# Global attractivity of a discrete cooperative system incorporating harvesting

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## Abstract

## Keywords

global attractivity cooperation equilibrium iterative method## MSC

34D23 92B05 34D40## 1 Introduction

*x*and

*y*denote the densities of two populations at time

*t*. The parameters \(r_{1}\), \(r_{2}\), \(a_{1}\), \(a_{2}\), \(b_{1}\), \(b_{2}\), \(k_{1}\), \(k_{2}\),

*E*,

*q*are all positive constants. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, they obtained sufficient conditions which ensure the persistent and stability of the positive equilibrium, respectively.

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].

*x*and

*y*at

*k*-generation. Throughout this paper, we assume that the coefficients of the system (1.2) satisfies:

- (H
_{1}) -
\(r_{i}\), \(b_{i}\), \(a_{i}\),

*E*,*q*, \(i=1, 2\) are all positive constants, \(r_{1}>Eq\).

_{1}), system (1.2) admits a unique positive equilibrium \((x^{*},y^{*})\). Indeed, the positive equilibrium of system (1.2) satisfies

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

*In addition to*(H

_{1}),

*further assume that*

- (H
_{2}) -
\(0< r_{1}-qE\leq1\), \(r_{2}\leq1\),

*hold*,

*then system*(1.2)

*admits a unique positive equilibrium*\((x^{*},y^{*})\)

*which is globally attractive*.

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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 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]

*Assume that the sequence*\(\{u(k) \}\)

*satisfies*

*where*

*α*

*and*

*β*

*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*\(u(k)\leq\frac{1}{\beta}\), \(k=2,3,\ldots\) .

### Lemma 2.3

[25]

*Suppose that the 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 following difference equations*:

*respectively*,

*and*\(x(0)\leq u(0)\) (\(x(0)\geq u(0)\)),

*then*

### 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

_{1}) and (H

_{2}) in Theorem 1.1. From Theorem 1.1, the unique positive equilibrium \(E_{+}(x_{1}^{*},x_{2}^{*})\) is globally attractive. Numeric simulations also support our finding (see Figures 1 and 2).

### Example 3.2

_{2}) in Theorem 1.1, and the stability property of this positive equilibrium could not be judged by Theorem 1.1. However, numeric simulations (see Figures 3 and 4) show that in this case, the positive equilibrium still is globally attractive.

