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Global behavior of a plantherbivore model
Advances in Difference Equations volume 2015, Article number: 119 (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.
Introduction and preliminaries
In mathematical biology, the model of the plantherbivore has attracted many researchers during the last few decades. There are several theoretical results for the mutual dependence between plant and herbivore populations. We consider here an open problem proposed in [1] related to the global character of solutions of the plantherbivore model
where \(x_{n}\) and \(y_{n}\) are the population biomasses of the plant and the herbivore in successive generations n and \(n+1\), respectively. Moreover, \(\alpha\in\mathbb{(}1,\infty)\), \(\beta\in\mathbb{(}0,\infty)\), and \(\gamma\in\mathbb{(}0,1)\) with \(\alpha+\beta>1+\frac{\beta}{\gamma}\) and initial conditions \(x_{0}\), \(y_{0}\) are positive real numbers. Furthermore, (1) has a unique positive equilibrium point under the following inequalities:
It is pointed out in [2] that the system (1) has a very complex behavior, which has been investigated mainly numerically and visually. Moreover, there is a range of the values of parameters for which the equilibrium points on the xaxis are globally asymptotically stable. The computer simulations indicate that for some parameter values there is chaotic behavior, but proofs are yet to be completed.
In 2002, Kulenović and Ladas [1] proposed the following open problem:

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
$$ x_{n+1}=\frac{\alpha x_{n}}{\beta x_{n}+e^{y_{n}}},\qquad y_{n+1}=\gamma(x_{n}+1)y_{n}. $$
The above model is for the interaction of the apple twig borer (an insect pest of the grape vine) and grapes in the Texas High Plains was developed and studied by Allen et al. [3]. See also Allen et al. [4] and Allen et al. [5].
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].
Let \((\bar{x},\bar{y})\) be an equilibrium point of the system (1), then
Hence, \(P_{0}=(0,0)\), \(P_{1}= (\frac{\alpha1}{\beta},0 )\) and \(P_{2}= (\frac{1\gamma}{\gamma},\ln(\frac{\alpha\gamma\beta+\beta \gamma}{\gamma}) )\) are equilibrium points of the system (1). Furthermore, \(P_{2}= (\frac{1\gamma}{\gamma},\ln(\frac{\alpha\gamma \beta+\beta\gamma}{\gamma}) )\) is a unique positive equilibrium point of the system (1) if and only if
The Jacobian matrix of linearized system of (1) about the fixed point \((\bar{x},\bar{y})\) is given by
Lemma 1.1
(Jury condition)
Consider the seconddegree polynomial equation
where p and q are real numbers. Then the necessary and sufficient condition for both roots of (2) to lie inside the open disk \(\lambda<1\) is
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
Assume that the initial conditions \(x_{0}\), \(y_{0}\) of (1) are positive, then every solution of (1) is positive. Assume that \(\{(x_{n},y_{n})\}\) is an arbitrary positive solution of the system (1). Then, from the first part of the system (1), one has
for all \(n=0,1,2,\ldots \) . Assume that \(x_{n}\le x_{n+1}\) for all \(n=0,1,\ldots \) . Moreover, from the first equation of the system (1), we obtain
From (3) and the second part of the system (1), we have
for all \(n=0,1,\ldots \) . Finally, we let \(F(x)=\ln (\alpha\beta x )\) such that \(0\le x<\frac{\alpha}{\beta}\). Then it follows that \(F'(x)=\frac{\beta}{\alpha\beta x}<0\) for all \(0\le x<\frac{\alpha}{\beta}\). Hence, \(F(x)\le F(0)\) for all \(0\le x<\frac{\alpha}{\beta}\), i.e., \(\ln (\alpha\beta x )\le\ln(\alpha)\) for all \(0\le x<\frac {\alpha}{\beta}\). Thus for any positive solution \(\{(x_{n},y_{n})\}\) of the system (1), we have
for all \(n=1,2,\ldots \) . The proof is therefore completed. □
Linearized stability
Theorem 3.1
Assume that \(\alpha\in(1,\infty)\) and \(\gamma\in(0,1)\), then the following statements are true:

(i)
The equilibrium point \(P_{0}=(0,0)\) of the system (1) is a saddle point.

(ii)
The equilibrium point \(P_{1}= (\frac{\alpha1}{\beta},0 )\) of the system (1) is locally asymptotically stable if and only if \(\beta>\frac{\gamma\alpha \gamma}{\gamma1}\).
