# Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton

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## Abstract

A two species non-autonomous competitive phytoplankton system with nonlinear inter-inhibition terms and one toxin producing phytoplankton is studied in this paper. Sufficient conditions which guarantee the extinction of a species and the global attractivity of the other one are obtained. Some parallel results corresponding to Yue (Adv. Differ. Equ. 2016:1, 2016, doi:10.1007/s11590-013-0708-4) are established. Numeric simulations are carried out to show the feasibility of our results.

### Keywords

extinction competition phytoplankton system### MSC

34D23 92D25 34D20 34D40## 1 Introduction

Given a function \(g(t)\), let \(g_{L}\) and \(g_{M}\) denote \(\inf_{-\infty< t <\infty}g(t)\) and \(\sup_{-\infty< t<\infty}g(t)\), respectively.

*t*, respectively. \(r_{i}(t)\), \(i=1,2\) are the intrinsic growth rates of species; \(a_{i}\) (\(i = 1,2\)) are the rates of intraspecific competition of the first and second species, respectively; \(b_{i}\) (\(i = 1,2\)) are the rates of interspecific competition of the first and second species, respectively. The second species could produce a toxic, while the first one has a non-toxic product.

*et al.*[17] and Bandyopadhyay [15] considered a Lotka-Volterra type of model for two interacting phytoplankton species, where one species could produce toxic, while the other one has a non-toxic product. The model takes the form

*et al.*[8] proposed the following two species discrete competition system:

*et al.*[23] proposed the following two species competition model:

*et al.*[8], the author obtained some sufficient conditions which guarantee the extinction of one of the components and the global attractivity of the other one.

It is well known that if the amount of the species is large enough, the continuous model is more appropriate, and this motivated us to propose the system (1.1). The aim of this paper is, by developing the analysis technique of [1, 8, 9], to investigate the extinction property of the system (1.1). The remaining part of this paper is organized as follows. In Section 2, we study the extinction of some species and the stability property of the rest of the species. Some examples together with their numerical simulations are presented in Section 3 to show the feasibility of our results. We give a brief discussion in the last section.

## 2 Main results

Following Lemma 2.1 is a direct corollary of Lemma 2.2 of Chen [10].

### Lemma 2.1

*If*\(a>0\), \(b>0\),

*and*\(\dot{x}\geq x(b-ax)\),

*when*\(t\geq{0}\)

*and*\(x(0)>0\),

*we have*

*If*\(a>0\), \(b>0\),

*and*\(\dot{x}\leq x(b-ax)\),

*when*\(t\geq{0}\)

*and*\(x(0)>0\),

*we have*

### Lemma 2.2

*Let*\(x(t)=(x_{1}(t),x_{2}(t))^{T}\)

*be any solution of system*(1.1)

*with*\(x_{i}(t_{0})>0\), \(i=1,2\),

*then*\(x_{i}(t)>0\), \(t\geq t_{0}\)

*and there exists a positive constant*\(M_{0}\)

*such that*

*i*.

*e*.,

*any positive solution of system*(1.1)

*are ultimately bounded above by some positive constant*.

### Proof

### Lemma 2.3

(Fluctuation lemma [34])

*Let*\(x(t)\)

*be a bounded differentiable function on*\((\alpha,\infty)\),

*then there exist sequences*\(\tau_{n}\rightarrow\infty\), \(\sigma_{n}\rightarrow\infty\)

*such that*

### Lemma 2.4

*Suppose that* \(r_{1}(t)\) *and* \(a_{1}(t)\) *are continuous functions bounded above and below by positive constants*, *then any positive solutions of equation* (2.5) *are defined on* \([0, +\infty)\), *bounded above and below by positive constants and globally attractive*.

Our main results are Theorems 2.1-2.5.

### Theorem 2.1

*Assume that*

*hold*,

*further assume that the inequality*

*holds*,

*then the species*\(x_{2}\)

*will be driven to extinction*,

*that is*,

*for any positive solution*\((x_{1}(t),x_{2}(t))^{T}\)

*of system*(1.1), \(x_{2}(t)\rightarrow0 \)

*as*\(t\rightarrow+\infty\).

