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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 17, Issue 1

Issues

Volume 13 (2015)

Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting

Xiaoyan Huang / Fengde Chen
  • Corresponding author
  • College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350108, P. R. China
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/ Xiangdong Xie / Liang Zhao
  • School of Information and Statistics, Guangxi University of Finance and Economics„, Nanning, Guangxi, 530003, P. R. China
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Published Online: 2019-07-31 | DOI: https://doi.org/10.1515/math-2019-0067

Abstract

The extinction property of a two species competitive stage-structured phytoplankton system with harvesting is studied in this paper. Several sets of sufficient conditions which ensure that one of the components will be driven to extinction are established. Our results supplement and complement the results of Li and Chen [Extinction in periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances, J. Comput. Appl. Math., 2009, 231(1), 143-153] and Liu, Chen, Luo et al. [Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 2002, 274(2), 667-684].

Keywords: extinction; competition; stage-structured; toxicology; harvesting

MSC 2010: 34-XX

1 Introduction

Throughout this paper, for a given function g(t), we let gL and gM denote inf–∞<t<∞ g(t) and sup–∞<t<∞g(t), respectively.

During the last two decades, ecosystem with stage structure become one of the most important research area, and some substantive progress has been made on this direction, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and the references cited therein. For example, Chen et al. [2] showed that stage structure plays important role on the persistent property of the cooperative system. For the system without stage structure, the system always admits a unique positive equilibrium, which means the stable coexistence of the two species. However, if the stage structure is enough large, despite the cooperation between the two species, the species may still be driven to extinction. Xiao et al. [3] investigated the Hopf bifurcation and stability property of a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge. Among those works, many scholars ([1], [6, 7, 8, 9, 10, 11]) done works on the stage structured competitive system. Also, competitive system with the effect of toxic substances is another important research area, many excellent results have been obtained, see [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] and the references cited therein. Li et al. [13] studied the stability property of a competitive system with the effect of toxic substances, they showed that the toxic substance have no influence to the stability property of the system, though it has influence on the position of the equilibrium. Their result is then generalized by Chen et al. [23] to the infinite delay case. Some scholars [17, 24, 29, 35] argued that it is better to describe the relationship between the competitive species by using the nonlinear function, and they obtained some interesting results, such as the extinction of the species, the existence, uniqueness and global stability of the periodic solution, etc.

Based on the traditional two species Lotka-Volterra competitive system, Liu et al. [6] first time proposed the following two-species competitive model with stage structure

x˙1(t)=b1ed1τ1x1(tτ1)a11x12a12x1x2,y˙1(t)=b1x1d1y1b1ed1τ1x1(tτ1),x˙2(t)=b2ed2τ2x2(tτ2)a22x22a21x1x2,y˙2(t)=b2x2d2y2b2ed2τ2x2(tτ2).(1.1)

As as pointed out by Liu et al. [6], to study the dynamic behaviors of the system (1.1), it is enough to study the asymptotic behavior of the following subsystem of system (1.1)

x˙1(t)=b1ed1τ1x1(tτ1)a11x12a12x1x2,x˙2(t)=b2ed2τ2x2(tτ2)a22x22a21x1x2.(1.2)

System (1.2) admits three non-negative equilibria.

E0(0,0),E1(b1ed1τ1a11,0),E2(0,b2ed2τ2a22).(1.3)

Concerned with the stability property of E1 and E2, the authors obtained the following results.

Theorem A

E1 is globally asymptotically stable provided

b1ed1τ1b2ed2τ2>a11a21andb1ed1τ1b2ed2τ2>a12a22.(1.4)

Theorem B

E2 is globally asymptotically stable provided

b1ed1τ1b2ed2τ2<a11a21andb1ed1τ1b2ed2τ2<a12a22.(1.5)

Liu et al. [7] proposed the following n-species nonautonomous stage-structured competitive system,

x˙i(t)=bi(tτi)etτitdi(s)dsxi(tτi)xi(t)j=1nxj(t),y˙i(t)=bi(t)xi(t)di(t)yi(t)bi(tτi)etτitdi(s)dsxi(tτi),(1.6)

where i = 1, 2, …, n, τi are nonnegative constants. bi(t), aij(t), di(t) (i, j = 1, 2, …, n) are all nonnegative continuous and ω-periodic functions. bi(t), aii(t), di(t) > 0 for all t ∈ [0, ω]. Set

Bi(t)=bi(tτi)etτitdi(s)ds.