Proof
(i) The proof follows from the fact that the Jacobian matrix of linearized system of (1) about the equilibrium point \((0,0)\) is given by
(ii) The Jacobian matrix of linearized system of (1) about the fixed point \((\frac{\alpha1}{\beta},0 )\) is given by
It is obvious that the roots of the characteristic polynomial of \(F_{J} (\frac{\alpha1}{\beta},0 )\) about \((\frac{\alpha1}{\beta},0 )\) are \(\lambda_{1}=\frac{1}{\alpha }<1\) for \(\alpha\in(1,\infty)\) and \(\lambda_{2}=\frac{\alpha \gamma+\beta \gamma\gamma}{\beta}<1\) if and only if \(\beta>\frac{\gamma\alpha \gamma}{\gamma1}\). □
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
The unique positive equilibrium point \(P_{2}= (\frac{1\gamma}{\gamma },\ln(\frac{\alpha\gamma\beta+\beta\gamma}{\gamma}) )\) of the system (1) is locally asymptotically stable if and only if the following holds:
Proof
The Jacobian matrix of linearized system of (1) about the positive equilibrium point \(P_{2}= (\frac{1\gamma}{\gamma},\ln(\frac {\alpha\gamma\beta+\beta\gamma}{\gamma}) )\) is given by
The characteristic polynomial of \(F_{J}(P_{2})\) about positive equilibrium point \(P_{2}= (\frac{1\gamma}{\gamma}, \ln(\frac{\alpha \gamma\beta+\beta\gamma}{\gamma}) )\) is given by
where \(\delta=(1\gamma) (\beta(\gamma1)+\alpha\gamma )\ln (\alpha+\beta\frac{\beta}{\gamma} )\). Set
Then (5) can be written as
Then it follows from Lemma 1.1 that both eigenvalues of \(F_{J}(P_{2})\) lie inside the unit disk \(\lambda<1\), if and only if
Inequality (7) is equivalent to the following three inequalities:
First, we will prove (8). This inequality is equivalent to
Assume that \(\ln (\alpha+\beta\frac{\beta}{\gamma} )<\frac {\beta}{\gamma(\alpha+\beta)\beta}\), then it follows that \(\beta(\gamma1)+\delta<0\). Next, we consider (9). This inequality is equivalent to
It is easy to see that under condition (4), one has \(2\beta (1\gamma)4\alpha\gamma\delta<0\). Finally, (10) is equivalent to the following inequality:
and this obviously holds due to the assumption that \(\alpha+\beta >1+\frac{\beta}{\gamma}\). Then it follows from Lemma 1.1 that the unique positive equilibrium point \(P_{2}= (\frac{1\gamma}{\gamma},\ln(\frac{\alpha \gamma\beta+\beta\gamma}{\gamma}) )\) of the system (1) is locally asymptotically stable if and only if \(\ln (\alpha+\beta\frac{\beta}{\gamma} )<\frac{\beta }{\gamma(\alpha+\beta)\beta}\). □
Global behavior
Lemma 4.1
[7]
Let \(I=[a,b]\) and \(J=[c,d]\) be real intervals, and let \(f:I\times J\to I\) and \(g:I\times J\to J\) be continuous functions. Consider the following system:
with initial conditions \((x_{0},y_{0})\in I\times J\). Suppose that the following statements are true:

(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 system
$$\begin{aligned}& m_{1}= f(m_{1},M_{2}),\qquad M_{1}=f(M_{1},m_{2}), \\& m_{2} = g(m_{1},m_{2}),\qquad M_{2}=g(M_{1},M_{2}), \end{aligned}$$such 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 \(\lim_{n\to\infty}(x_{n},y_{n})=(\bar {x},\bar{y})\).
Theorem 4.1
The positive equilibrium point \(P_{2}\) of the system (1) is a global attractor.
Proof
Let \(f(x,y)=\frac{\alpha x}{\beta x+e^{y}}\) and \(g(x,y)=\gamma(x+1)y\). Then it is easy to see that \(f(x,y)\) is nondecreasing in x and nonincreasing in y. Moreover, \(g(x,y)\) is nondecreasing in both x and y. Let \((m_{1},M_{1},m_{2},M_{2})\) be a positive solution of the system
Then one has
and
Furthermore, we make the assumption as in the proof of Theorem 1.16 of [15]; it suffices to suppose that
From (12), one has
From (13), one has
It follows from (15) that
Moreover, from (14) and (16), we obtain
Hence, from Lemma 4.1 the equilibrium point \(P_{2}\) is a global attractor. □
Lemma 4.2
Under the condition (4) the unique positive equilibrium point \(P_{2}\) of the system (1) is globally asymptotically stable.
Proof
Conjecture
Now we consider the following conjecture related to special form of the system (1).

Conjecture 6.10.6 ([1], p.128) Assume that \(x_{0}, y_{0}\in (0,\infty)\) and that
$$\alpha\in(2,3),\qquad \beta=1,\qquad \gamma=0.5. $$Show that the positive equilibrium of the system (1) is globally asymptotically stable.
