- Research
- Open Access
Permanence, partial survival, extinction, and global attractivity of a nonautonomous harvesting Lotka–Volterra commensalism model incorporating partial closure for the populations
- Yu Liu^{1},
- Xiangdong Xie^{2}Email author and
- Qifa Lin^{2}
https://doi.org/10.1186/s13662-018-1662-3
© The Author(s) 2018
- Received: 9 February 2018
- Accepted: 18 May 2018
- Published: 19 June 2018
Abstract
We propose and study a nonautonomous harvesting Lotka–Volterra commensalism model incorporating partial closure for the populations. By using the differential inequality theory we obtain sufficient conditions that ensure the extinction, partial survival, and permanence of the system. By applying the fluctuation lemma we establish sufficient conditions that ensure the extinction of one of the components and the stability of the the other one. For the permanent case, by constructing a suitable Lyapunov function we obtain some sufficient conditions for the globally attractivity of the positive solution of the system. Examples, together with their numeric simulations, show the feasibility of the main results. To ensure the stable coexistence of the two species, the harvesting area should be carefully restricted.
Keywords
- Commensalism model
- Harvesting
- Partial closure
- Different inequality
- Extinction
- Permanence
- Partial survival
- Lyapunov function
- Fluctuation lemma
MSC
- 34C25
- 92D25
- 34D20
- 34D40
1 Introduction
During the last decade, many scholars [1–11] investigated the dynamic behavior of the mutualism model, and many excellent results were obtained. For example, Chen, Xie, and Chen [1] showed that the stage structure of the species can lead to the extinction of the mutualism model, despite the cooperation between the species; Chen, Chen, and Li [3] showed that the feedback control variables have no influence on the persistent property of a kind of mutualism model, and in this direction, some similar results was established in [6, 8]; several scholars [2, 4, 7, 10, 11] investigated the stability property of the positive equilibrium of the cooperative system, Xie, Chen, and Xue [10] showed that if the harvesting effort is limited, then the cooperative system admits a unique positive equilibrium, which is globally attractive.
Commensalism, which describes a symbiotic interaction between two populations where one population gets benefit from the other while the other is neither harmed nor benefited due to the interaction with the previous species [12], has not arisen the attention of the scholars, since the model seems simple and can be seen as a particular case of the mutualism model. Only recently scholars paid attention to such a kind of relationship; see [12–20] and the references therein. Topics such as the existence of the positive periodic solution [17], the existence of a positive almost periodic solution [14], the existence and stability of the positive equilibrium [16], the influence of the impulsive [15] were investigated, and many excellent results were obtained. However, as was pointed out by Georgescu and Maxin [20], “One would think that the stability of the coexisting equilibria for two-species models of commensalism would follow immediately from the corresponding results for models of mutualism, when these results are available, …, However, this is not actually the case”. Hence, it is necessary to do some further works on commensalism model.
As for as an ecosystem is concerned, there are the most important three topics: permanence, extinction, and global attractivity, which reflect the existence of the species in the long run, the extinction of the species, and the species maintained in a stable state. During the last decades, there are many excellent results on these three topics; see [28–37] and the references therein. For example, Shi, Li, and Chen [30] studied the extinction property of a competition system with infinite delay and feedback controls; Chen, Xie, and Li [31] investigated the partial extinction of the predator–prey model with stage structure; Chen, Chen, and Huang [32] investigated the extinction property of the nonlinear competition system with Beddington–DeAngelis functional response; Xie, Xue, Wu et al. [33] studied the extinction property of a nonlinear toxic substance competition system; Chen, Ma, and Zhang [34] showed that if the refuge is restricted to suitable area, then the Lotka–Volterra predato–prey system can admit a unique positive equilibrium, which is globally attractive. In this paper, we also focus our attention on the persistency, extinction, and stability of system (1.4).
The paper is arranged as follows. We will investigate the extinction, partial survival and persistency of system (1.4) in the next section. In Sect. 3, we investigate the global stability property of the solutions of the system. Two examples, together with their numeric simulations, are presented in Sect. 4 to show the feasibility of the main results. We end this paper by a brief discussion.
2 Extinction and persistency of the system
Lemma 2.1
([28])
Lemma 2.2
The domain \(R^{2}_{+}=\{(x,y)|x>0,y>0\}\) is invariant with respect to (1.4).
Proof
Theorem 2.1
Proof
Theorem 2.2
Proof
Theorem 2.3
Proof
Theorem 2.4
Proof
3 Global attractivity
In Sect. 2, we discussed the persistent or extinction property of the system, which means that the solutions of the system are bounded above and below by some positive constants or the species will be driven to extinction. One of the interesting problems is to give sufficient conditions to ensure the global attractivity of the positive solution of the system. Before we state the main results of this section, we need to introduce two useful lemmas.
Lemma 3.1
(Fluctuation lemma, [35, Lemma 4])
Lemma 3.2
Suppose that \(r(t)\) and \(a(t)\) are bounded above and below by positive constants. Then any positive solutions of Eq. (3.1) are defined on \([0, +\infty )\), bounded above and below by positive constants, and globally attractive.
