- Research
- Open Access
The influence of commensalism on a Lotka–Volterra commensal symbiosis model with Michaelis–Menten type harvesting
- Baoguo Chen^{1}Email author
https://doi.org/10.1186/s13662-019-1989-4
© The Author(s) 2019
- Received: 13 February 2018
- Accepted: 23 January 2019
- Published: 31 January 2019
Abstract
Keywords
- Commensal symbiosis model
- Michaelis–Menten type harvesting
- Differential inequality theory
- Global attractivity
MSC
- 34C25
- 92D25
- 34D20
- 34D40
1 Introduction
Mutualism means that the different species exist in a relationship in which each species benefits from the activity of the other species. During the last decade, many scholars [1–17] investigated the dynamic behaviors of the mutualism model and some essential progress on persistent, extinction and stability of the system are obtained. Some scholars [1–7] focused on the persistent property of the cooperation system. Li and Yang [6] proposed a discrete model of mutualism with infinite deviating arguments, they showed that the system is permanent. Li and Zhang [1], Chen, Chen and Li [2], Chen and Xie [3], Chen, Xie, Chen [4] and Yang and Li [5] had studied the persistent property of the mutualism model with feedback control, and in [4], by applying a difference inequality of Fan and Wang, Chen, Xie and Chen showed that feedback control variables have influence on the permanence of the discrete N-species cooperative system, Li and Zhang [1] also obtained some similar results. Some scholars [8–12] focused on the stability property of the positive equilibrium of the mutualism model. For example, Xie, Chen, Yang et al. [12] showed that the unique positive equilibrium of an integrodifferential model of mutualism is globally attractive. Based on a difference inequality which was established by Fengde Chen, Xie, Xue and Wu [11] also investigated the stability property of the positive equilibrium of a discrete mutualism model with infinite deviating arguments. Some scholars [13–15] argued that non-autonomous case is more suitable, and such topics as the existence of the positive periodic solution and the persistence of the system were investigated. Recently, Chen, Xie and Chen [16] and Yang, Miao and Chen [17] focused on the extinction property of the mutualism model, in [16], Chen, Xie and Chen showed that the stage structure of the species plays important roles in the extinction of the species, despite the cooperation of the species. Yang, Miao, Chen et al. [17] proposed a mutualism model with single feedback control, and they found that the system admits more complicated dynamic behaviors, for example, by choosing suitable coefficients, the species may be driven to extinction.
On the other hand, to obtain a resource for humans, harvesting of species is necessary. Already, many scholars investigated the influence of the harvesting on the population system [9, 10, 30–38].
There are three types of harvesting: (1) constant harvesting [39]; (2) linear harvesting [9, 10, 27, 31, 32]; (3) nonlinear harvesting [30, 33–37]. As is well known, nonlinear harvesting is more realistic from the biological and economical points of view [30]. Clark [11] proposed a harvesting term \(h= \frac{qEx}{cE+lx}\), which is named the Michealis–Menten type functional form of the catch rate. Generally speaking, such kind of harvesting may lead to the complexity dynamic behaviors of the system; for example, Idlangoa, Shepherd and Gear [40] showed that the logistic model with Holling type II harvesting term may admit zero, one or two positive equilibria. Hu and Cao [35] showed that the predator–prey model with Michealis–Menten type harvesting in predator species may admit a rich bifurcation phenomenon.
The paper is arranged as follows. The local and global stability property of the equilibria of system (1.2) is investigated in Sects. 2 and 3, respectively. The extinction property of the system is investigated in Sect. 4. Some examples together with their numeric simulations are presented in Sect. 5 to show the feasibility of the main results. We end this paper by a brief discussion.
2 Local stability of the equilibria
The aim of this section is to investigate the existence and local stability property of the equilibrium of system (1.2).
Lemma 2.1
Proof
Lemma 2.2
We shall now investigate the local stability property of the above equilibria.
Theorem 2.1
Assume that (2.8) and (2.9) hold, then \(E_{1}(0,0)\) and \(E_{3}(0,K_{2})\) are all unstable; \(E_{4}(x^{*}, y ^{*})\) is locally asymptotically stable. Assume further that (2.1) holds, then \(E_{2}(x_{1},0)\) is unstable.
