# Dynamic behaviors of a stage structure amensalism system with a cover for the first species

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## Abstract

In this paper, we propose and study a two-species stage structured amensalism model with a cover for the first species. By developing a new analysis technique or, more precisely, by combining the differential inequality theory and the Lyapunov function method, we obtain sufficient conditions ensuring the global attractivity of positive and boundary equilibria, respectively. Our study shows that the final density of the first species is an increasing function of the partial cover, and if the stage structured species is globally asymptotically stable, then there exists a threshold such that if the cover is greater than this threshold, the species can still exist in the long run, whereas if the cover is too small, then the first species is driven to extinction.

## Keywords

Stage structure Amensalism Lyapunov function Differential inequality Global stability## MSC

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

*γ*are positive constants, \(x_{1}(t)\) and \(x_{2}(t)\) are the densities of the immature and mature first species at time

*t*,

*y*is the density of the second species at time

*t*, and \(k\in(0,1)\) is a cover provided for the first species. The following assumptions are made in model (1.1):

- 1.The first species has two-stage structure, immature and mature. Its dynamic behavior is described by the equation systemWe refer to Khajanchi and Banerjee [1] for more background of this equation system.$$\begin{aligned} &\frac{dx_{1}}{dt}=\alpha x_{2}-\beta x_{1}- \delta_{1} x_{1}, \\ &\frac{dx_{2}}{dt}=\beta x_{1}-\delta_{2} x_{2}- \gamma x_{2}^{2}. \end{aligned}$$
- 2.
There is a partial cover (represented by

*k*) for the first species to protect it from the second species. - 3.
Both relationships between the immature species and the second species and between the mature species and the second species are bilinear: (\(d_{1}(1-k)x_{1}y\) and \(d_{2}(1-k)x_{2}y\)).

- 4.
The second species satisfies the logistic model.

*t*. Recently, Khajanchi and Banerjee [1] proposed the following stage structure predator–prey model with ratio dependent functional response:

It brings to our attention that, to this day, still no scholars propose and study the dynamic behaviors of the amensalism model with stage structure. This motivated us to propose system (1.1). We mention here that at first sight, system (1.1) is very simple, However, the third equation is independent of \(x_{1}\) and \(x_{2}\), and hence it is impossible to investigate the stability property of the system by constructing a suitable Lyapunov function. Also, since this is a three-dimensional system, we cannot investigate the stability property of the system by using the Dulac criterion.

The paper is arranged as follows. We investigate the existence and locally stability property of the equilibria of system (1.1) in Sect. 2. In Sect. 3, by applying the differential inequality theory and constructing some suitable Lyapunov function we are able to investigate the global attractivity of the positive and boundary equilibria. We then discuss the influence of partial cover to the final density of the first species in Sect. 4, and in Sect. 5, we present an example together with its numerical simulations to show the feasibility of the main results. We end this paper by a brief discussion.

## 2 Local stability of the equilibria

*γ*are positive constants. The following lemma is Theorems 4.1 and 4.2 of [36].

### Lemma 2.1

*If*

*then the boundary equilibrium*\(O(0,0)\)

*of system*(2.1)

*is globally stable*.

*If*

*then the positive equilibrium*\(B(x_{1}^{*},x_{2}^{*})\)

*of system*(2.1)

*is globally stable*,

*where*

Now we are in position to investigate the local stability property of system (1.1).

We will now investigate the local stability of the above equilibria.

### Theorem 2.1

\(A_{1}(0,0,0)\)*is unstable*.

### Proof

### Theorem 2.2

*If*

*then*\(A_{2}(0,0,\frac{b_{2}}{a_{2}})\)

*is locally asymptotically stable*.

*If*

*then*\(A_{2}(0,0,\frac{b_{2}}{a_{2}})\)

*is unstable*.

### Proof

### Theorem 2.3

\(A_{3}(x_{1}^{*},x_{2}^{*}, 0)\)*is unstable*.