Theorem 3.1
Proof
Theorem 3.2
Proof
Let \(N(t)=(N_{1}(t),N_{2}(t))^{T}\) be any positive solution of system (1.4). By Theorem 2.3 the species \(N_{1}\) will be driven to extinction, that is, \(N_{1}(t)\rightarrow 0 \) as \(t\rightarrow +\infty \). On the other hand, noting that the second equation of system (1.4) is independent of \(N_{1}(t)\), from Lemma 3.2 it immediately follows that \(N_{2}(t)\rightarrow N_{2}^{*}(t)\) as \(t\rightarrow + \infty \), where \(N_{2}^{*}(t)\) is any positive solution of system (3.4). This ends the proof of Theorem 3.2. □
Theorem 3.3
Proof
4 Numerical simulations
Example 4.1
Example 4.2
5 Discussion
Recently, many scholars [12–21] studied the dynamic behavior of the commensalism model; however, none of them consider the influence of harvesting. Stimulated by the recent works of Chakraborty, Das, and Kar [24], we propose a nonautonomous nonselective commensalism model incorporating partial closure to the population.
We focus our attention on the persistent and extinction property of the system, Theorems 2.1–2.4 show that, depending on the area that can be harvested, the the system may exhibit permanent, extinction, or partial survival phenomenon, that is, the introducing of harvesting makes the dynamic behavior of the system complicated. Theorem 2.4 shows that if the harvesting area is small enough (i.e., m is small enough), then two species can coexist in the long run. If we further assume that the intrinsic competition rate (\(e(t)\)) is larger than the cooperative between the two species (\(c(t)\)), then two species can coexist in a stable state. Such a result may help us in designing the reserve area of the species.
It seems interesting to incorporate the time delay to system (1.4) and study the influence of the time delay. We leave this for future study.
Declarations
Acknowledgements
The authors are grateful to anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper.
Funding
This work is supported by the National Natural Science Foundation of China under Grant (11601085) and the Natural Science Foundation of Fujian Province (2017J01400).
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
References
- Chen, F.D., Xie, X.D., Chen, X.F.: Dynamic behaviors of a stage-structured cooperation model. Commun. Math. Biol. Neurosci. 2015, Article ID 4 (2015) Google Scholar
- Chen, F.D., Xie, X.D.: Study on the Dynamic Behaviors of Cooperation Population Modeling. Science Press, Beijing (2014) Google Scholar
- Chen, L.J., Chen, L.J., Li, Z.: Permanence of a delayed discrete mutualism model with feedback controls. Math. Comput. Model. 50, 1083–1089 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Yang, K., Miao, Z.S., Chen, F.D., Xie, X.D.: Influence of single feedback control variable on an autonomous Holling-II type cooperative system. J. Math. Anal. Appl. 435(1), 874–888 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Chen, L.J., Xie, X.D.: Permanence of an N-species cooperation system with continuous time delays and feedback controls. Nonlinear Anal., Real World Appl. 12, 34–38 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Chen, L.J., Xie, X.D., Chen, L.J.: 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
- Yang, K., Xie, X., Chen, F.: Global stability of a discrete mutualism model. Abstr. Appl. Anal. 2014, Article ID 709124 (2014) MathSciNetGoogle Scholar
- Li, Y.K., Zhang, T.W.: Permanence of a discrete n-species cooperation system with time-varying delays and feedback controls. Math. Comput. Model. 53, 1320–1330 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Z.J., Wu, J.H., Tan, R.H., Chen, Y.P.: Modeling and analysis of a periodic delayed two-species model of facultative mutualism. Appl. Math. Comput. 217, 893–903 (2010) MathSciNetMATHGoogle Scholar
- Xie, X.D., Chen, F.D., Xue, Y.L.: Note on the stability property of a cooperative system incorporating harvesting. Discrete Dyn. Nat. Soc. 2014, Article ID 327823 (2014) MathSciNetGoogle Scholar
- Xie, X.D., Chen, F.D., Yang, K., Xue, Y.L.: Global attractivity of an integrodifferential model of mutualism. Abstr. Appl. Anal. 2014, Article ID 928726 (2014) MathSciNetGoogle Scholar
- Hari Prasad, B., Pattabhi Ramacharyulu, N.Ch.: Discrete model of commensalism between two species, I. J. Mod. Educ. Comput. Sci. 8, 40–46 (2012) Google Scholar
- Wu, R.X., Li, L., Lin, Q.F.: A Holling type commensal symbiosis model involving Allee effect. Commun. Math. Biol. Neurosci. 2018, Article ID 6 (2018) Google Scholar
- Xue, Y.L., Xie, X.D., Chen, F.D., et al.: Almost periodic solution of a discrete commensalism system. Discrete Dyn. Nat. Soc. 2015, Article ID 295483 (2015) MathSciNetGoogle Scholar