Proof
- (1)
For the steady-state solution \(E_{1}(0,0)\), \(\lambda _{1}=r_{1}- \frac{q}{m_{1}}\), \(\lambda _{2}=r_{2}>0\), so \(E_{1}(0,0)\) is unstable.
- (2)
For the steady-state solution \(E_{3}(0,K_{2})\), \(\lambda _{1}=r _{1} (1+ \frac{\alpha K_{2}}{K_{1}} )- \frac{q}{m_{1}}>0\), \(\lambda _{2}= -r_{2}<0\), and so, \(E_{3}(0,K_{2})\) is unstable.
- (3)Noting that the positive equilibrium \(E_{4}(x^{*}, y^{*})\) satisfiesBy using (2.17), the Jacobian of the system about the equilibrium point \(E_{4}(x^{*},y^{*})\) is given by$$ \begin{gathered} r_{1} \biggl(1- \frac{x^{*}}{K_{1}}+\alpha \frac{y^{*}}{K_{1}} \biggr)- \frac{qE}{m_{1}E+m_{2}x^{*}}=0, \\ r_{2} \biggl(1- \frac{y^{*}}{K_{2}} \biggr)=0. \end{gathered} $$(2.17)Under the assumption (2.1) and (2.2), the two eigenvalues of the matrix satisfies$J({x}^{\ast},{y}^{\ast})=\left(\begin{array}{cc}-\frac{{r}_{1}{x}^{\ast}}{{K}_{1}}+\frac{qE{x}^{\ast}{m}_{2}}{{(E{m}_{1}+{m}_{2}{x}^{\ast})}^{2}}& \frac{{r}_{1}{x}^{\ast}\alpha}{{K}_{1}}\\ 0& -\frac{{r}_{2}{y}^{\ast}}{{K}_{2}}\end{array}\right).$(2.18)Consequently, \(E_{4}(x^{*},y^{*})\) is locally asymptotically stable.$$\begin{aligned}& \lambda _{1}=- \frac{r_{1}x^{*}}{K_{1}} + \frac{qEx^{*}m_{2}}{(Em_{1}+m_{2}x^{*})^{2}}< x^{*} \biggl(-\frac{r_{1}}{K _{1}} + \frac{qm_{2}}{Em_{1}^{2}} \biggr)< 0, \\& \lambda _{2}=-\frac{r_{2}y^{*}}{K_{2}} < 0. \end{aligned}$$
- (4)Noting that \(x_{1}\) satisfiesThe Jacobian of the system about the equilibrium point \(E_{2}(x_{1},0)\) is given by$$ r_{1} \biggl(1- \frac{x_{1}}{K_{1}} \biggr)- \frac{qE}{m_{1}E+m_{2}x_{1}}=0. $$Under the assumption (2.1), (2.8) and (2.9), the two eigenvalues of the matrix satisfies \(\lambda _{1}=- \frac{r_{1}x_{1}}{K_{1}} + \frac{qEx_{1}m_{2}}{(Em_{1}+m_{2}x_{1})^{2}}<0\), \(\lambda _{2}=r_{2}>0\). Consequently, \(E_{2}(x_{1}, 0)\) is unstable.$\left(\begin{array}{cc}-\frac{{r}_{1}{x}_{1}}{{K}_{1}}+\frac{qE{x}_{1}{m}_{2}}{{(E{m}_{1}+{m}_{2}{x}_{1})}^{2}}& \frac{{r}_{1}{x}_{1}\alpha}{{K}_{1}}\\ 0& {r}_{2}\end{array}\right).$(2.19)
The proof of Theorem 2.1 is finished. □
3 Global stability
The aim of this section is to investigate the global stability property of the equilibrium of system (1.2).
Theorem 3.1
Assume that (2.8) and (2.9) hold, then the positive equilibrium \(E_{4}(x^{*},y^{*})\) of system (1.2) is globally stable.