## 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, 3, 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.

## Notes

### 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 (XRC-1438).

## References

- 1.Wei, FY, Li, CY: Permanence and globally asymptotic stability of cooperative system incorporating harvesting. Adv. Pure Math.
**3**, 627-632 (2013) CrossRefGoogle Scholar - 2.Xie, XD, Chen, FD, Xue, YL: Note on the stability property of a cooperative system incorporating harvesting. Discrete Dyn. Nat. Soc.
**2014**, Article ID 327823 (2014) MathSciNetGoogle Scholar - 3.Yang, K, Xie, XD, Chen, FD: Global stability of a discrete mutualism model. Abstr. Appl. Anal.
**2014**, Article ID 709124 (2014) MathSciNetGoogle Scholar - 4.Chen, GY, Teng, ZD: On the stability in a discrete two-species competition system. J. Appl. Math. Comput.
**38**, 25-36 (2012) MathSciNetCrossRefMATHGoogle Scholar - 5.Li, YK, Xu, GT: Positive periodic solutions for an integrodifferential model of mutualism. Appl. Math. Lett.
**14**, 525-530 (2001) MathSciNetCrossRefMATHGoogle Scholar - 6.Chen, LJ, Chen, LJ, Li, Z: Permanence of a delayed discrete mutualism model with feedback controls. Math. Comput. Model.
**50**, 1083-1089 (2009) MathSciNetCrossRefMATHGoogle Scholar - 7.Chen, LJ, Xie, XD: Permanence of an
*n*-species cooperation system with continuous time delays and feedback controls. Nonlinear Anal., Real World Appl.**12**, 34-38 (2001) MathSciNetCrossRefMATHGoogle Scholar - 8.Li, YK, Zhang, T: Permanence of a discrete
*N*-species cooperation system with time-varying delays and feedback controls. Math. Comput. Model.**53**, 1320-1330 (2011) MathSciNetCrossRefMATHGoogle Scholar - 9.Chen, LJ, Xie, XD: Feedback control variables have no influence on the permanence of a discrete
*N*-species cooperation system. Discrete Dyn. Nat. Soc.**2009**, Article ID 306425 (2009) MathSciNetMATHGoogle Scholar - 10.Chen, FD, Liao, XY, Huang, ZK: The dynamic behavior of
*N*-species cooperation system with continuous time delays and feedback controls. Appl. Math. Comput.**181**, 803-815 (2006) MathSciNetMATHGoogle Scholar - 11.Chen, FD: Permanence of a discrete
*N*-species cooperation system with time delays and feedback controls. Appl. Math. Comput.**186**, 23-29 (2007) MathSciNetMATHGoogle Scholar - 12.Chen, FD: Permanence for the discrete mutualism model with time delays. Math. Comput. Model.
**47**, 431-435 (2008) MathSciNetCrossRefMATHGoogle Scholar - 13.Chen, FD, Yang, JH, Chen, LJ, Xie, XD: On a mutualism model with feedback controls. Appl. Math. Comput.
**214**, 581-587 (2009) MathSciNetMATHGoogle Scholar - 14.Chen, FD, Xie, XD, Chen, XF: Dynamic behaviors of a stage-structured cooperation model. Commun. Math. Biol. Neurosci.
**2015**, Article ID 4 (2015) Google Scholar - 15.Chen, FD, Xie, XX: Study on the Dynamic Behaviors of Cooperative System. Science Press, Beijing (2014) Google Scholar
- 16.Chen, FD, Pu, LQ, Yang, LY: Positive periodic solution of a discrete obligate Lotka-Volterra model. Commun. Math. Biol. Neurosci.
**2015**, Article ID 14 (2015) Google Scholar - 17.Li, YK: Positive periodic solutions of a discrete mutualism model with time delays. Int. J. Math. Math. Sci.
**2005**(4), 499-506 (2005) MathSciNetCrossRefMATHGoogle Scholar - 18.Liu, ZJ, Tan, RH, Chen, YP, Chen, LS: On the stable periodic solutions of a delayed two-species model of facultative mutualism. Appl. Math. Comput.
**196**, 105-117 (2008) MathSciNetMATHGoogle Scholar - 19.Li, XP, Yang, WS: Permanence of a discrete model of mutualism with infinite deviating arguments. Discrete Dyn. Nat. Soc.
**2010**, Article ID 931798 (2010) MathSciNetMATHGoogle Scholar - 20.Li, Z: Permanence for the discrete mutualism model with delays. J. Math. Study
**43**(1), 51-54 (2010) MathSciNetGoogle Scholar - 21.Muhammadhaji, A, Teng, ZD: Global attractivity of a periodic delayed
*n*-species model of facultative mutualism. Discrete Dyn. Nat. Soc.**2013**, Article ID 580185 (2013) MathSciNetCrossRefMATHGoogle Scholar - 22.Xu, CJ, Wu, YS: Permanence in a discrete mutualism model with infinite deviating arguments and feedback controls. Discrete Dyn. Nat. Soc.
**2013**, Article ID 397382 (2013) MathSciNetGoogle Scholar - 23.Xie, XD, Chen, FD, Yang, K, Xue, Y: Global attractivity of an integrodifferential model of mutualism. Abstr. Appl. Anal.
**2014**, Article ID 928726 (2014) MathSciNetGoogle Scholar - 24.Xie, XD, Miao, ZS, Xue, YL: Positive periodic solution of a discrete Lotka-Volterra commensal symbiosis model. Commun. Math. Biol. Neurosci.
**2015**, Article ID 2 (2015) Google Scholar - 25.Wang, L, Wang, MQ: Ordinary Difference Equation. Xinjing University Press, Urmuqi (1989) Google Scholar
- 26.Yang, WS, Li, XP: Permanence of a discrete nonlinear
*N*-species cooperation system with time delays and feedback controls. Appl. Math. Comput.**218**, 3581-3586 (2011) MathSciNetMATHGoogle Scholar - 27.Xue, YL, Xie, XD, Chen, FD, Han, RY: Almost periodic solution of a discrete commensalism system. Discrete Dyn. Nat. Soc.
**2015**, Article ID 295483 (2015) MathSciNetGoogle Scholar - 28.Miao, ZS, Xie, XD, Pu, LQ: Dynamic behaviors of a periodic Lotka-Volterra commensal symbiosis model with impulsive. Commun. Math. Biol. Neurosci.
**2015**, Article ID 3 (2015) Google Scholar - 29.Wu, RX, Li, L, Zhou, XY: A commensal symbiosis model with Holling type functional response. J. Math. Comput. Sci.
**16**(3), 364-371 (2016) Google Scholar - 30.Yang, K, Miao, ZS, Chen, FD, Xie, XD: Influence of single feedback control variable on an autonomous Holling-II type cooperative system. J. Math. Anal. Appl.
**435**(1), 874-888 (2016) MathSciNetCrossRefMATHGoogle Scholar - 31.Beddington, JR, May, RM: Harvesting natural populations in a randomly fluctuating environment. Science
**197**, 463-465 (1997) CrossRefGoogle Scholar - 32.Liu, M, Bai, C: Analysis of a stochastic tri-trophic food-chain model with harvesting. J. Math. Biol.
**73**, 597-625 (2016) MathSciNetCrossRefMATHGoogle Scholar - 33.Liu, M, Bai, C: Optimal harvesting of a stochastic mutualism model with Levy jumps. Appl. Math. Comput.
**276**, 301-309 (2016) MathSciNetGoogle Scholar

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