### Proof

### Theorem 2.4

*If*

*then*\(A_{4}(x_{1}^{**},x_{2}^{**},y^{**})\)

*is locally asymptotically stable*.

### Proof

## 3 Global stability

As was shown in the previous section, under some suitable conditions, \(A_{2}\) and \(A_{4}\) can be locally asymptotically stable. In this section, we obtain some sufficient conditions that for the global asymptotical stability of the equilibria \(A_{2}\) and \(A_{4}\).

### Theorem 3.1

*If*

*then*\(A_{2}(0,0,\frac{b_{2}}{a_{2}})\)

*is globally attractive*,

*that is*,

### Proof

*ε*, it follows from (3.1) that

### Remark 3.1

Under the assumption \(\alpha\beta<\delta_{2}(\beta +\delta_{1})\), it follows from Lemma 2.1 that the first species will be driven to extinction. In this case, for all \(0< k<1\), inequality (3.1) holds, and it follows from Theorem 3.1 that \(A_{2}(0,0,\frac{b_{2}}{a_{2}})\) is globally attractive, which means that the first species is still driven to extinction.

### Remark 3.2

### Remark 3.3

At first sight, system (1.1) is not complicate, and we may conjecture that it is easy to investigate the stability of the equilibrium by constructing a suitable Lyapunov function as that of An and Lei [36]; however, this is impossible, since the term \(-d_{1}(1-k)x_{1}y\) in the first equation of system (1.1) cannot be dealt with directly. Here, by combining the differential inequality theory and the Lyapunov function we give a strict proof of Theorem 3.1. Such a method possibly could be applied to other situations.

### Theorem 3.2

*If*

*then*\(A_{4}(x_{1}^{**},x_{2}^{**},y^{**})\)

*is globally attractive*.

### Proof

### Remark 3.4

Condition (3.14) is necessary to ensure the existence of the positive equilibrium. Theorem 3.2 shows that if the positive equilibrium exists, then it is globally asymptotically stable, and hence it is impossible for the system to have the bifurcation phenomenon.

### Remark 3.5

If \(\alpha\beta>\delta_{2}(\beta+\delta _{1}) \), then for large enough *k* (*k* is close to 1) inequality (3.14) can hold, and from Lemma 2.1 we know that in this case, system (2.1) admits a unique positive equilibrium. In other words, if system (2.1) admits a unique positive equilibrium, then for the amensalism model, if the influence of the second species to the first species is limited, then the system still admits a unique globally asymptotically stable positive equilibrium.

## 4 The influence of the partial cover

From (2.8) we easily see that the final density of the immature and mature species are relevant to the partial cover, and hence one interesting issue is to find out a relationship between the final density of the species and the partial cover.

## 5 Example

Now let us consider the following example.

### Example 5.1

## 6 Conclusion

During the lase decade, many scholars [13, 14, 15, 16, 17, 18, 19] studied the dynamic behavior of the amensalism model; however, only recently, scholars [14, 16, 18] studied the influence of the partial cover to the traditional two-species amensalism model. In this paper, for first time, we propose a two-species stage-structured amensalism model with a cover for the first species.

Though at first sight, the system seems very simple, note that the third equation of system (1.1) is independent of \(x_{1}\) and \(x_{2}\), and thus the Lyapunov method cannot be applied directly to investigate the stability property of system (1.1). By combining the differential inequality theory and the Lyapunov function method we are able to investigate the global stability property of the boundary and positive equilibrium. Theorem 3.2 shows that if the positive equilibrium exists, then it is globally attractive, and the final density of the first species is an increasing function of the partial cover.

We mention here that the method used in this paper can be applied to investigate the stability property of the other ecosystem. We leave this for the future study.

## Notes

### Acknowledgements

I would like to thank Dr. Xuelin Xie for useful discussion during the period of writing this paper.

### Authors’ contributions

All authors read and approved the final manuscript.

### 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).

### Competing interests

The author declares that there is no conflict of interests.

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