# Dynamic behaviors of a non-selective harvesting Lotka–Volterra amensalism model incorporating partial closure for the populations

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## Abstract

A non-selective harvesting Lotka–Volterra amensalism model incorporating partial closure for the populations is proposed and studied in this paper. Local and global stability of the boundary and interior equilibria are investigated. By introducing the harvesting, the dynamic behaviors of the system become complicated. Depending on the fraction of the stock available for harvesting, the system maybe extinction, partial survival or two species may coexist in a stable state. Our results supplement and complement the main results of Xiong, Wang, and Zhang (Adv. Appl. Math. 5(2):255-261, 2016).

## Keywords

Amensalism model Lyapunov function Global stability## MSC

34C25 92D25 34D20 34D40## 1 Introduction

## Theorem A

- (1)
\(A(0,0)\)

*is unstable*; - (2)
\(B(P_{1}, 0) \)

*is a saddle point*,*thus is unstable*; - (3)
*if*\(u<\frac{P_{1}}{P_{2}}\), \(C(0, P_{2})\)*is a saddle point and consequently unstable*;*if*\(u>\frac{P_{1}}{P_{2}}\), \(C(0, P_{2})\)*is a stable node*; - (4)
*if*\(u<\frac{P_{1}}{P_{2}}\), \(D(P_{1}-uP_{2}, P_{2})\)*is a stable node*.

On the other hand, as was pointed out by Chakraborty et al. [16], the study of resource management, including fisheries, forestry, and wildlife management, has great importance. They argued that it is necessary to harvest the population, but harvesting should be regulated so that both the ecological sustainability and conservation of the species can be implemented in a long run. Already, they proposed a non-selective harvesting predator–prey system incorporating partial closure for the populations, they investigated the local and global stability property of the system, and some interesting results related to the optimal harvesting were obtained.

*u*, \(i=1, 2\), are all positive constants. \(r_{i}(P_{i})\) represents the intrinsic growth rate (environmental carrying capacity) of the

*i*th species,

*E*is the combined fishing effort used to harvest and \(m(0< m<1)\) is the fraction of the stock available for harvesting. One could refer to [1, 16] for more background and the adjustment of system (1.2).

As far as system (1.2) is concerned, one interesting issue is the following:

*Find out the influence of the parameter* *m, which reflects the fraction of the stock available for harvesting.*

The paper is arranged as follows. We investigate the existence and locally stability property of the equilibrium solutions of system (1.2) in the next section. In Sect. 3, by constructing some suitable Lyapunov function, we investigate the global stability property of the equilibria. The influence of the parameter *m* is then discussed 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 with a brief discussion.

## 2 Local stability of the equilibria

The system always admits the boundary equilibrium \(A(0,0)\).

If \(r_{1}>Emq_{1}\) holds, the system admits the boundary equilibrium \(B(N_{10}, 0)\), where \(N_{10}= \frac{P_{1}(r_{1}-Emq _{1})}{r_{1}}\).

If \(r_{2}>Emq_{2}\) holds, the system admits the boundary equilibrium \(C(0, N_{20})\), where \(N_{20}= \frac{P_{2}(r_{2}-Emq _{2})}{r_{2}}\).

## Theorem 2.1

- (1)
*Assume that*$$ m>\max \biggl\{ \frac{r_{1}}{Eq_{1}}, \frac{r_{2}}{Eq_{2}} \biggr\} $$(2.2)*holds*,*then*\(A(0,0)\)*is locally stable*,*otherwise it is unstable*; - (2)
*Assume that*$$ \frac{r_{2}}{Eq_{2}}< m< \frac{r_{1}}{Eq_{1}} $$(2.3)*holds*,*then*\(B(N_{10}, 0)\)*is locally stable*,*otherwise it is unstable*; - (3)
*Assume that*$$ \frac{r_{1}r_{2}(P_{1}-uP_{2})}{r_{2}q_{1}EP_{1}- r_{1}uEP_{2}q _{2}} < m< \frac{r_{2}}{Eq_{2}} $$(2.4)*holds*,*then*\(C(0, N_{20})\)*is locally stable*,*otherwise it is unstable*; - (4)
*Assume that*$$ m< \min \biggl\{ \frac{r_{2}}{Eq_{2}}, \frac{r_{1}r_{2}(P_{1}-uP_{2})}{r _{2}q_{1}EP_{1}- r_{1}uEP_{2}q_{2}} \biggr\} $$(2.5)*holds*,*then*\(D(N_{1}^{*}, N_{2}^{*})\)*is locally stable*.