Then, Liu et al. [7] obtained the following results.

Theorem C

For system (1.6) in the case n = 2, assume

B1L>B2Ma12Ma22LandB1L>B2Ma11Ma21L.(1.7)

Then limt+ x2(t) = limt+ y2(t) = 0.

Theorem D

For system (1.6) in the case n = 2, assume

B2L>B1Ma22Ma12LandB2L>B1Ma21Ma11L.(1.8)

Then limt+x1(t) = limt+y1(t) = 0.

One could easily see that Theorem C and D generalize Theorem A and B to the non-autonomous case.

Based on the works of [5] and [6], Li and Chen [10] proposed the following two species periodic competitive stage-structured system with the effects of toxic substances:

x˙1(t)=b1(tτ1)etτ1tr1(s)dsx1(tτ1)a11(t)x12(t)a12(t)x1(t)x2(t)d1(t)x12(t)x2(t),y˙1(t)=b1(t)x1(t)r1(t)y1(t)b1(tτ1)etτ1tr1(s)dsx1(tτ1),x˙2(t)=b2(tτ2)etτ2tr2(s)dsx2(tτ2)a22(t)x22(t)a21(t)x1(t)x2(t)d2(t)x1(t)x22(t),y˙2(t)=b2(t)x2(t)r2(t)y2(t)b2(tτ2)etτ2tr2(s)dsx2(tτ2),(1.9)

where xi(t) and yi(t) (i = 1, 2) represent the density of mature and immature species at time t > 0, respectively; bi(t), aij(t), ri(t), di(t)(i, j = 1, 2) are all nonnegative continuous and ω-periodic functions. Li and Chen [10] obtained the following result.

Theorem E

If the coefficients of system (1.9) satisfy

b1Ler1Mτ1b2Mer2Lτ2>a12Ma22L,b1Ler1Mτ1b2Mer2Lτ2a11Ma21L,b1Ler1Mτ1b2Mer2Lτ2d1Md2L.(1.10)

Then second species will be driven to extinction while the first one is global attractive to a positive periodic solution of a stage-structured single species system.

Comparing Theorem A, C and E, one could see that the first two inequalities of Theorem E is the same as that of the Theorem C. Noting that the authors of [10] is to investigated the dynamic behaviors of a stage-structured system with toxic substance, hence, one could see that the idea behind that of Theorem E is to assume that the second species in the system without toxic substance is driven to extinction, and to find out the suitable restrictions on the coefficients of toxic substances term, to ensure the second species still be driven to extinction.

Now, one of the interesting issue proposed: What would happen if the first two inequalities in Theorem E hold, while the third inequality does not holds?

To bring some hints on this issue, let’s consider the following example.

Example 1.1

Consider the following equations

x˙1(t)=3e0.2x1(t0.2)(1.5+0.5cos(t))x12(t)(2+sin(t))x1(t)x2(t)0.2x12(t)x2(t),y˙1(t)=3x1(t)y1(t)3e0.2x1(t0.2),x˙2(t)=2e0.2x2(t0.2)(3.5+0.5cos(t))x22(t)2x1(t)x2(t)0.1x1(t)(x2(t))2,y˙2(t)=2x2(t)y2(t)2e0.2x2(t0.2),(1.11)

where τ1 = 0.2, τ2 = 0.2, b1(t) = 4, r1(t) = 1, a11(t) = 1.5 + 0.5 cos(t), a12(t) = 2 + sin(t), d1(t) = 0.2, d2(t) = 0.1, b2(t) = 2, r2(t) = 1, a21(t) = 2, a22(t) = 3.5 + 0.5 cos(t).

One could easily see that

b1Ler1Mτ1b2Mer2Lτ2=3e0.22e0.2=32>33=1=a12Ma22L,b1Ler1Mτ1b2Mer2Lτ2=32>22=1=a11Ma21L,b1Ler1Mτ1b2Mer2Lτ2=32<0.20.1=2=d1Md2L.(1.12)

Inequalities (1.12) show that the coefficients of the system (1.11) satisfies the first two inequalities in (1.10), while the third inequality no longer holds. Numeric simulation (Fig. 1) shows that in this case, species 2 will be driven to extinction while species 1 is globally attractive.

