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Global attractivity of a discrete competition model
Advances in Difference Equations volumeÂ 2016, ArticleÂ number:Â 273 (2016)
Abstract
We study a discrete competition model of the form
where \(x_{i}(k)\) (\(i=1,2\)) are the population density of the ith species at kgeneration. \(r_{i}\), \(K_{i}\), \(i=1, 2\), are all positive constants such that \(K_{i}>\alpha_{i}\), \(i=1, 2\). By using the iterative method and the comparison principle of difference equations we obtain sufficient conditions that ensure the global attractivity of the interior equilibrium of the system. Our result supplements and complements the main result of Yang et al. (Abstr. Appl. Anal. 2014:709124, 2014).
1 Introduction
Yang et al. [1] studied the dynamic behavior of the following autonomous discrete cooperative system (1.1):
where \(x_{i}(k)\) (\(i=1,2\)) are the population density of the ith species at kgeneration. They showed that if
 (\(H_{1}\)):

\(r_{i}\), \(K_{i}\), \(\alpha_{i}\) (\(i=1, 2\)) are all positive constants, and \(\alpha_{i}>K_{i}\) (\(i=1,2\)),
and
 (\(H_{2}\)):

\(r_{i}\alpha_{i}\leq1\) (\(i=1,2\)),
then system (1.1) admits a unique positive equilibrium \((x_{1}^{*},x_{2}^{*})\), which is globally asymptotically stable.
It brought to our attention that the main results of [1] deeply depend on the assumption \(\alpha_{i}>K_{i}\), \(i=1,2\). Now, an interesting issue is proposed: Is it possible for us to investigate the stability property of system (1.1) under the assumption \(K_{i}>\alpha_{i}\), \(i=1, 2\)?
Note that in this case, the first equation of system (1.1) can be rewritten as
Similarly to the above analysis, the second equation of system (1.1) can be rewritten as
From (1.2) and (1.3) we can easily see that both species have negative effect to the other species, that is, under the assumption \(K_{i}>\alpha _{i}\), \(i=1, 2\), the relationship between two species is competition. Also, we mention here that under the assumption \(K_{i}>\alpha_{i}\), \(i=1, 2\), system (1.1) admits a unique positive equilibrium \((x_{1}^{*},x_{2}^{*})\). Indeed, the positive equilibrium of system (1.1) satisfies
which is equivalent to
where
From \(A_{1}>0\), \(A_{3}<0\), \(B_{1}>0\), \(B_{3}<0\) we can easily see that system (1.5) admits a unique positive solution
Since the relationship between two species is competition, and the system admits a unique positive equilibrium, it is natural to seek some suitable conditions that ensure the global attractivity of the positive equilibrium. Since the existence of stable positive equilibrium represents the stable coexistence of the two species, establishing some similar result as that of [1] becomes a challenging problem.
The aim of this paper is to obtain a set of sufficient conditions to ensure the global attractivity of the interior equilibrium of system (1.1). More precisely, we will prove the following result.
Theorem 1.1
Assume that
 (\(A_{1}\)):

\(r_{i}\), \(K_{i}\), \(\alpha_{i}\) (\(i=1, 2\)) are all positive constants, and \(K_{i}>\alpha_{i}\) (\(i=1,2\)),
and
 (\(A_{2}\)):

\(r_{i}K_{i}\leq1\) (\(i=1,2\)).
Then system (1.1) admits a unique positive equilibrium \((x_{1}^{*},x_{2}^{*})\), which is globally attractive.
The rest of the paper is arranged as follows. With the help of several useful lemmas, we will prove TheoremÂ 1.1 in SectionÂ 2. An example, together with its numeric simulations, is presented in SectionÂ 3 to show the feasibility of our results. We end this paper by a brief discussion. For more work on cooperative or competitive systems, we refer to [1â€“33] and the references therein.
2 Global attractivity
Before we prove TheoremÂ 1.1, we need to introduce several useful lemmas.
Lemma 2.1
[12]
Let \(f(u)=u\exp(\alpha\beta u)\), where Î± and Î² are positive constants. Then \(f(u)\) is nondecreasing for \(u\in(0,\frac{1}{\beta}]\).
Lemma 2.2
[12]
Assume that a sequence \(\{u(k) \}\) satisfies
where Î± and Î² are positive constants, and \(u(0)>0\). Then

(i)
if \(\alpha<2\), then \(\lim_{k\rightarrow+\infty}{u(k)}=\frac{\alpha}{\beta}\);

