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

- 589 Downloads

## 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).

## References

- 1.Yue, Q: Extinction for a discrete competition system with the effect of toxic substances. Adv. Differ. Equ.
**2016**, 1 (2016). doi: 10.1186/s13662-015-0739-5 MathSciNetCrossRefGoogle Scholar - 2.Chattopadhyay, J: Effect of toxic substances on a two-species competitive system. Ecol. Model.
**84**, 287-289 (1996) CrossRefGoogle Scholar - 3.Li, Z, Chen, FD: Extinction in two dimensional discrete Lotka-Volterra competitive system with the effect of toxic substances. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms
**15**(2), 165-178 (2008) MathSciNetMATHGoogle Scholar - 4.Li, Z, Chen, FD: Extinction in two dimensional nonautonomous Lotka-Volterra systems with the effect of toxic substances. Appl. Math. Comput.
**182**, 684-690 (2006) MathSciNetMATHGoogle Scholar - 5.Li, Z, Chen, FD: Extinction in periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances. J. Comput. Appl. Math.
**231**(1), 143-153 (2009) MathSciNetCrossRefMATHGoogle Scholar - 6.Li, Z, Chen, FD, He, MX: Global stability of a delay differential equations model of plankton allelopathy. Appl. Math. Comput.
**218**(13), 7155-7163 (2012) MathSciNetMATHGoogle Scholar - 7.Li, Z, Chen, FD, He, MX: Asymptotic behavior of the reaction-diffusion model of plankton allelopathy with nonlocal delays. Nonlinear Anal., Real World Appl.
**12**(3), 1748-1758 (2011) MathSciNetCrossRefMATHGoogle Scholar - 8.Chen, F, Gong, X, Chen, W: Extinction in two dimensional discrete Lotka-Volterra competitive system with the effect of toxic substances (II). Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms
**20**, 449-461 (2013) MathSciNetMATHGoogle Scholar - 9.Chen, FD, Xie, XD, Miao, ZS, Pu, LQ: Extinction in two species nonautonomous nonlinear competitive system. Appl. Math. Comput.
**274**(1), 119-124 (2016) MathSciNetGoogle Scholar - 10.Chen, FD: On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay. J. Comput. Appl. Math.
**80**(1), 33-49 (2005) CrossRefMATHGoogle Scholar - 11.Chen, F, Li, Z, Huang, Y: Note on the permanence of a competitive system with infinite delay and feedback controls. Nonlinear Anal., Real World Appl.
**8**, 680-687 (2007) MathSciNetCrossRefMATHGoogle Scholar - 12.Chen, FD, Li, Z, Chen, X, Laitochová, J: Dynamic behaviors of a delay differential equation model of plankton allelopathy. J. Comput. Appl. Math.
**206**(2), 733-754 (2007) MathSciNetCrossRefMATHGoogle Scholar - 13.Chen, L, Chen, F: Extinction in a discrete Lotka-Volterra competitive system with the effect of toxic substances and feedback controls. Int. J. Biomath.
**8**(1), 1550012 (2015) MathSciNetCrossRefMATHGoogle Scholar - 14.Chen, LJ, Sun, JT, Chen, FD, Zhao, L: Extinction in a Lotka-Volterra competitive system with impulse and the effect of toxic substances. Appl. Math. Model.
**40**(3), 2015-2024 (2016) MathSciNetCrossRefGoogle Scholar - 15.Bandyopadhyay, M: Dynamical analysis of a allelopathic phytoplankton model. J. Biol. Syst.
**14**(2), 205-217 (2006) CrossRefMATHGoogle Scholar - 16.Shi, C, Li, Z, Chen, F: Extinction in nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls. Nonlinear Anal., Real World Appl.
**13**(5), 2214-2226 (2012) MathSciNetCrossRefMATHGoogle Scholar - 17.Solé, J, Garca-Ladona, E, Ruardij, P, Estrada, M: Modelling allelopathy among marine algae. Ecol. Model.
**183**, 373-384 (2005) CrossRefGoogle Scholar - 18.He, MX, Chen, FD, Li, Z: Almost periodic solution of an impulsive differential equation model of plankton allelopathy. Nonlinear Anal., Real World Appl.
