Global behavior of a plantherbivore model
 Qamar Din^{1}Email author
https://doi.org/10.1186/s136620150458y
© Din; licensee Springer. 2015
Received: 22 September 2014
Accepted: 6 April 2015
Published: 16 April 2015
Abstract
The present work deals with an analysis of the local asymptotic stability and global behavior of the unique positive equilibrium point of the following discretetime plantherbivore model: \(x_{n+1}=\frac{\alpha x_{n}}{\beta x_{n}+e^{y_{n}}}\), \(y_{n+1}=\gamma(x_{n}+1)y_{n}\), where \(\alpha\in\mathbb{(}1,\infty)\), \(\beta\in\mathbb{(}0,\infty)\), and \(\gamma\in\mathbb{(}0,1)\) with \(\alpha+\beta>1+\frac{\beta}{\gamma}\) and initial conditions \(x_{0}\), \(y_{0}\) are positive real numbers. Moreover, the rate of convergence of positive solutions that converge to the unique positive equilibrium point of this model is also discussed. In particular, our results solve an open problem and a conjecture proposed by Kulenović and Ladas in their monograph (Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, 2002). Some numerical examples are given to verify our theoretical results.
Keywords
plantherbivore system steadystates local stability global behavior rate of convergenceMSC
39A10 40A051 Introduction and preliminaries

Open Problem 6.10.13 ([1], p.128) (a plantherbivore system) Assume that \(\alpha\in\mathbb{(}1,\infty)\), \(\beta\in\mathbb{(}0,\infty)\), and \(\gamma\in\mathbb{(}0,1)\) with \(\alpha+\beta>1+\frac{\beta}{\gamma}\). Obtain conditions for the global asymptotic stability of the positive equilibrium of the system$$ x_{n+1}=\frac{\alpha x_{n}}{\beta x_{n}+e^{y_{n}}},\qquad y_{n+1}=\gamma(x_{n}+1)y_{n}. $$
In this paper, our aim is to investigate the local asymptotic stability, the global asymptotic character of the unique positive equilibrium point, and the rate of convergence of positive solutions of the system (1). For some interesting results related to the qualitative behavior of discrete dynamical systems, we refer to [6–14].
Lemma 1.1
(Jury condition)
2 Boundedness
The following theorem shows that every positive solution of the system (1) is bounded.
Theorem 2.1
Assume that \(x_{n}\le x_{n+1}\) for all \(n=0,1,\ldots \) , then every positive solution \(\{(x_{n},y_{n})\}\) of the system (1) is bounded.
Proof
3 Linearized stability
Theorem 3.1
Proof
The following theorem shows a necessary and sufficient condition for local asymptotic stability of unique positive equilibrium point of the system (1).
Theorem 3.2
Proof
4 Global behavior
Lemma 4.1
[7]
 (i)
\(f(x,y)\) is nondecreasing in x and nonincreasing in y.
 (ii)
\(g(x,y)\) is nondecreasing in both arguments.
 (iii)If \((m_{1},M_{1},m_{2},M_{2})\in I^{2}\times J^{2}\) is a solution of the systemsuch that \(m_{1}=M_{1}\) and \(m_{2}=M_{2}\). Then there exists exactly one positive equilibrium point \((\bar{x},\bar{y})\) of the system (11) such that \(\lim_{n\to\infty}(x_{n},y_{n})=(\bar {x},\bar{y})\).$$\begin{aligned}& m_{1}= f(m_{1},M_{2}),\qquad M_{1}=f(M_{1},m_{2}), \\& m_{2} = g(m_{1},m_{2}),\qquad M_{2}=g(M_{1},M_{2}), \end{aligned}$$
Theorem 4.1
The positive equilibrium point \(P_{2}\) of the system (1) is a global attractor.
Proof
5 Conjecture
Theorem 5.1
The unique positive equilibrium point \((1,\ln(\alpha1) )\) of the system (17) is globally asymptotically stable if and only if \(2<\alpha<R_{0}\), where \(R_{0}\) is a root of the function defined in (18) and \(R_{0}\approx3.3457507549227654\).
