}}({{f}_{2}}(p,T_{2}^{*}))=0\) with respect to the same value of the parameter p. Therefore, if \(T< T_{1}^{*}\) the predator-extinction periodic solution is globally attractive and if \(T>T_{2}^{*}\) the system has permanence. If system (1) is without time delay, according to [14], we know that there would be a threshold \(T_{\max} \). If \(T < T_{\max}\), then the prey- (or predator)-eradication periodic solution is locally asymptotically stable; if \(T > T_{\max} \) the system is permanent. But we get two thresholds \(T_{1}^{*}\) and \(T_{2}^{*}\), and there is no information as regards the system when \(T_{1}^{*}< T< T_{2}^{*}\). This is essentially different when system (1) is with or without time delay.
4 Numerical analysis
Numerical experiments are carried out to integrate the system by using the DDE23 algorithm method in MATLAB.
4.1 Example 1
4.2 Example 2
5 Conclusion
In this paper, we have investigated a prey-predator system with constant gestation time delay and impulsive perturbation on the prey in detail. We have shown that there exists a globally attractive predator-extinction periodic solution when the impulsive period T is less than the critical value \(T_{1}^{*}\). The system is permanent when the impulsive period T is larger than the critical value \(T_{2}^{*}\). Therefore, we get two thresholds \(T_{1}^{*}\) and \(T_{2}^{*}\), and there is no information as regards the system (1) when \(T_{1}^{*}< T< T_{2}^{*}\). If the system (1) is without time delay, then \(T_{1}^{*}=T_{\max}=T_{2}^{*}\) would be the unique threshold. This is essentially different when system (1) is with or without time delay.
Numerical examples show that system (1) have various kinds of periodic oscillations, including high-order periodic and quasi-periodic oscillations, chaotic oscillations. These results imply that the parameters of impulsive period T, time delay τ, and impulsive perturbation proportionality constant p would be important factors to affect the dynamic behaviors of the system (1), and make the system (1) subject to complex dynamical behaviors. That large time delay could stabilize the system, an impulsive effect could destabilize the system. Therefore, the dynamical behaviors would be more complex when the system is subject to both time delay and an impulsive effect.
Declarations
Acknowledgements
This work was supported by the Joint Natural Science Foundation of Guizhou Province (Nos. LKQS[2013]12 and LH[2014]7437); Foundation of Qiannan Normal College for Nationalities (Nos. 2014ZCSX03 and 2014ZCSX11).
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
References
- Li, S, Xue, Y, Liu, W: Hopf bifurcation and global periodic solutions for a three-stage-structured prey-predator system with delays. Int. J. Inf. Syst. Sci. 8(1), 142-156 (2012) MathSciNetGoogle Scholar
- Li, S, Liu, W, Xue, X: Bifurcation analysis of a stage-structured prey-predator system with discrete and continuous delays. Appl. Math. 4(7), 1059-1064 (2013) View ArticleGoogle Scholar
- Li, S, Xiong, Z: Bifurcation analysis of a predator-prey system with sex-structure and sexual favoritism. Adv. Differ. Equ. 2013, Article ID 219 (2013) View ArticleMathSciNetGoogle Scholar
- Wang, L, Xu, R, Feng, G: A stage-structured predator-prey system with time delay and Holling type-III functional response. Int. J. Pure Appl. Math. 48(2), 53-66 (2008) MathSciNetGoogle Scholar
- Zou, W, Xie, J, Xiong, Z: Stability and Hopf bifurcation for an eco-epidemiology model with Holling-III functional response and delays. Int. J. Biomath. 1(3), 377-389 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Zhang, X, Xu, R, Gan, Q: Periodic solution in a delayed predator prey model with Holling type III functional response and harvesting term. World J. Model. Simul. 7(1), 70-80 (2011) Google Scholar