In this conjecture, we modify the interval of α for which the unique positive equilibrium point of the system (1) is globally asymptotically stable. In this case the system (1) can be written as
where \(\alpha\in(2,R_{0})\) such that \(R_{0}\) is a root of the function defined by
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
The characteristic polynomial of \(F_{J}(\bar{x},\bar{y})\) about positive equilibrium point \((\bar{x},\bar{y})= (1,\ln(\alpha1) )\) for the system (17) is given by
Then it follows from Lemma 1.1 that the unique positive equilibrium point \((1,\ln(\alpha1) )\) of the system (17) is locally asymptotically stable if and only if
It is easy to see that \(\vert \frac{1}{\alpha}2\vert <2\frac {1}{\alpha}+\frac{ (\frac{\alpha}{2}\frac{1}{2} ) \ln (\alpha1)}{\alpha}\) for all \(\alpha>2\). Hence, it is enough to show that \(2\frac{1}{\alpha}+\frac{ (\frac{\alpha}{2}\frac {1}{2} ) \ln(\alpha1)}{\alpha}<2\), or equivalently
Moreover, \(F(2)=\frac{1}{2}\), \(F(3.4)=0.0148713\), and \(F'(\alpha)=\frac {\alpha+\ln(\alpha1)+2}{2 \alpha^{2}}>0\) for all \(\alpha>2\). From this it follows that \(F(\alpha)\) has the unique root \(R_{0}\) in \((2,3.4)\). With the help of Mathematica this unique root of \(F(\alpha)\) is approximated by \(R_{0}\approx3.3457507549227654\). It is easy to see that \(F(\alpha)<0\) if and only if \(2<\alpha<R_{0}\). Hence, the unique positive equilibrium point \((1,\ln(\alpha 1) )\) of the system (17) is locally asymptotically stable if and only if \(\alpha\in (2,R_{0} )\). Moreover, Theorem 4.1 shows that the unique positive equilibrium point \((1,\ln(\alpha 1) )\) of the system (17) is a global attractor. Hence, the proof is completed. □
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).
The following result gives the rate of convergence of solutions of a system of difference equations:
where \(X_{n}\) is an mdimensional vector, \(A\in C^{m\times m}\) is a constant matrix, and \(B:\mathbb{Z}^{+}\to C^{m\times m}\) is a matrix function satisfying
as \(n\to\infty\), where \(\\cdot\\) denotes any matrix norm which is associated with the vector norm
Proposition 6.1
(Perron’s theorem) [16]
Suppose that condition (21) holds. If \(X_{n}\) is a solution of (20), then either \(X_{n}=0\) for all large n or
exists and is equal to the modulus of one of the eigenvalues of matrix A.
Proposition 6.2
[16]
Suppose that condition (21) holds. If \(X_{n}\) is a solution of (20), then either \(X_{n}=0\) for all large n or
exists and is equal to the modulus of one of the eigenvalues of matrix A.
Let \(\{(x_{n},y_{n})\}\) be any solution of the system (1) such that \(\lim_{n\to\infty}x_{n}=\bar{x}\) and \(\lim_{n\to\infty}y_{n}=\bar{y}\). To find the error terms, one has the following system:
and
Let \(e_{n}^{1}=x_{n}\bar{x}\) and \(e_{n}^{2}=y_{n}\bar{y}\), then one has
and
where
Moreover,
Now the limiting system of error terms can be written as
which is similar to the linearized system of (1) about the equilibrium point \(P_{2}\).
Using Proposition 6.1, one has the following result.
Theorem 6.1
Assume that \(\{(x_{n},y_{n})\}\) is a positive solution of the system (1) such that \(\lim_{n\to\infty}x_{n}=\bar{x}\) and \(\lim_{n\to\infty}y_{n}=\bar{y}\), where
Then the error vector \(e_{n} = \bigl ( {\scriptsize\begin{matrix} e_{n}^{1} \cr e_{n}^{2} \end{matrix}} \bigr )\) of every solution of (1) satisfies both of the following asymptotic relations:
where \(\lambda_{1,2}F_{J}(\bar{x},\bar{y})\) are the characteristic roots of Jacobian matrix \(F_{J}(\bar{x},\bar{y})\).
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
Let \(\alpha=24\), \(\beta=39\), \(\gamma=0.81\). Then the system (1) can be written as
with initial conditions \(x_{0}=0.23\), \(y_{0}=2.7\). In Figure 1, a plot of \(x_{n}\) is shown in Figure 1(a), a plot of \(y_{n}\) is shown in Figure 1(b), and an attractor of the system (24) is shown in Figure 1(c). In this case, \(\ln (\alpha+\beta\frac{\beta}{\gamma} )=2.69812\) and \(\frac{\beta}{\gamma(\alpha+\beta)\beta}=3.2419\). Hence, condition (4) of Theorem 3.2 is satisfied, i.e., \(\ln (\alpha+\beta\frac{\beta}{\gamma} )<\frac{\beta }{\gamma(\alpha+\beta)\beta}\). The unique positive equilibrium point of the system (24) is given by \((\frac{1\gamma }{\gamma}, \ln(\frac{\alpha\gamma\beta+\beta\gamma}{\gamma}) )=(0.234568, 2.69812)\).