Proof
6 Rate of convergence
In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1).
Proposition 6.1
(Perron’s theorem) [16]
Proposition 6.2
[16]
Using Proposition 6.1, one has the following result.
Theorem 6.1
7 Examples
In order to verify our theoretical results we consider some interesting numerical examples in this section. These examples represent different types of qualitative behavior of the system (1). First two examples show that positive equilibrium point of the system (1) is locally asymptotically stable, i.e., condition (4) of Theorem 3.2 is satisfied. Meanwhile, the third example shows that the positive equilibrium point of the system (1) is unstable, i.e., condition (4) of Theorem 3.2 does not hold. All plots in this section are drawn with Mathematica.
Example 7.1
Example 7.2
Example 7.3
Declarations
Acknowledgements
The author thanks the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
 Kulenović, MRS, Ladas, G: Dynamics of Second Order Rational Difference Equations. Chapman & Hall/CRC, Boca Raton (2002) MATHGoogle Scholar
 Kocic, VL, Ladas, G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht (1993) View ArticleMATHGoogle Scholar
 Allen, LJS, Hannigan, MK, Strauss, MJ: Mathematical analysis of a model for a plantherbivore system. Bull. Math. Biol. 55(4), 847864 (1993) View ArticleMATHGoogle Scholar
 Allen, LJS, Hannigan, MK, Strauss, MJ: Development and analysis of mathematical model for a plantherbivore system. In: Proceedings of the First World Congress on World Congress of Nonlinear Analysts. WCNA’92, vol. 4, pp. 37233732 (1995) Google Scholar
 Allen, LJS, Strauss, MJ, Tnorvilson, HG, Lipe, WN: A preliminary mathematical model of the apple twig borer (Coleoptera: Bostricidae) and grapes on the Texas High Planes. Ecol. Model. 58, 369382 (1991) View ArticleGoogle Scholar
 Din, Q: Dynamics of a discrete LotkaVolterra model. Adv. Differ. Equ. (2013). doi:https://doi.org/10.1186/16871847201395 Google Scholar
 Din, Q, Donchev, T: Global character of a hostparasite model. Chaos Solitons Fractals 54, 17 (2013) View ArticleMathSciNetGoogle Scholar
 Din, Q, Elsayed, EM: Stability analysis of a discrete ecological model. Comput. Ecol. Softw. 4(2), 89103 (2014) Google Scholar
 Din, Q, Ibrahim, TF, Khan, KA: Behavior of a competitive system of secondorder difference equations. Sci. World J. 2014, Article ID 283982 (2014) View ArticleGoogle Scholar
 Din, Q: Global stability of a population model. Chaos Solitons Fractals 59, 119128 (2014) View ArticleMathSciNetGoogle Scholar
 Din, Q: Stability analysis of a biological network. Netw. Biol. 4(3), 123129 (2014) Google Scholar
 Din, Q, Khan, KA, Nosheen, A: Stability analysis of a system of exponential difference equations. Discrete Dyn. Nat. Soc. 2014, Article ID 375890 (2014) View ArticleMathSciNetGoogle Scholar
 Din, Q: Asymptotic behavior of an anticompetitive system of secondorder difference equations. J. Egypt. Math. Soc. (2014). doi:https://doi.org/10.1016/j.joems.2014.08.008 Google Scholar
 Din, Q: Qualitative nature of a discrete predatorprey system. Contemp. Methods Math. Phys. Gravit. (Online) 1(1), 2742 (2015) Google Scholar
 Grove, EA, Ladas, G: Periodicities in Nonlinear Difference Equations. Chapman & Hall/CRC, Boca Raton (2004) View ArticleGoogle Scholar
 Pituk, M: More on Poincaré’s and Perron’s theorems for difference equations. J. Differ. Equ. Appl. 8, 201216 (2002) View ArticleMATHMathSciNetGoogle Scholar