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Threshold dynamics of a delayed predator–prey model with impulse via the basic reproduction number
- Xiangsen Liu^{1}Email author and
- Binxiang Dai^{2}
https://doi.org/10.1186/s13662-018-1895-1
© The Author(s) 2018
- Received: 14 May 2018
- Accepted: 19 November 2018
- Published: 7 December 2018
Abstract
In this paper, we study a delayed predator–prey model with impulse and, in particular, the existence of the predator-free periodic solution. We employ the approach and techniques coming from epidemiology and calculate the basic reproduction number for the predator. Using the basic reproduction number, we consider the global attraction of the predator-free periodic solution and uniform persistence of the predator. Our results improve the results by Li and Liu (Adv. Differ. Equ. 2016:42, 2016), where they left the open problem of finding a threshold value that determines the eradication and uniform persistence of the predator. Furthermore, we give some numerical simulations to illustrate our results.
Keywords
- Predator–prey model
- Delay
- Impulse
- Basic reproduction number
1 Introduction
In the natural world, many species usually pass through a number of stages during their life cycle. So it is practical to introduce time delay into models of theoretical ecology. In particular, it is often important to take into account the processes of gestation and maturation to make an abstract model more biologically realistic [2–4]. The characteristic property of population models with time delay is their oscillatory behavior: for a sufficiently large maturation period, an initially stable equilibrium becomes unstable, and the system exhibits sustained oscillations [2, 5]. Additionally, impulsive differential equations have been extensively used as models in biology, physics, chemistry, engineering, and other sciences with particular emphasis on population dynamics [6–9]. In [9] the authors discussed an impulsive predator–prey system with stage structure and generalized functional response. Sufficient conditions are established for the existence of a predator-free positive periodic solution and the permanence of the system. Numerical simulation shows that impulses and functional response affect the dynamics of the system.
In [1], sufficient conditions for the global attraction of a predator-free periodic solution are obtained by the theory of impulsive differential equations, that is, \(T< T_{1}^{\ast}\). The conditions for the permanence of the system are investigated, that is, \(T>T_{2}^{\ast}\). Note that \(T_{1}^{\ast}< T_{2}^{\ast}\) always holds. It is obvious that if \(T \in(T_{1}^{\ast}, T_{2}^{\ast})\), then we cannot determine whether the predator can persist or not. In the present paper, we give a thorough global dynamics of (1.1), which completely solves the question left in [1]. To do this, we employ the approach coming from epidemiology [13]. As far as we know, there are no papers employing this approach in ecology. Throughout the present paper, roughly speaking, the basic reproduction number \(R_{0}\) may be thought as the number of predators one predator gives rise during its life, when introduced in a prey population [14]. A similar threshold value for the coexistence of a predator–prey system has previously been formulated and explained by Pielou [15], among others but, to the best of our knowledge, has not been termed a “basic reproduction number.” In ecology, many authors have investigated the autonomous predator–prey systems using the basic reproduction number [16, 17]. For example, in [16] the authors considered a stage-structured predator–prey model with nonlinear predation rate. They discussed the stability of the system using the basic reproduction number of the predator population. In contrast, there have been few papers discussing the nonautonomous, delayed, or impulsive predator–prey systems using the basic reproduction number (except for [18]). In [18] the authors considered an ecoepidemiological model with Holling type-III functional response and time delay. They used the ecological and disease basic reproduction numbers to determine the persistence of the system. In this paper, using the basic reproduction number of the predator population and approach in [13], we wish to find a threshold value to determine whether the predator can exist or not.
The remainder of this paper is organized as follows. In the next section, we discuss the existence of a predator-free periodic solution and boundedness of system (1.1). In Sect. 3, we employ the approach coming from epidemiology and calculate the basic reproduction number for the predator. In Sect. 4, using the basic reproduction number, we consider the global attraction of the predator-free periodic solution and persistence of the predator in (1.1). In Sect. 5, we give some numerical simulations to illustrate our results. Finally, we give some concluding remarks.
2 The existence of a predator-free periodic solution and boundedness of system (1.1)
For system (2.1), we have the following result.
Lemma 2.1
([1])
According to Lemma 2.1, we obtain the following result.
Theorem 2.1
If \((1-p)e^{rT}>1\), then system (1.1) has a predator-free periodic solution \((x^{\ast}(t), 0)\), where \(x^{\ast}(t)\) is shown in (2.2).
Next, we will show that all solutions of (1.1) are uniformly upper bounded.
Theorem 2.2
If \((1-p)e^{rT}>1\), then all solutions of (1.1) are uniformly upper bounded.
Proof
3 The basic reproduction number of the predator
Following [19], the basic reproduction number of the predator is defined as \(R_{0}\triangleq r(L)\), the spectral radius of L.