(ii)
if \(\alpha\leq1\), then \(u(k)\leq\frac{1}{\beta}\), \(k=2,3,\ldots\)â€‰.
Lemma 2.3
[24]
Suppose that functions \(f,g:Z_{+}\times[0,\infty)\rightarrow[0,\infty)\) satisfy \(f(k,x)\leq g(k,x)\) \((f(k,x)\geq g(k,x))\) for \(k\in Z_{+}\) and \(x\in[0,\infty)\) and \(g(k,x)\) is nondecreasing with respect to x. If \(\{x(k) \}\) and \(\{u(k) \}\) are the nonnegative solutions of the difference equations
respectively, and \(x(0)\leq u(0)\) \((x(0)\geq u(0))\). Then
Proof of TheoremÂ 1.1
Let \((x_{1}(k),x_{2}(k))\) be an arbitrary solution of system (1.1) with \(x_{1}(0)>0\) and \(x_{2}(0)>0\). Denote
We claim that \(U_{1}=V_{1}={\overline{x}_{1}}\) and \(U_{2}=V_{2}={\overline{x}_{2}}\).
From (1.2) we obtain
Consider the auxiliary equation
Because of \(0< r_{1}K_{1}\leq1\), according to (ii) of LemmaÂ 2.2, we can obtain \(u(k)\leq\frac{1}{r_{1}}\) for all \(k\geq2\), where \(u(k)\) is an arbitrary positive solution of (2.2) with initial value \(u(0)>0\). By LemmaÂ 2.1, \(f(u)=u\exp(r_{1} K_{1}r_{1}u)\) is nondecreasing for \(u\in(0,\frac{1}{r_{1}}]\). According to LemmaÂ 2.3, we obtain \(x_{1}(k)\leq u(k)\) for all \(k\geq2\), where \(u(k)\) is the solution of (2.2) with initial value \(u(2)= x_{1}(2)\). According to (i) of LemmaÂ 2.2, we obtain
From (1.3) we obtain
Similarly to the above analysis, we have
Then, for a sufficiently small constant \(\varepsilon>0\), there is an integer \(k_{1}>2\) such that
According to (1.2), we obtain
Consider the auxiliary equation
Since \(0< r_{1}\alpha_{1}\leq r_{1}K_{1}\leq1\), according to (ii) of LemmaÂ 2.2, we obtain \(u(k)\leq\frac{1}{r_{1}}\) for all \(k\geq2\), where \(u(k)\) is an arbitrary positive solution of (2.8) with initial value \(u(0)>0\). By LemmaÂ 2.1, \(f(u)=u\exp(r_{1}\alpha_{1}r_{1}u)\) is nondecreasing for \(u\in(0,\frac{1}{r_{1}}]\). According to LemmaÂ 2.3, we obtain \(x_{1}(k)\geq u(k)\) for all \(k\geq2\), where \(u(k)\) is the solution of (2.8) with initial value \(u(k_{1})= x_{1}(k_{1})\). According to (i) of LemmaÂ 2.2, we have
From (1.3) we obtain
Similarly to the analysis of (2.7)(2.9), we have
Then, for a sufficiently small constant \(\varepsilon>0\) (without loss of generality, we may assume that \(\varepsilon< \frac{1}{2}\{\alpha_{1}, \alpha_{2}\}\)), there exists an integer \(k_{2}>k_{1}\) such that
The second inequality in (2.12), combined with (1.2), leads to
Noting that
similarly to the analysis of (2.1)(2.3), we have
Obviously,
For \(k>k_{2}\), the second inequality in (2.12), combined with (1.3), leads to
By applying LemmaÂ 2.3 to the above inequality we obtain
Obviously, we have
Then, for a sufficiently small constant \(\varepsilon>0\), there is an integer \(k_{3}>k_{2}\) such that
The second inequality in (2.19), combined with (1.2), leads to
Noting that
from this we finally obtain
From the monotonicity of the function \(g(x)=\frac{x}{1+x}\) and (2.18) we have
The first inequality in (2.19), combined with (1.3), leads to
From this inequality we obtain
From the monotonicity of the function \(g(x)=\frac{x}{1+x}\) and (2.15) we have
Then, for a sufficiently small constant \(\varepsilon>0\), there is an integer \(k_{4}>k_{3}\) such that
Continuing the above steps, we get four sequences \(\{M_{k}^{x_{1}} \}\), \(\{M_{k}^{x_{2}} \}\), \(\{m_{k}^{x_{1}} \}\), and \(\{m_{k}^{x_{2}} \}\) such that
Clearly, we have
Now, we will prove by induction that \(\{M_{k}^{x_{i}} \}\) (\(i=1,2\)) is monotonically decreasing and \(\{m_{k}^{x_{i}} \}\) (\(i=1,2\)) is monotonically increasing.