### Proof

*α*,

*β*such that

*t*(\(\geq T_{1}\)), it follows that

### Theorem 2.2

*In addition to*(2.6),

*further assume that the inequality*

*holds*,

*then the species*\(x_{2}\)

*will be driven to extinction*,

*that is*,

*for any positive solution*\((x_{1}(t),x_{2}(t))^{T}\)

*of system*(1.1), \(x_{2}(t)\rightarrow0 \)

*as*\(t\rightarrow+\infty\).

### Proof

*α*,

*β*such that

*t*(\(\geq T_{2}\)), it follows that

### Theorem 2.3

*In addition to*(2.6),

*further assume that the inequality*

*holds*,

*then the species*\(x_{2}\)

*will be driven to extinction*,

*that is*,

*for any positive solution*\((x_{1}(t),x_{2}(t))^{T}\)

*of system*(1.1), \(x_{2}(t)\rightarrow0 \)

*as*\(t\rightarrow+\infty\).

### Proof

*α*,

*β*such that

*t*(\(\geq T_{3}\)), it follows that

### Lemma 2.5

*Under the assumption of Theorem*2.1

*or*2.2

*or*2.3,

*let*\(x(t)=(x_{1}(t),x_{2}(t))^{T}\)

*be any positive solution of system*(1.1),

*then there exists a positive constant*\(m_{0}\)

*such that*

*where*\(m_{0}\)

*is a constant independent of any positive solution of system*(1.1),

*i*.

*e*.,

*the first species*\(x_{1}(t)\)

*of system*(1.1)

*is permanent*.

### Theorem 2.4

*Assume that the conditions of Theorem * 2.1 *or* 2.2 *or* 2.3 *hold*, *let* \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) *be any positive solution of system* (1.1), *then the species* \(x_{2}\) *will be driven to extinction*, *that is*, \(x_{2}(t)\rightarrow0 \) *as* \(t\rightarrow +\infty\), *and* \(x_{1}(t)\rightarrow x_{1}^{*}(t)\) *as* \(t\rightarrow+\infty\), *where* \(x_{1}^{*}(t)\) *is any positive solution of system* (2.5).

### Proof

By applying Lemmas 2.3 and 2.4, the proof of Theorem 2.4 is similar to that of the proof of Theorem in [4]. We omit the details here. □

Another interesting thing is to investigate the extinction property of species \(x_{1}\) in system (1.1). For this case, we have the following.

### Theorem 2.5

*Assume that*

*hold*,

*then the species*\(x_{1}\)

*will be driven to extinction*,

*that is*,

*for any positive solution*\((x_{1}(t),x_{2}(t))^{T}\)

*of system*(1.1), \(x_{1}(t)\rightarrow0 \)

*as*\(t\rightarrow+\infty\)

*and*\(x_{2}(t)\rightarrow x_{2}^{*}(t)\)

*as*\(t\rightarrow+\infty\),

*where*\(x_{2}^{*}(t)\)

*is any positive solution of system*\(\dot{x}_{2}(t)=x_{2}(t)(r_{2}(t)-b_{2}(t)x_{2}(t))\).

### Proof

*α*,

*β*, and a small enough positive constant \(\varepsilon_{4}\), such that

*t*(\(\geq T_{4}\)), it follows that

## 3 Numeric example

Now let us consider the following example.

### Example 1

## 4 Conclusion

Stimulated by the work of Yue [1], in this paper, a two species non-autonomous competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton is proposed and studied. Series conditions which ensure the extinction of one species and the global attractivity of the other species are established.

We mention here that in system (1.1), we did not consider the influence of delay, we leave this for future investigation.

## Notes

### Acknowledgements

The authors are grateful to anonymous referees for their excellent suggestions, which greatly improved 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).

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