Remark 3.1
4 The global dynamics of system (1.1)
In this section, we study the global dynamics of system (1.1) in terms of its basic reproduction number \(R_{0}\). To this end, we first introduce some lemmas for our main results.
For any \(\psi\in C([-\tau,0],\mathbb{R})\), let \(P(t)\psi=u_{t}(\psi)\) be the unique solution of (3.1) satisfying \(u_{0}=\psi\). Then \(P\triangleq P(T)\) is the Poincaré map of (3.1).
Lemma 4.1
([20])
- (1)
\(R_{0}=1\) if and only if \(r(P)=1\);
- (2)
\(R_{0}>1\) if and only if \(r(P)>1\);
- (3)
\(R_{0}<1\) if and only if \(r(P)<1\).
Lemma 4.2
[13]. Let \(\mu=\ln\frac{r(P)}{T}\). Then there exists a positive T-periodic function \(v(t)\) such that \(e^{\mu t}v(t)\) is a solution of (4.1).
For the predator-free periodic solution \((x^{\ast}(t), 0)\) of (1.1), we have the following result.
Theorem 4.1
Assume that \((1-p)e^{rT}>1\). If \(R_{0}<1\), then the predator-free periodic solution \((x^{\ast}(t), 0)\) of (1.1) is globally attracting, where \(R_{0}\) is defined in (3.5).
Proof
By Lemma 4.1 we see that \(R_{0}<1\) if and only if \(r(P)<1\), where P is the Poincaré map of (3.1). Since \(\lim_{\epsilon\rightarrow0}r(P_{\epsilon})=r(P)<1\), we may fix a small enough \(\epsilon>0\) such that \(r(P_{\epsilon})<1\). By Lemma 4.2 there is a positive T-periodic function \(v_{\epsilon}(t)\) such that \(e^{\mu_{\epsilon}t}v_{\epsilon}(t)\) is a positive solution of (4.2), where \(\mu_{\epsilon}=\frac{\ln r(P_{\epsilon})}{T}<0\).
By the comparison principle, for \(x(0+)\geq z_{2}(0+)\) and \(t>\max\{ T_{1}, T_{4}\}\), we have \(x(t)\geq z_{2}(t)\), and \(z_{2}(t)-z_{2}^{\ast }(t)\rightarrow0\) as \(t\rightarrow+\infty\). Meanwhile, \(z_{2}^{\ast }(t)-x^{\ast}(t)\rightarrow0\) as \(\epsilon_{1}\rightarrow0\). Based on this analysis and (4.3), we see that \(x(t)-x^{\ast}(t)\rightarrow0\) as \(t\rightarrow+\infty\). Therefore the predator-free periodic solution \((x^{\ast}(t), 0)\) of (1.1) is globally attracting. The proof is completed. □
Theorem 4.2
Let \((1-p)e^{rT}>1\). If \(R_{0}>1\), then there exists \(q>0\) such that every positive solution \((x(t), y(t))\) of (1.1) satisfies \(y(t)\geq q\) for t large enough.
Proof
Since \(\lim_{\eta\rightarrow0}r(M_{\eta})=r(P)>1\), we can fix a small positive number η such that \(r(M_{\eta})>1\) and \(\eta<\inf_{t\geq0}x^{\ast}(t)\).
By Lemma 4.2 there is a positive T-periodic function \(v_{\eta }(t)\) such that \(e^{\mu_{\eta}t}v_{\eta}(t)\) is a positive solution of (4.5), where \(\mu_{\eta}=\frac{\ln r(M_{\eta})}{T}>0\).
- (H1)
\(y(t)\geq\gamma\) for all large t.
- (H2)
\(y(t)\) oscillates about γ for all large t.
Note that the function \(y(t)\) for \(t\geq0\) is uniformly continuous since its derivative is bounded for all \(t\geq0\). Hence there exists \(T'\) (\(0< T'<\tau\) is independent of the choice of \(\underline{t}\)) such that \(y(t)>\frac{\gamma}{2}\) for \(t\in[\underline{t}, \underline{t}+T']\). Let us consider the following three cases:
Case (\(B_{1}\)) \(\bar{t}-\underline{t}\leq T'\). Then \(y(t)>\frac{\gamma }{2}\) for all \(t\in[\underline{t}, \bar{t}]\).
Case (\(B_{2}\)) \(T'<\bar{t}-\underline{t}\leq\tau\).
Case (\(B_{3}\)) \(\bar{t}-\underline{t}> \tau\).