Proof
- (1)There is a unique \(u_{1\varepsilon }^{*} \), such that \(F_{3}(u _{1\varepsilon }^{*}) = 0\), where, by simple computation,$$\begin{aligned}& u_{1\varepsilon }^{*}= \frac{-B_{21}+\sqrt{B_{21}^{2}-4A_{2}C_{21}}}{2A_{2}}. \end{aligned}$$(3.8)$$\begin{aligned}& \begin{aligned} &A_{2}=m_{2}r_{1}, \qquad B_{21}= Em_{1}r_{1}-(K_{2}+ \varepsilon )\alpha m_{2}r_{1}-K_{1}m_{2}r _{1}, \\ &C_{21}=EK_{1}q-EK_{1}m_{1}r_{1}-E(K_{2}+ \varepsilon )\alpha m_{1}r _{1}. \end{aligned} \end{aligned}$$(3.9)
- (2)
For all \(u_{1\varepsilon }^{*} > u_{1} > 0\), \(F_{3}(u_{1}) > 0\).
- (3)
For all \(u_{1} > u_{1\varepsilon }^{*} > 0\), \(F_{3}(u_{1}) < 0\).
- (1)There is a unique \(v_{1\varepsilon }^{*} \), such that \(F_{4}(v _{1\varepsilon }^{*}) = 0\), where, by simple computation,here$$ v_{1\varepsilon }^{*}= \frac{-B_{22}+\sqrt{B_{22}^{2}-4A_{2}C_{22}}}{2A_{2}}, $$(3.15)$$ \begin{gathered} A_{2}=m_{2}r_{1}, \qquad B_{22}= Em_{1}r_{1}-(K_{2}- \varepsilon )\alpha m_{2}r_{1}-K_{1}m_{2}r _{1}, \\ C_{22}=EK_{1}q-EK_{1}m_{1}r_{1}-E(K_{2}- \varepsilon )\alpha m_{1}r _{1}. \end{gathered} $$
- (2)
For all \(v_{1\varepsilon }^{*} > v_{1} > 0\), \(F_{4}(v_{1}) > 0\).
- (3)
For all \(v_{1} > v_{1\varepsilon }^{*} > 0\), \(F_{4}(v_{1}) < 0\).
One could easily see that (2.1) implies (2.8), hence, as a direct corollary of Theorem 3.1, we have the following.
Corollary 3.1
Assume that (2.1) and (2.2) hold, then system (1.2) admits a unique positive equilibrium \(E_{4}(x^{*}, y^{*})\), which is globally stable.
Remark 3.1
Corollary 3.1 shows that if the harvesting is limited, the catchability is small enough (i.e., q is small enough), then the two species could coexist in a stable state.
Remark 3.2
Theorem 3.1 shows that, for the catchability large enough (i.e., q is large), then the cooperative intensity of the two species becomes the most important factor, if α is large enough, then two species could also coexist in a stable state.
4 Extinction of the first species
In Sect. 2, assumption (2.8) and (2.9) implies that the catchability coefficient q should be limited or the cooperative effect (α) should be large enough. One may be curious as to what would happen if the harvesting effort is large enough and the cooperative effect is limited. In this case, will the species be driven to extinction?
We will give an affirmative answer to this question. Indeed, we have the following result.
Theorem 4.1
Proof
5 Numerical simulations
Example 5.1
Example 5.2
6 Discussion
Discussing the influence of harvesting is one of the main topics in the study of population dynamics, and many scholars (see [30–40]) have done work in this direction. Specially, recently, many scholars (see [30, 33–36]) studied the ecosystem with nonlinear harvesting term.
Stimulated by the work of [30, 33–36], in this paper, we try to incorporate the Michaelis–Menten type harvesting term for the first species of a Lotka–Volterra commensalism model, this seems more interesting and necessary, since more and more species become endangered due to the over harvesting by humans. It is natural to ask: Could the commensalism of the second species to the first species could avoid the extinction of the species? Theorem 3.1 shows that if the cooperative intensity is large enough, then the two species could really coexist in a stable state. However, Theorem 4.1 shows that if the cooperative effect is limited, the first species may still be driven to extinction due to the over harvesting.
Our study shows that to ensure the long run existence of the species, the harvesting effort should be limited.
Declarations
Acknowledgements
The authors would like to thank two referees for their useful comments, which greatly improved the paper.
Funding
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).
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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