Global attractivity of a discrete cooperative system incorporating harvesting
 Fengde Chen^{1},
 Huiling Wu^{2, 3} and
 Xiangdong Xie^{4}Email author
https://doi.org/10.1186/s136620160996y
© Chen et al. 2016
Received: 14 August 2016
Accepted: 13 October 2016
Published: 22 October 2016
Abstract
Keywords
MSC
1 Introduction
Recently, Xie et al. [2] revisited the dynamic behaviors of the system (1.1). By using the iterative method, they showed that the condition which ensures the existence of a unique positive equilibrium is enough to ensure the globally attractive of the positive equilibrium. Their result significantly improves the corresponding results of Wei and Li [1].
 (H_{1}):

\(r_{i}\), \(b_{i}\), \(a_{i}\), E, q, \(i=1, 2\) are all positive constants, \(r_{1}>Eq\).
The aim of this paper is, by further developing the analysis technique of Xie et al. [2], Yang et al. [3], and Chen and Teng [4], 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
 (H_{2}):

\(0< r_{1}qE\leq1\), \(r_{2}\leq1\),
The rest of the paper is arranged as follows. With the help of several useful lemmas, we will prove Theorem 1.1 in Section 2. Two examples together with their numeric simulations are presented in Section 3 to show the feasibility of our results. We end this paper by a brief discussion. For more work about cooperative systems, we can refer to [1–30] and the references therein.
2 Global attractivity
We will give a strict proof of Theorem 1.1 in this section. To achieve this objective, we introduce several useful lemmas.
Lemma 2.1
[4]
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 2.2
[4]
 (i)
If \(\alpha<2\), then \(\lim_{k\rightarrow+\infty}{u(k)}=\frac{\alpha}{\beta}\).
 (ii)
If \(\alpha\leq1\), then \(u(k)\leq\frac{1}{\beta}\), \(k=2,3,\ldots\) .
Lemma 2.3
[25]
Proof of Theorem 1.1
Now, we will prove \(\{M_{k}^{x} \}\), \(\{M_{k}^{y} \}\) is monotonically decreasing, \(\{m_{k}^{x} \}\), \(\{m_{k}^{y} \}\) is monotonically increasing by means of inductive method.
3 Examples
In this section, we shall give two examples to illustrate the feasibility of the main result.
Example 3.1
Example 3.2
4 Discussion
In [2], Xie et al. studied the stability property of the system (1.1), their result shows that once the system (1.1) admits a unique positive equilibrium, it is globally attractive. In this paper, we try to consider the discrete type of system (1.1), we first establish the corresponding system (1.2), then, by developing the analysis technique of [2–4], we also obtain a set of sufficient conditions which ensure the global attractivity of the positive equilibrium. Our result shows that the intrinsic growth rate plays an important role in the stability property of the system.
It brings to our attention that conditions for the continuous system are very simple (one only requires \(r_{1}>qE\)), while conditions for the discrete one is very strong, since one requires \(r_{1}qE\leq1\) and \(r_{2}\leq1\). This motivated us to study the case \(r_{i}>1\), numeric simulation (Example 3.2) shows that in this case, the system still possible admits a unique globally attractive positive equilibrium, and we conjecture that Theorem 1.1 still holds under the condition \(r_{1}Eq<2\), \(r_{2}<2\); we leave this for future study.
At the end of the paper, we would like to point out that one of the reviewers of this paper said ‘Population models with stochastic noises may also be important and interesting. In fact, many authors have studied stochastic population models with stochastic noises, for example, Beddington and May [31], Liu and Bai [32, 33]. I suggest the authors take stochastic noises into account in their future study.’ We do agree with the opinion of the reviewers, and we hope we could do some relevant work in the future.
Declarations
Acknowledgements
The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. The research was supported by the Natural Science Foundation of Fujian Province (2015J01012, 2015J01019, 2015J05006) and the Scientific Research Foundation of Fuzhou University (XRC1438).
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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