Above example enlighten us to revisit the dynamic behaviors of the system (1.9), and to find out some new sufficient conditions which ensure the extinction of some of the species in system (1.9).

On the other hand, based on the traditional two species competitive system with toxic substance, Kar and Chaudhuri [36] proposed the following non-selective harvesting system

dxdt=r1x(1xk1)α1xyγ1x2yq1Ex,dydt=r2y(1yk2)α2xyγ2xy2q2Ey,(1.13)

where q1, q2 are the catchability coefficients of the two species. The authors gave a thoroughly investigation of the dynamical behaviour about system.

Recently, Gupta et al. [37] made the following assumption: the two species are being harvested by different agencies, both the species are harvested with harvesting efforts E1 and E2, respectively. This leads to the following modeling

dxdt=r1x(1xk1)α1xyγ1x2yq1E1x,dydt=r2y(1yk2)α2xyγ2xy2q2E2y,(1.14)

The authors showed that the system (1.14) may exists two saddle-node bifurcations for different bifurcation parameters.

Now stimulated by the works of [36, 37], it is natural to incorporating the harvesting efforts to system (1.9), here, without loss of generality, we may assume that we only harvest the mature species, and this leads to the following system:

x˙1(t)=b1(tτ1)etτ1tr1(s)dsx1(tτ1)a11(t)x12(t)a12(t)x1(t)x2(t)d1(t)x12(t)x2(t)q1(t)E1(t)x1(t),y˙1(t)=b1(t)x1(t)r1(t)y1(t)b1(tτ1)etτ1tr1(s)dsx1(tτ1),x˙2(t)=b2(tτ2)etτ2tr2(s)dsx2(tτ2)a22(t)x22(t)a21(t)x1(t)x2(t)d2(t)x1(t)x22(t)q2(t)E2(t)x2(t),y˙2(t)=b2(t)x2(t)r2(t)y2(t)b2(tτ2)etτ2tr2(s)dsx2(tτ2),(1.15)

where xi(t) and yi(t) (i = 1, 2) represent the density of mature and immature species at time t > 0, respectively; bi(t), aij(t), ri(t), di(t), qi(t), Ei(t)(i, j = 1, 2) are all nonnegative continuous and ω-periodic functions.

Already, there are many scholars investigated the extinction property of the competitive system with toxic substance, see [12, 13, 17, 18, 19, 20, 21, 22, 23, 24], however, all of those works did not consider the influence of harvesting.

The aim of this paper is, by further developing the analysis technique of Li and Chen [10], Chen et al. [35] and Montes De Oca and Vivas [32], to investigate the extinction property of the system (1.15).

The initial conditions for system (1.15) take the form

xi(θ)=ϕi(θ)>0,yi(θ)=ψi(θ)>0,τθ0,i=1,2,(1.16)

where τ = max{τ1, τ2}. For the continuity of the solutions of system (1.15), in this paper, we always assume

yi(0)=ψi(0)=τi0bi(s)ϕi(s)es0ri(u)duds,i=1,2.(1.17)

The organization of this paper is as follows. In Section 2, we introduce some useful lemmas. In Section 3, we study the extinction property of system (1.15). In Section 4, several numeric examples are carried out to illustrate the feasibility of the main results. We end this paper by a briefly discussion.

2 Preliminaries

Now let us state several lemmas which will be useful in the proof of our main results.

Lemma 2.1

Solutions of system (1.15) with initial conditions (1.16) and (1.17) are positive for all t > 0.

Proof

The proof of Lemma 2.1 is similar to that of Lemma 3.1 [5], and we omit the detail proof here.

Lemma 2.2

[7] Consider the following equations:

x(t)=bx(tδ)a1x(t)a2x2(t),x(t)=ϕ(t)>0,δt0,

and assume that b, a2 > 0, a1 ≥ 0 and δ ≥ 0 are constants, then:

  1. If ba1, then limt+x(t)=ba1a2;

  2. If ba1, then limt+ x(t) = 0.

Lemma 2.3

Let (x1(t), y1(t), x2(t), y2(t))T be any solution of system (1.15) with initial conditions (1.16) and (1.17). Then for i = 1, 2

lim supt+xi(t)Mi,lim supt+yi(t)Ni,(2.1)

where

Mi=biMeriLτiqiLEiLaiiL,Ni=biMMiriL(1eriMτi).(2.2)

Proof

It follows from the first or third equation of system (1.15) that

x˙i(t)bi(tτ1)etτitri(s)dsxi(tτi)aii(t)xi2(t)qi(t)Ei(t)xi(t)biMeriLτixi(tτi)aiiLxi2(t)qiLEiLxi(t).