Example 7.2
Let \(\alpha=11\), \(\beta=12\), \(\gamma=0.77\). Then the system (1) can be written as
with initial conditions \(x_{0}=0.3\), \(y_{0}=2\). In Figure 2, a plot of \(x_{n}\) is shown in Figure 2(a), a plot of \(y_{n}\) is shown in Figure 2(b), and an attractor of the system (25) is shown in Figure 2(c). In this case, \(\ln (\alpha+\beta\frac{\beta}{\gamma} )=2.00358\) and \(\frac{\beta}{\gamma(\alpha+\beta)\beta}=2.10158\). Hence, condition (4) of Theorem 3.2 is satisfied, i.e., \(\ln (\alpha+\beta\frac{\beta}{\gamma} )<\frac{\beta }{\gamma(\alpha+\beta)\beta}\). The unique positive equilibrium point of the system (25) is given by \((\frac{1\gamma}{\gamma}, \ln(\frac{\alpha\gamma\beta+\beta\gamma}{\gamma}) )=(0.298701, 2.00358)\).
Example 7.3
Let \(\alpha=2.5\), \(\beta=0.69\), \(\gamma=0.55\). Then the system (1) can be written as
with initial conditions \(x_{0}=0.8\), \(y_{0}=0.6\). In Figure 3, a plot of \(x_{n}\) is shown in Figure 3(a), a plot of \(y_{n}\) is shown in Figure 3(b), and a phase portrait of the system (26) is shown in Figure 3(c). In this case, \(\ln (\alpha+\beta\frac{\beta}{\gamma} )=0.660342\) and \(\frac{\beta}{\gamma(\alpha+\beta)\beta}=0.648192\). Hence, condition (4) of Theorem 3.2 does not hold, i.e., \(\ln (\alpha+\beta\frac{\beta}{\gamma} )>\frac {\beta}{\gamma(\alpha+\beta)\beta}\).
References
 1.
Kulenović, MRS, Ladas, G: Dynamics of Second Order Rational Difference Equations. Chapman & Hall/CRC, Boca Raton (2002)
 2.
Kocic, VL, Ladas, G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht (1993)
 3.
Allen, LJS, Hannigan, MK, Strauss, MJ: Mathematical analysis of a model for a plantherbivore system. Bull. Math. Biol. 55(4), 847864 (1993)
 4.
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)
 5.
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)
 6.
Din, Q: Dynamics of a discrete LotkaVolterra model. Adv. Differ. Equ. (2013). doi:10.1186/16871847201395
 7.
Din, Q, Donchev, T: Global character of a hostparasite model. Chaos Solitons Fractals 54, 17 (2013)
 8.
Din, Q, Elsayed, EM: Stability analysis of a discrete ecological model. Comput. Ecol. Softw. 4(2), 89103 (2014)
 9.
Din, Q, Ibrahim, TF, Khan, KA: Behavior of a competitive system of secondorder difference equations. Sci. World J. 2014, Article ID 283982 (2014)
 10.
Din, Q: Global stability of a population model. Chaos Solitons Fractals 59, 119128 (2014)
 11.
Din, Q: Stability analysis of a biological network. Netw. Biol. 4(3), 123129 (2014)
 12.
Din, Q, Khan, KA, Nosheen, A: Stability analysis of a system of exponential difference equations. Discrete Dyn. Nat. Soc. 2014, Article ID 375890 (2014)
 13.
Din, Q: Asymptotic behavior of an anticompetitive system of secondorder difference equations. J. Egypt. Math. Soc. (2014). doi:10.1016/j.joems.2014.08.008
 14.
Din, Q: Qualitative nature of a discrete predatorprey system. Contemp. Methods Math. Phys. Gravit. (Online) 1(1), 2742 (2015)
 15.
Grove, EA, Ladas, G: Periodicities in Nonlinear Difference Equations. Chapman & Hall/CRC, Boca Raton (2004)
 16.
Pituk, M: More on Poincaré’s and Perron’s theorems for difference equations. J. Differ. Equ. Appl. 8, 201216 (2002)
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.
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Din, Q. Global behavior of a plantherbivore model. Adv Differ Equ 2015, 119 (2015). https://doi.org/10.1186/s136620150458y
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MSC
 39A10
 40A05
Keywords
 plantherbivore system
 steadystates
 local stability
 global behavior
 rate of convergence