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Stability analysis of a discrete competitive system with nonlinear interinhibition terms
- Jinhuang Chen^{1} and
- Xiangdong Xie^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1362-4
© The Author(s) 2017
- Received: 19 June 2017
- Accepted: 13 September 2017
- Published: 20 September 2017
Abstract
We obtain some conditions for the local stability of the equilibria. Using the iterative method and the comparison principle of a difference equation, we also obtain a set of sufficient conditions that ensure the global stability of the interior equilibrium. Numeric simulations show the feasibility of the main results. Our results supplement and complement some known results.
Keywords
- local stability
- competitive
- global stability
- iterative method
MSC
- 34D23
- 92B05
- 34D40
1 Introduction
Concerned with the persistent property and stability property of the system, the authors obtained the following results (Theorems 2.4 and 3.3 in [2]).
Theorem A
Theorem B
As a direct corollary of Theorems A and B, for the autonomous case of system (1.3) (i.e., all the coefficients of the system are positive constants), we have the following result.
Theorem C
Note that conditions (1.6) are sufficient conditions. We propose an interesting issue: whether the conditions are good enough or the conditions still have room to improve? To give some hint on this problem, let us consider the following example.
Example 1.1
Theorem D
- (i)$$\begin{aligned} a_{2}-b_{2}+r_{2}\neq0; \end{aligned}$$(1.10)
- (ii)Then system (1.8) admits a unique positive equilibrium on the rectangle \((0, \frac{r_{1}}{a_{1}})\times(0, \frac {r_{2}}{a_{2}})\), which is globally stable.$$\begin{aligned} a_{2}-b_{2}+r_{2}=0, \qquad a_{1}r_{2}-a_{2}r_{1}>0. \end{aligned}$$(1.11)
Lemma 1.1
Assume that \((H_{1})\) holds. Then system (1.1) admits a unique positive equilibrium \((x^{*}_{1},x^{*}_{2})\) on the rectangle \((0, \frac{r_{1}}{a_{1}})\times(0, \frac{r_{2}}{a_{2}})\).
Proof
Since we focus on the positive equilibrium of system (1.1), we only need to consider the case \(x_{1}>0, x_{2}>0\). To ensure the first equality in (1.12), \(x_{1}\) should lie in the interval \((0, \frac {r_{1}}{a_{1}})\). Similarly, to ensure the second equality of (1.12), \(x_{2}\) should lie in the interval \((0, \frac {r_{2}}{a_{2}})\). We will further investigate the positive equilibrium of system (1.1) on the rectangle \((0, \frac{r_{1}}{a_{1}})\times (0, \frac{r_{2}}{a_{2}})\).
Therefore, from the continuity of the function \(F(x_{1})\), \(F(x_{1})=0\) has at least one positive solution on the interval \((0, \frac {r_{1}}{a_{1}})\). We now prove that the equation \(F(x_{1})=0\) has at most one positive solution on the interval \((0, \frac {r_{1}}{a_{1}})\). We discuss this in three cases.
Case 1. If \(A_{1}>0\), then \(F(+\infty)=F(-\infty)=+\infty \). Since \(F(0)<0\), it follows that \(F(x_{1})=0\) has at least one solution on the intervals \((-\infty,0)\) and \((0,+\infty)\), respectively. Therefore \(F(x_{1})=0\) has only one solution on the interval \((0, \frac{r_{1}}{a_{1}})\);
Case 2. If \(A_{1}=0\), then, since \(F(x_{1})\) is a linear function of \(x_{1}\), \(F( \frac{r_{1}}{a_{1}})>0\), and \(F(0)<0\), it follows that \(F(x_{1})=0\) has only one solution on the interval \((0, \frac{r_{1}}{a_{1}})\);
Case 3. If \(A_{1}<0\), then \(F(+\infty)=-\infty\), and since \(F( \frac{r_{1}}{a_{1}})>0\) and \(F(0)<0\), it follows that \(F(x_{1})\) has at least one solution on the intervals \((0, \frac{r_{1}}{a_{1}})\) and \(( \frac {r_{1}}{a_{1}},+\infty)\). So \(F(x_{1})=0\) has only one solution on the interval \((0, \frac{r_{1}}{a_{1}})\).