- Das, U, Kar, TK: Bifurcation analysis of a delayed predator-prey model with Holling type III functional response and predator harvesting. J. Nonlinear Dyn. 2014, Article ID 543041 (2014) MathSciNetGoogle Scholar
- Zhang, Z, Yang, H, Fu, M: Hopf bifurcation in a predator-prey system with Holling type III functional response and time delays. J. Appl. Math. Comput. 44(1-2), 337-356 (2014) View ArticleMathSciNetMATHGoogle Scholar
- Fan, Y, Wang, L: Multiplicity of periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response and harvesting terms. J. Math. Anal. Appl. 365(2), 525-540 (2010) View ArticleMathSciNetMATHGoogle Scholar
- Cai, Z, Huang, L, Chen, H: Positive periodic solution for a multi species competition-predator system with Holling III functional response and time delays. Appl. Math. Comput. 217(10), 4866-4878 (2011) View ArticleMathSciNetMATHGoogle Scholar
- Li, G, Yan, J: Positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling type III functional response. Appl. Math. Comput. 218(8), 4341-4348 (2011) View ArticleMathSciNetGoogle Scholar
- Xu, C, Wu, Y, Lu, L: Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays. Adv. Differ. Equ. 2014, Article ID 208 (2014) View ArticleMathSciNetGoogle Scholar
- Li, S: Complex dynamical behaviors in a predator-prey system with generalized group defense and impulsive control strategy. Discrete Dyn. Nat. Soc. 2013, Article ID 358930 (2013) Google Scholar
- Li, S, Xiong, Z, Wang, X: The study of a predator-prey system with group defense and impulsive control strategy. Appl. Math. Model. 34(9), 2546-2561 (2010) View ArticleMathSciNetMATHGoogle Scholar
- Su, H, Dai, B, Chen, Y, et al.: Dynamic complexities of a predator-prey model with generalized Holling type III functional response and impulsive effects. Comput. Math. Appl. 56(7), 1715-1725 (2008) View ArticleMathSciNetMATHGoogle Scholar
- Fan, X, Jiang, F, Zhang, H: Dynamics of multi-species competition-predator system with impulsive perturbations and Holling type III functional responses. Nonlinear Anal., Theory Methods Appl. 74(10), 3363-3378 (2011) View ArticleMathSciNetMATHGoogle Scholar
- Liu, Z, Zhong, S: An impulsive periodic predator-prey system with Holling type III functional response and diffusion. Appl. Math. Model. 36(12), 5976-5990 (2012) View ArticleMathSciNetGoogle Scholar
- Yan, C, Dong, L, Liu, M: The dynamical behaviors of a nonautonomous Holling III predator-prey system with impulses. J. Appl. Math. Comput. 47(1), 193-209 (2015) View ArticleMathSciNetGoogle Scholar
- Tan, R, Liu, W, Wang, Q: Uniformly asymptotic stability of almost periodic solutions for a competitive system with impulsive perturbations. Adv. Differ. Equ. 2014, Article ID 2 (2014) View ArticleMathSciNetGoogle Scholar
- Xu, L, Wu, W: Dynamics of a nonautonomous Lotka-Volterra predator-prey dispersal system with impulsive effects. Adv. Differ. Equ. 2014, Article ID 264 (2014) View ArticleGoogle Scholar
- Ju, Z, Shao, Y, Kong, W: An impulsive prey-predator system with stagestructure and Holling II functional response. Adv. Differ. Equ. 2014, Article ID 280 (2014) View ArticleMathSciNetGoogle Scholar
- Jia, J, Cao, H: Dynamic complexities of Holling type II functional response predator-prey system with digest delay and impulsive. Int. J. Biomath. 2(2), 229-242 (2009) View ArticleMathSciNetGoogle Scholar
- Bainov, DD, Simeonov, DD: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow (1993) MATHGoogle Scholar
- Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) View ArticleMATHGoogle Scholar
- Kuang, Y: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993) MATHGoogle Scholar