**11**(4), 2296-2301 (2010) MathSciNetCrossRefMATHGoogle Scholar - 19.Liu, Z, Chen, L: Periodic solution of a two-species competitive system with toxicant and birth pulse. Chaos Solitons Fractals
**32**(5), 1703-1712 (2007) MathSciNetCrossRefMATHGoogle Scholar - 20.Liu, C, Li, Y: Global stability analysis of a nonautonomous stage-structured competitive system with toxic effect and double maturation delays. Abstr. Appl. Anal.
**2014**, Article ID 689573 (2014) MathSciNetGoogle Scholar - 21.Pu, LQ, Xie, XD, Chen, FD, Miao, ZS: Extinction in two-species nonlinear discrete competitive system. Discrete Dyn. Nat. Soc.
**2016**, Article ID 2806405 (2016) MathSciNetCrossRefGoogle Scholar - 22.Wu, RX, Li, L: Extinction of a reaction-diffusion model of plankton allelopathy with nonlocal delays. Commun. Math. Biol. Neurosci.
**2015**, Article ID 8 (2015) Google Scholar - 23.Wang, QL, Liu, ZJ, Li, ZX: Existence and global asymptotic stability of positive almost periodic solutions of a two-species competitive system. Int. J. Biomath.
**7**(4), 1450040 (2014) MathSciNetCrossRefMATHGoogle Scholar - 24.Wang, Q, Liu, Z: Uniformly asymptotic stability of positive almost periodic solutions for a discrete competitive system. J. Appl. Math.
**2013**, Article ID 182158 (2013) MATHGoogle Scholar - 25.Yu, S: Permanence for a discrete competitive system with feedback controls. Commun. Math. Biol. Neurosci.
**2015**, Article ID 16 (2015) Google Scholar - 26.He, MX, Li, Z, Chen, FD: Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses. Nonlinear Anal., Real World Appl.
**11**(3), 1537-1551 (2010) MathSciNetCrossRefMATHGoogle Scholar - 27.Shi, CL, Li, Z, Chen, FD: The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays. Nonlinear Anal., Real World Appl.
**8**(5), 1536-1550 (2007) MathSciNetCrossRefMATHGoogle Scholar - 28.Chen, FD: Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model. Nonlinear Anal., Real World Appl.
**7**(4), 895-915 (2006) MathSciNetCrossRefMATHGoogle Scholar - 29.Chen, FD: Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays. Nonlinear Anal., Real World Appl.
**7**(5), 1205-1222 (2006) MathSciNetCrossRefMATHGoogle Scholar - 30.Yang, K, Miao, ZS, Chen, FD, Xie, XD: Influence of single feedback control variable on an autonomous Holling-II type cooperative system. J. Math. Anal. Appl.
**435**(1), 874-888 (2016) MathSciNetCrossRefMATHGoogle Scholar - 31.Li, Z, Han, MA, Chen, FD: Influence of feedback controls on an autonomous Lotka-Volterra competitive system with infinite delays. Nonlinear Anal., Real World Appl.
**14**(1), 402-413 (2013) MathSciNetCrossRefMATHGoogle Scholar - 32.Chen, LJ, Chen, FD, Wang, YQ: Influence of predator mutual interference and prey refuge on Lotka-Volterra predator-prey dynamics. Commun. Nonlinear Sci. Numer. Simul.
**18**(11), 3174-3180 (2013) MathSciNetCrossRefMATHGoogle Scholar - 33.Chen, LJ, Chen, FD, Chen, LJ: Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge. Nonlinear Anal., Real World Appl.
**11**(1), 246-252 (2010) MathSciNetCrossRefMATHGoogle Scholar - 34.Montes De Oca, F, Vivas, M: Extinction in two dimensional Lotka-Volterra system with infinite delay. Nonlinear Anal., Real World Appl.
**7**(5), 1042-1047 (2006) MathSciNetCrossRefMATHGoogle Scholar - 35.Zhao, JD, Chen, WC: The qualitative analysis of
*N*-species nonlinear prey-competition systems. Appl. Math. Comput.**149**, 567-576 (2004) MathSciNetMATHGoogle Scholar

## Copyright information

**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.