The above analysis shows that \(F(x_{1})=0\) has only one solution on the interval \((0, \frac{r_{1}}{a_{1}})\). We denote it as \(x^{*}_{1}\). Similarly, we can prove that there exists \(x^{*}_{2} \) in the interval \((0, \frac{r_{2}}{a_{2}})\) that satisfies \(B_{1}x_{2}^{2}+B_{2}x_{2}+B_{3}=0\). Then system (1.1) admits a unique positive equilibrium \((x^{*}_{1},x^{*}_{2})\) on the rectangle \((0, \frac{r_{1}}{a_{1}})\times(0, \frac {r_{2}}{a_{2}})\). This ends the proof of Lemma 1.1. □
The rest of the paper is arranged as follows. With the help of several useful lemmas, we investigate the local stability in Section 2 and prove the global stability result (Theorem 3.1) in Section 3. Four examples, together with their numeric simulations, are presented in Section 4 to show the feasibility of our results. We end this paper by a brief discussion. For more work about competitive systems, we can refer to [2, 4–23] and the references cited.
2 Local stability
We give a strict proof of the local stability in this section. From the biological background of system (1.1), we assume that initial values \(x_{1}(0)>0\) and \(x_{2}(0)>0\) in system (1.1). It is clear that any solution of system (1.1) is defined on \(N=\{0, 1, 2, \ldots\}\) and remains positive for all \(n\geq0\). Now let us state several useful lemmas.
Lemma 2.1
[1]
- 1.
\(\vert \lambda_{1} \vert <1\) and \(\vert \lambda _{2} \vert <1\) if and only if \(F(-1)>0\) and \(C<1\);
- 2.
\(\vert \lambda_{1} \vert >1\) and \(\vert \lambda _{2} \vert >1\) if and only if \(F(-1)>0\) and \(C>1\);
- 3.
\(\vert \lambda_{1} \vert <1\) and \(\vert \lambda _{2} \vert >1\) if and only if \(F(-1)<0\);
- 4.
\(\lambda_{1}=-1\) and \(\vert \lambda_{2} \vert \neq1\) if and only if \(F(-1)=0\) and \(B\neq0,2\);
- 5.
\(\lambda_{1}\) and \(\lambda_{2}\) are a pair of conjugate complex roots and \(\vert \lambda_{1} \vert = \vert \lambda_{2} \vert =1\) if and only if \(B^{2}-4C<0\) and \(C=1\).
- 1.
If \(\vert \lambda_{1} \vert <1\) and \(\vert \lambda _{2} \vert <1\), then \(J(x,y)\) is called a sink and is locally asymptotic stable;
- 2.
If \(\vert \lambda_{1} \vert >1\) and \(\vert \lambda _{2} \vert >1\), then \(J(x,y)\) is called a source and is unstable;
- 3.
If \(\vert \lambda_{1} \vert >1\) and \(\vert \lambda _{2} \vert <1\) (or \(\vert \lambda _{1} \vert <1\) and \(\vert \lambda_{2} \vert >1\)), then \(J(x,y)\) is called a saddle and is unstable;
- 4.
If \(\lambda_{1}=1\) or \(\vert \lambda_{2} \vert =1\), then \(J(x,y)\) is called nonhyperbolic.
We first discuss the existence of the equilibria of model (1.1). Obviously, \(E_{1}(0,0)\), \(E_{2}( \frac{r_{1}}{a_{1}},0)\), and \(E_{3}(0, \frac {r_{2}}{a_{2}})\) are three equilibria of model (1.1). If \((H_{1})\) holds, system (1.1) admits a unique positive equilibrium \(E_{4}(x^{*}_{1},x^{*}_{2})\).
Now we are in the position of discussing the local stability of the equilibria of model (1.1).
- 1.
If \(0< r_{1}<2\), then \(E_{2}( \frac{r_{1}}{a_{1}},0)\) is a saddle.
- 2.
If \(r_{1}=2\), then \(E_{2}( \frac{r_{1}}{a_{1}},0)\) is nonhyperbolic.
- 3.
If \(r_{1}>2\), then \(E_{2}( \frac{r_{1}}{a_{1}},0)\) is a source.
- 1.
If \(0< r_{2}<2\), then \(E_{3}(0, \frac{r_{2}}{a_{2}})\) is a saddle.
- 2.
If \(r_{2}=2\), then \(E_{3}(0, \frac{r_{2}}{a_{2}})\) is nonhyperbolic.
- 3.
If \(r_{2}>2\), then \(E_{3}(0, \frac{r_{2}}{a_{2}})\) is a source.