- Miao, Z.S., Xie, X.D., Pu, L.Q.: Dynamic behaviors of a periodic Lotka–Volterra commensal symbiosis model with impulsive. Commun. Math. Biol. Neurosci. 2015, Article ID 3 (2015) Google Scholar
- Wu, R.X., Lin, L., Zhou, X.Y.: A commensal symbiosis model with Holling type functional response. J. Math. Comput. Sci. 16, 364–371 (2016) View ArticleGoogle Scholar
- Lin, Q.F.: Dynamic behaviors of a commensal symbiosis model with non-monotonic functional response and non-selective harvesting in a partial closure. Commun. Math. Biol. Neurosci. 2018, Article ID 4 (2018) Google Scholar
- Sun, G.C., Sun, H.: Analysis on symbiosis model of two populations. J. Weinan Normal Univ. 28(9), 6–8 (2013) Google Scholar
- Zhu, Z.F., Chen, Q.L.: Mathematic analysis on commensalism Lotka–Volterra model of populations. J. Jixi Univ. 8(5), 100–101 (2008) MathSciNetGoogle Scholar
- Georgescu, P., Maxin, D.: Global stability results for models of commensalism. Int. J. Biomath. 10(3), 1750037 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Xue, Y.L., Han, R.Y., Yang, L.Y., Chen, F.D.: On the existence and stability of positive periodic solution of a nonautonomous commensal symbiosis model of two populations. J. Shangming Univ. 32(2), 32–37 (2015) Google Scholar
- Chen, B.G.: Dynamic behaviors of a non-selective harvesting Lotka–Volterra amensalism model incorporating partial closure for the populations. Adv. Differ. Equ. 2018, 111 (2018) MathSciNetView ArticleGoogle Scholar
- Chen, L., Chen, F.: Global analysis of a harvested predator–prey model incorporating a constant prey refuge. Int. J. Biomath. 3(02), 177–189 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Chakraborty, K., Das, S., Kar, T.K.: On non-selective harvesting of a multispecies fishery incorporating partial closure for the populations. Appl. Math. Comput. 221, 581–597 (2013) MathSciNetMATHGoogle Scholar
- Kar, T.K., Chaudhuri, K.S.: On non-selective harvesting of two competing fish species in the presence of toxicity. Ecol. Model. 161, 125–137 (2003) View ArticleGoogle Scholar
- Leard, B., Rebaza, J.: Analysis of predator–prey models with continuous threshold harvesting. Appl. Math. Comput. 217(12), 5265–5278 (2011) MathSciNetMATHGoogle Scholar
- Chakraborty, K., Jana, S., Kar, T.K.: Global dynamics and bifurcation in a stage structured prey–predator fishery model with harvesting. Appl. Math. Comput. 218(18), 9271–9290 (2012) MathSciNetMATHGoogle Scholar
- Chen, F., Li, Z., Huang, Y.J.: Note on the permanence of a competitive system with infinite delay and feedback controls. Nonlinear Anal., Real World Appl. 8, 680–687 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Chen, F.D., Xie, X.D., Miao, Z.S., et al.: Extinction in two species nonautonomous nonlinear competitive system. Appl. Math. Comput. 274, 119–124 (2016) MathSciNetGoogle Scholar
- Shi, C., Li, Z., Chen, F.: Extinction in a nonautonomous Lotka–Volterra competitive system with infinite delay and feedback controls. Nonlinear Anal., Real World Appl. 13(5), 2214–2226 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Chen, F., Xie, X., Li, Z.: Partial survival and extinction of a delayed predator–prey model with stage structure. Appl. Math. Comput. 219(8), 4157–4162 (2012) MathSciNetMATHGoogle Scholar
- Chen, F., Chen, X., Huang, S.: Extinction of a two species non-autonomous competitive system with Beddington–DeAngelis functional response and the effect of toxic substances. Open Math. 14(1), 1157–1173 (2016) MathSciNetMATHGoogle Scholar
- Xie, X., Xue, Y., Wu, R., et al.: Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton. Adv. Differ. Equ. 2016, Article ID 258 (2016) MathSciNetView ArticleGoogle Scholar
- Chen, F., Ma, Z., Zhang, H.: Global asymptotical stability of the positive equilibrium of the Lotka–Volterra prey–predator model incorporating a constant number of prey refuges. Nonlinear Anal., Real World Appl. 13(6), 2790–2793 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Montes De Oca, F., Vivas, M.: Extinction in two dimensional Lotka–Volterra system with infinite delay. Nonlinear Anal., Real World Appl. 7(5), 1042–1047 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Zhao, J.D., Chen, W.C.: The qualitative analysis of N-species nonlinear prey-competition systems. Appl. Math. Comput. 149, 567–576 (2004) MathSciNetMATHGoogle Scholar
- Lin, Q.X., Xie, X.D., et al.: Dynamical analysis of a logistic model with impulsive Holling type-II harvesting. Adv. Differ. Equ. 2018, 112 (2018) MathSciNetView ArticleGoogle Scholar