## Proof

The proof of Theorem 2.1 is finished. □

## 3 Global stability

One interesting problem is to further investigate the global stability property of the equilibria of system (1.2), since the global one means that despite the random initial condition, the finial dynamic behaviors of the system could be forecasted. In this aspect, we could obtain the following result.

## Theorem 3.1

- (1)
*Assume that*$$ m>\max \biggl\{ \frac{r_{1}}{Eq_{1}}, \frac{r_{2}}{Eq_{2}} \biggr\} $$(3.1)*holds*,*then*\(A(0,0)\)*is globally asymptotically stable*; - (2)
*Assume that*$$ \frac{r_{2}}{Eq_{2}}< m< \frac{r_{1}}{Eq_{1}} $$(3.2)*holds*,*then*\(B(N_{10}, 0)\)*is globally asymptotically stable*; - (3)
*Assume that*$$ \frac{r_{2}}{Eq_{2}}>m> \frac{r_{1}}{Eq_{1}} $$(3.3)*holds*,*then*\(C(0, N_{20})\)*is globally asymptotically stable*; - (4)
*Assume that*$$ m< \min \biggl\{ \frac{r_{2}}{Eq_{2}}, \frac{r_{1}r_{2}(P_{1}-uP_{2})}{r _{2}q_{1}EP_{1}- r_{1}uEP_{2}q_{2}} \biggr\} $$(3.4)*holds*,*then*\(D(N_{1}^{*}, N_{2}^{*})\)*is globally asymptotically stable*.

## Proof

We will prove Theorem 3.1 by constructing some suitable Lyapunov functions.

*η*is a suitable constant to be determined in the subsequent steps. One could easily see that the function \(V_{2}\) is zero at the equilibrium \(B(N_{10},0)\) and is positive for all other positive values of \(N_{1}\) and \(N_{2}\). By applying (3.6), the time derivative of \(V_{2}\) along the trajectories of (1.2) is

## Remark 3.1

Theorems 2.1 and 3.1 show that if system (1.2) admits the unique positive equilibrium, then the positive equilibrium is globally asymptotically stable.

## Remark 3.2

Compared with Theorems 2.1 and 3.1, one could see that in three cases, the local stability of the equilibrium also implies the global one. However, to ensure \(C(0,N_{20})\) is globally stable, we need assumption (3.3) since our condition is a set of sufficient conditions, maybe it is not the necessary one. Whether (2.4) is enough to ensure the globally attractivity of \(C(0,N_{20})\) or not is still unknown. Obviously, we could not deal with this problem by constructing a suitable Lyapunov function.

## Remark 3.3

From Theorem 3.1(4) and the biological meaning of the parameter *m*, we can draw the conclusion: if the fraction of the stock available for harvesting is limited, then two species could coexist in the long run, despite the initial state.

## 4 The influence of the parameter *m*

- (1)
If \(P_{2}q_{2}r_{1}u>P_{1}q_{1}r_{2}\), then \(\frac{dN_{1}^{*}}{dt}>0\), and \(N_{1}^{*}\) is the strictly increasing function of

*m*; - (2)
If \(P_{2}q_{2}r_{1}u< P_{1}q_{1}r_{2}\), then \(\frac{dN_{1}^{*}}{dt}<0\), and \(N_{1}^{*}\) is the strictly decreasing function of

*m*.

*m*.

## 5 Numerical simulations

## Example 5.1

- (1)
- (2)
- (3)
- (4)

## 6 Discussion

With the aim of the ecological sustainability and conservation of the species to be implemented in a long run, in this paper, we propose a non-selective harvesting Lotka–Volterra amensalism model incorporating partial closure for the populations, i.e., system (1.2), which can be seen as the generalization of system (1.1), and the model is more suitable for the real situation.

With the introducing of harvesting, the dynamic behaviors of the system become very complicated. Depending on the fraction of the stock that could be harvested, the system may have positive equilibrium, which is globally asymptotically stable, which means that two species could coexist in a stable state; or one of the species will be driven to extinction, or both of the species could be driven to extinction.

To sum up, to ensure the conservation of the species, we need to restrict the harvesting to a limited area. Otherwise, although we can afford the area which could not be harvested, the species may still be driven to extinction. Theorem 2.1 and 3.1 give some threshold on *m*, which ensures the coexistence of the two species. The results obtained in this paper maybe useful in designing the natural protection area.

## Notes

### Acknowledgements

The author is grateful to anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. This work is supported by the 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.

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