- 1.
If \(b_{1}b_{2}< K_{2}\), then \(E_{4}(x^{*}_{1},x^{*}_{2})\) is a source.
- 2.
If \(K_{2}< b_{1}b_{2}< K_{1}\), then \(E_{4}(x^{*}_{1},x^{*}_{2})\) is a saddle.
- 3.
If \(K_{2}=b_{1}b_{2}< K_{1}\), then \(E_{4}(x^{*}_{1},x^{*}_{2})\) is nonhyperbolic.
3 Global stability
Previously, we have discussed the local stability of the equilibria of system (1.1). In this section, we give a set of sufficient conditions that ensure the global attractivity of the unique positive equilibrium on the rectangle \((0, \frac{r_{1}}{a_{1}})\times(0, \frac {r_{2}}{a_{2}})\).
Theorem 3.1
Now let us state several lemmas, which will be useful in the proof of Theorem 3.1.
Lemma 3.1
[1]
Let \(f(u)=u\exp(\alpha -\beta u)\), where α and β are positive constants. Then \(f(u)\) is nondecreasing for \(u\in(0, \frac{1}{\beta}]\).
Lemma 3.2
[1]
- (i)
If \(\alpha<2\), then \(\lim_{n\rightarrow+\infty} u(n)= \frac{\alpha}{\beta}\).
- (ii)
If \(\alpha\leq1\), then \(u(n)\leq\frac{1}{\beta}, n=2,3,\ldots\) .
Lemma 3.3
[15]
Proof of Theorem 3.1
4 Numeric simulations
In this section, we give four examples to illustrate the feasibility of the main results.
Example 4.1
Example 4.2
Example 4.3
Example 4.4
Examples 4.1-4.3 show that the coefficients satisfy the conditions of Theorem 3.1. In Example 4.1, we can verify that \(A_{1}>0\), which represents case 1 in Lemma 1.1, whereas Examples 4.2 and 4.3 represent cases 2 and 3 in Lemma 1.1, respectively (\(A_{1}=0\), \(A_{1}<0\)). As for Example 4.4, though the coefficients of system (4.4) do not satisfy \((H_{2})\) in Theorem 3.1, Figure 8 also implies that the positive equilibrium is still globally attractive.
5 Disscussion
Chen and Teng [1] studied the local and global stability of positive equilibrium of system (1.2). In this paper, we studied the dynamic behavior of system (1.1) adding the nonlinear interinhibition terms into the model. When \(c_{1}=c_{2}=0\), system (1.1) reduces to (1.2), and conditions reduce to \(r_{1}<1, r_{2}<1, a_{1}r_{2}>b_{2}r_{1}, a_{2}r_{1}>b_{1}r_{2}\), and those conditions are equivalent to the conditions \(r_{1}<1, r_{2}<1, 1-\mu_{1}K_{1}>0, 1-\mu _{2}K_{2}>0\) in [1].
From \((H_{1})\) we obtain \(r_{1}> \frac {b_{1}r_{2}}{a_{2}+r_{2}}\) and \(r_{2}> \frac{b_{2}r_{1}}{a_{1}+r_{1}}\). Our results show that the intrinsic growth rate plays an important role in the stability property of the system. We know that when the intrinsic growth rates of the two species are fixed, if the rates of interspecific competitive coefficients are small enough, then condition \((H_{1})\) always holds, and consequently, two species can coexist in a stable state. This means that smaller interspecific competitive coefficients have positive effect to the stability property of the system.
By developing the analysis technique of [3, 24] we also obtain a set of sufficient conditions that ensure the global attractivity of the positive equilibrium. We relax the conditions in [3] and [2]. Numeric simulations also support our findings. However, Example 4.4 does not satisfy all the conditions of Theorem 3.1, and the system still admits a unique globally stable positive equilibrium. We conjecture that the conditions \(r_{i}\leq1\) in Theorem 3.1 can be relaxed to \(0< r_{i}<2\ (i=1,2)\), that is, the conditions of Theorem 3.1 still have room to improve. However, at present, we have difficulty in proving this conjecture. With the change of \(r_{i}\), we found the bifurcations in the above figures, wherein a \(2^{k}\)-cycle loses stability. We leave these two problems for future study.
Declarations
Acknowledgements
The research was supported by the National Natural Science Foundation of China under Grant (11601085) and the Natural Science Foundation of Fujian Province (2015J01019).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of 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
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