Consequently, we get \(y(t)\geq q\) for \(t\in[\underline{t}, \bar{t}]\). Since this kind of interval \([\underline{t}, \bar{t}]\) is chosen arbitrarily, we get \(y(t)\geq q\) for t large enough. This completes the proof. □
5 Numerical simulation
6 Conclusion
In this paper, we mainly discuss the extinction and permanence of the predator for system (1.1). Using the basic reproduction number coming from epidemiology, we may find the threshold value \(R_{0}\) such that if \(R_{0}<1\), then the predator is extinct, whereas if \(R_{0}>1\), then it will persist. Thus we improve the results of [1]. As far as we know, this is the first paper employing this approach of [13] in ecology.
Declarations
Acknowledgements
We would like to thank the anonymous referees very much for their valuable comments and suggestions.
Funding
The research was supported by the National Natural Foundation of China (11271371, 51479215, 11571324).
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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.Y., Liu, W.W.: A delayed Holling type III functional response predator–prey system with impulsive perturbation on the prey. Adv. Differ. Equ. 2016, 42 (2016). https://doi.org/10.1186/s13662-016-0768-8 MathSciNetView ArticleMATHGoogle Scholar
- Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993) MATHGoogle Scholar
- May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (2001) MATHGoogle Scholar
- Murdoch, W.W., Briggs, C.J., Nisbet, R.M.: Consumer-Resource Dynamics. Princeton University Press, Princeton (2003) Google Scholar
- Ruan, S.G.: On nonlinear dynamics of predator–prey models with discrete delay. Math. Model. Nat. Phenom. 4, 140–188 (2009) MathSciNetView ArticleGoogle Scholar
- Hui, J., Zhu, D.: Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects. Chaos Solitons Fractals 29, 233–251 (2006) MathSciNetView ArticleGoogle Scholar
- Liu, B., Zhang, Y., Chen, L.S.: The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management. Nonlinear Anal., Real World Appl. 6, 227–243 (2005) MathSciNetView ArticleGoogle Scholar
- Zhang, H., Georgescu, P., Chen, L.S.: An impulsive predator–prey system with Beddington–Deangelis functional response and time delay. Int. J. Biomath. 1, 1–17 (2008) MathSciNetView ArticleGoogle Scholar
- Shao, Y.F., Li, Y.: Dynamical analysis of a stage structured predator–prey system with impulsive diffusion and generic functional response. Appl. Math. Comput. 220, 472–481 (2013) MathSciNetMATHGoogle Scholar
- Liu, X., Ballinger, G.: Boundedness for impulsive delay differential equations and applications to population growth models. Nonlinear Anal., Theory Methods Appl. 53, 1041–1062 (2003) MathSciNetView ArticleGoogle Scholar
- Leonid, B., Elena, B.: Linearized oscillation theory for a nonlinear delay impulsive equation. J. Comput. Appl. Math. 161, 477–495 (2003) MathSciNetView ArticleGoogle Scholar
- Yan, J.: Stability for impulsive delay differential equations. Nonlinear Anal., Theory Methods Appl. 63, 66–80 (2005) MathSciNetView ArticleGoogle Scholar
- Bai, Z.G.: Threshold dynamics of a time-delayed SEIRS model with pulse vaccination. Math. Biosci. 269, 178–185 (2015) MathSciNetView ArticleGoogle Scholar
- Garrione, M., Rebelo, C.: Persistence in seasonally varying predator–prey systems via the basic reproduction number. Nonlinear Anal., Real World Appl. 30, 73–98 (2016) MathSciNetView ArticleGoogle Scholar
- Pielou, E.C.: Introduction to Mathematical Ecology. Wiley-Interscience, New York (1969) MATHGoogle Scholar
- Paul, G., Hsieh, Y.H., Zhang, H.: A Lyapunov functional for a stage-structured predator–prey model with nonlinear predation rate. Nonlinear Anal., Real World Appl. 11, 3653–3665 (2010) MathSciNetView ArticleGoogle Scholar
- Bate, A.M., Hilker, F.M.: Predator–prey oscillations can shift when diseases become endemic. J. Theor. Biol. 316, 1–8 (2013) MathSciNetView ArticleGoogle Scholar
- Xu, R., Tian, X.H.: Global dynamics of a delayed eco-epidemiological model with Holling type-III functional response. Math. Methods Appl. Sci. 37, 2120–2134 (2014) MathSciNetView ArticleGoogle Scholar
- Bacaër, N., Guernaoui, S.: The epidemic threshold of vector-borne diseases with seasonality. J. Math. Biol. 53, 421–436 (2006) MathSciNetView ArticleGoogle Scholar
- Zhao, X.Q.: Basic reproduction ratios for periodic compartmental models with time delay. J. Dyn. Differ. Equ. (2015). https://doi.org/10.1007/s10884-015-9425-2 View ArticleGoogle Scholar