First of all, it is clear that \(M_{2}^{x_{i}}< M_{1}^{x_{i}}\) and \(m_{2}^{x_{i}}> m_{1}^{x_{i}}\) (\(i=1, 2\)). For \(i\geq2\), we assume that \(M_{i}^{x_{1}}< M_{i1}^{x_{1}}\) and \(m_{i}^{x_{1}}>m_{i1}^{x_{1}}\). Then, since the function \(g(x)=\frac {x}{1+x}\) is increasing, we have
and so,
Inequalities (2.30)(2.33) show that \(\{M_{k}^{x_{1}} \}\) and \(\{M_{k}^{x_{2}} \}\) are decreasing and \(\{m_{k}^{x_{1}} \}\) and \(\{m_{k}^{x_{2}} \}\) are increasing. Consequently, \(\lim_{k\rightarrow+\infty} \{M_{k}^{x_{i}} \}\) and \(\lim_{k\rightarrow+\infty} \{m_{k}^{x_{i}} \}\) (\(i=1,2\)) both exist. Let
From (2.27) and (2.28) we have
Equalities (2.35) and (2.36) are equivalent to
Equalities (2.37) and (2.38) show that \((\overline{X}_{1}, \underline {X}_{2})\) and \((\underline{X}_{1}, \overline{X}_{2})\) both are solutions of system (1.4). However, under the assumption of TheoremÂ 1.1, system (1.4) has a unique positive solution \((x_{1}^{*},x_{2}^{*})\). Therefore,
that is, \(E_{+}(x_{1}^{*},x_{2}^{*})\) is globally attractive. This ends the proof of TheoremÂ 1.1.â€ƒâ–¡
3 Example
In this section, we give an example to illustrate the feasibility of the main result.
Example
Considering the following competition system:
For system (1.1), we have \(r_{1}=2\), \(r_{2}=3\), \(K_{1}=0.3\), \(K_{2}=0.2\), \(\alpha _{1}=0.1\), \(\alpha_{2}=0.1\). By calculation we have that the positive equilibrium \(E_{+}(x_{1}^{*},x_{2}^{*})\approx(0.27,0.18)\), \(r_{1}K_{1}=0.6<1\), \(r_{2}K_{2}=0.6<1\), \(K_{i}>\alpha_{i}\) (\(i=1,2\)), and the coefficients of system (3.1) satisfy (\(A_{1}\)) and (\(A_{2}\)) in TheoremÂ 1.1. By TheoremÂ 1.1 the unique positive equilibrium \(E_{+}(x_{1}^{*},x_{2}^{*})\) is globally attractive. Numeric simulations also support our finding (see FiguresÂ 1 and 2).
4 Discussion
Yang et al. [1] proposed system (1.1) under the assumption \(\alpha_{i}>K_{i}\), \(i=1, 2\), and showed that if \(r_{i}\alpha _{i}\leq1\), then the mutualism model admits a unique globally asymptotically stable positive equilibrium.
In this paper, we try to deal with the case \(K_{i}>\alpha_{i}\). We first show that under this assumption, the relationship between two species is competition, and then we show that if \(r_{i}K_{i}\leq1\), then two species can coexist in a stable state.
We mention here that a more suitable model should consider the past state of the species, and this leads to the competition system with delay; indeed, delay is one of the most important factors to influence the dynamic behavior of the competition system ([31â€“37]). It seems interesting to incorporate the time delay to system (1.1) and investigate the dynamic behavior of the system. We leave this for future study.
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Acknowledgements
The author is grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. This work is supported by National Social Science Foundation of China (16BKS132), Humanities and Social Science Research Project of Ministry of Education Fund (15YJA710002), and the Natural Science Foundation of Fujian Province (2015J01283).
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Chen, B. Global attractivity of a discrete competition model. Adv Differ Equ 2016, 273 (2016). https://doi.org/10.1186/s1366201610006
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DOI: https://doi.org/10.1186/s1366201610006