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

*Advances in Difference Equations*
**volumeÂ 2016**, ArticleÂ number:Â 258 (2016)

## 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.

## 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.

The aim of this paper is to investigate the extinction property of the following two species non-autonomous competitive phytoplankton system with nonlinear inter-inhibition terms and one toxin producing phytoplankton:

where \(r_{i}(t)\), \(a_{i}(t)\), \(b_{i}(t)\), \(i=1, 2\), \(c_{1}(t)\) are assumed to be continuous and bounded above and below by positive constants, and \(x_{1}(t)\), \(x_{2}(t)\) are population density of species \(x_{1}\) and \(x_{2}\) at time *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.

The traditional two species Lotka-Volterra competition model takes the form:

Chattopadhyay [2] studied a two species competition model, each species produces a substance toxic to the other only when the other is present. The model takes the form

He investigated the local stability and global stability of the equilibrium. Obviously, system (1.3) is more realistic than that of (1.2). After the work of Chattopadhyay [2], the competitive system with toxic substance became one of the most important topic in the study of population dynamics, see [1â€“22] and the references cited therein. Li and Chen [4] studied the non-autonomous case of system (1.2), a set of sufficient conditions which guarantee the extinction of the second species and the globally attractive of the first species are obtained. Li and Chen [3] studied the extinction property of the following two species discrete competitive system:

Recently, SolÃ© *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

Corresponding to system (1.5), Chen *et al.* [8] proposed the following two species discrete competition system:

They investigated the extinction property of the system.

Some scholars argued that the more appropriate competition model should with nonlinear inter-inhibition terms. Indeed, Wang *et al.* [23] proposed the following two species competition model:

By using a differential inequality, the module containment theorem, and the Lyapunov function, the authors obtained sufficient conditions which ensure the existence and global asymptotic stability of positive almost periodic solutions.

Again, corresponding to system (1.7), several scholars [24, 25] investigated the dynamic behaviors of the discrete type two species competition system with nonlinear inter-inhibition terms,

Wang and Liu [24] studied the almost periodic solution of the system (1.8). Yu [25] further incorporated the feedback control variables to the system (1.8) and investigated the persistent property of the system.

During the lase decade, many scholars [3â€“5, 8, 12â€“14, 26â€“33] investigated the extinction property of the competition system. Maybe stimulating by this fact, Yue [1] proposed the following two species discrete competitive phytoplankton system with nonlinear inter-inhibition terms and one toxin producing phytoplankton:

By further developing the analysis technique of Chen *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 SectionÂ 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 SectionÂ 3 to show the feasibility of our results. We give a brief discussion in the last section.

## 2 Main results

Following LemmaÂ 2.1 is a direct corollary of LemmaÂ 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

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

From the first equation of system (1.1), we have

By applying LemmaÂ 2.1 to differential inequality (2.2), it follows that

Similarly to the analysis of (2.2) and (2.3), from the second equation of system (1.1), we have

Set \(M_{0}=\max\{M_{1}, M_{2}\}\), then the conclusion of LemmaÂ 2.2 follows. This ends the proof of LemmaÂ 2.2.â€ƒâ–¡

### 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*

For the logistic equation

From LemmaÂ 2.1 of Zhao and Chen [35], we have the following.

### 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

It follows from (2.7) that one could choose a small enough positive constant \(\varepsilon_{1}>0\) such that

Equation (2.8) is equivalent to

Therefore, there exist two constants *Î±*, *Î²* such that

That is,

Let \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) be a solution of system (1.1) with \(x_{i}(0)>0\), \(i=1,2\). For the above \(\varepsilon_{1}>0\), from LemmaÂ 2.2 there exists a large enough \(T_{1}\) such that

From (1.1) we have

Let

From (2.11), (2.12), and (2.13), for \(t\geq T_{1}\), it follows that

Integrating this inequality from \(T_{1}\) to *t* (\(\geq T_{1}\)), it follows that

By LemmaÂ 2.2 we know that there exists \(M>M_{0}>0\) such that

Therefore, (2.14) implies that

where

Consequently, we have \(x_{2}(t)\rightarrow0\) exponentially as \(t\rightarrow +\infty\).â€ƒâ–¡

### 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

Equation (2.18) is equivalent to

It follows from (2.19) that one could choose a small enough \(\varepsilon_{2}>0\) such that

It follows from (2.20) and (2.6) that there exist two constants *Î±*, *Î²* such that

That is,

Let \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) be a solution of system (1.1) with \(x_{i}(0)>0\), \(i=1,2\). For the above \(\varepsilon_{2}>0\), from LemmaÂ 2.2 there exists a large enough \(T_{2}\) such that

Let

From (2.22) and (2.23), for \(t\geq T_{2}\), it follows that

Integrating this inequality from \(T_{2}\) to *t* (\(\geq T_{2}\)), it follows that

From (2.24), similarly to the analysis of (2.15)-(2.16), we can draw the conclusion that \(x_{2}(t)\rightarrow0\) exponentially as \(t\rightarrow +\infty\).â€ƒâ–¡

### 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

Equation (2.25) is equivalent to

It follows from (2.26) that one could choose a small enough \(\varepsilon_{3}\) such that

It follows from (2.6) and (2.27) that there exist two constants *Î±*, *Î²* such that

That is,

Let \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) be a solution of system (1.1) with \(x_{i}(0)>0\), \(i=1,2\). For the above \(\varepsilon_{3}>0\), from LemmaÂ 2.2 there exists a large enough \(T_{3}\) such that

Let

From (2.29) and (2.30), for \(t\geq T_{3}\), it follows that

Integrating this inequality from \(T_{3}\) to *t* (\(\geq T_{3}\)), it follows that

From (2.31), similarly to the analysis of (2.15)-(2.16), we can draw the conclusion that \(x_{2}(t)\rightarrow0\) exponentially as \(t\rightarrow +\infty\).â€ƒâ–¡

### Lemma 2.5

*Under the assumption of TheoremÂ *
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*.

### Proof

The proof of LemmaÂ 2.5 is similar to that of LemmaÂ 3.5 in [4], we omit the details here.â€ƒâ–¡

### Theorem 2.4

*Assume that the conditions of TheoremÂ *
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 TheoremÂ 2.4 is similar to that of the proof of Theorem in [4]. We omit the details here.â€ƒâ–¡

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

Condition (2.32) implies that there exist two constants *Î±*, *Î²*, and a small enough positive constant \(\varepsilon_{4}\), such that

That is,

For the above \(\varepsilon_{4}>0\), from LemmaÂ 2.2 there exists a large enough \(T_{1}\) such that

Let

It follows from (2.34) that

Integrating this inequality from \(T_{4}\) to *t* (\(\geq T_{4}\)), it follows that

From this, similarly to the analysis of (2.15)-(2.16), we have \(x_{1}(t)\rightarrow0\) exponentially as \(t\rightarrow+\infty\). The rest of the proof of TheoremÂ 2.5 is similar to that of the proof of Theorem in [4]. We omit the details here.â€ƒâ–¡

## 3 Numeric example

Now let us consider the following example.

### Example 1

Corresponding to system (1.1), one has

And so,

consequently

hold. Also,

and so

It follows from TheoremÂ 2.1 that the first species of the system (3.1) is globally attractive, and the second species will be driven to extinction; numeric simulations (FiguresÂ 1 and 2) also support these finds.

## 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.

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## 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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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**, 258 (2016). https://doi.org/10.1186/s13662-016-0974-4

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DOI: https://doi.org/10.1186/s13662-016-0974-4