Consider the following equation

u˙i(t)=biMeriLτiui(tτi)aiiLui2(t)qiLEiLui(t)

with ui(t) = xi(t)(–τt ≤ 0), i = 1, 2. By Lemma 2.2, limt+ui(t)=biMeriLτiqiLEiLaiiL, and so,

lim supt+xi(t)biMeriLτiqiLEiLaiiL=defMi,i=1,2.(2.3)

The rest of the proof is similar to that of the proof of Lemma 2.3 in [10], and we omit the detail here.

Remark 2.1

If biMeriLτiqiLEiL,i=1,2, then limt+ xi(t) = 0, and consequently, limt+ yi(t) = 0, i = 1, 2. That is, overfishing will lead to the extinction of both species.

Lemma 2.4

[32] (Fluctuation lemma) Let x(t) be a bounded differentiable function on (α, ∞), then there exist sequences τn → ∞, σn → ∞ such that

(a)x˙(τn)0andx(τn)lim suptx(t)=x¯asn,(b)x˙(σn)0andx(σn)lim inftx(t)=x_asn.

3 Main results

As indicated by the Remark 2.1, overfishing will leads to the extinction of both species, hence, from now on, we make the following assumption:

biLeriMτi>qiMEiM,i=1,2.(3.1)

Before stating the main results of this section, we introduce a set of conditions

b1Ler1Mτ1q1ME1Mb2Mer2Lτ2q2LE2L>max{a12Ma22L,a11Ma21L,d1Md2L}(3.2)

b1Ler1Mτ1q1ME1Mb2Mer2Lτ2q2LE2L>max{a12Ma22L,a11M+d1MM2a21L}(3.3)

b1Ler1Mτ1q1ME1Mb2Mer2Lτ2q2LE2L>max{a12M+d1MM1a22L,a11Ma21L}(3.4)

b2Ler2Mτ2q2ME2Mb1Mer1Lτ1q1LE1L>max{a21Ma11L,a22Ma12L,d2Md1L}(3.5)

b2Ler2Mτ2q2ME2Mb1Mer1Lτ1q1LE1L>max{a21Ma11L,a22M+d2MM1a12L}(3.6)

b2Ler2Mτ2q2ME2Mb1Mer1Lτ1q1LE1L>max{a21M+d2MM2a11L,a22Ma12L}(3.7)

where Mi, i = 1, 2 are defined by (2.2).

Before we begin to prove the main results, we need several Lemmas again.

Lemma 3.1

Let (x1(t), y1(t), x2(t), y2(t))T be any solution of system (1.15) with initial conditions (1.16) and (1.17). Assume that (3.2) or (3.3) or (3.4) holds, then there exists a α > 0 such that x1(t) ≥ α for all t ≥ 0.

Proof

We first show that the conclusion of Lemma 3.1 holds under the assumption (3.2). It follows from Lemma 2.3 that lim supt+x2(t)b2Mer2Lτ1q2LE2La22L. Given ε=12(b1Ler1Mτ1q1ME1Ma12Mb2Mer2Lτ2q2ME2Ma22L), there exists a T ≥ 0 such that for all tT

x2(t)b2Mer2Lτ1q2LE2La22L+ε=12(b1Ler1Mτ1q1ME1Ma12M+b2Mer2Lτ2q2ME2Ma22L).

So, for all tT, from the first equation of system (1.15), it follows that

x1˙(t)b1Ler1Mτ1x1(tτ1)a11Mx12(t)a12Mx1(t)x2(t)d1Mx12(t)x2(t)q1ME1Mx1(t)b1Ler1Mτ1x1(tτ1)(a11M+d1M2(b1Ler1Mτ1q1ME1Ma12M+b2Mer2Lτ2q2ME2Ma22L))x12(t)(12(b1Ler1Mτ1q1ME1M+a12Mb2Mer2Lτ2q2ME2Ma22L)+q1ME1M)x1(t)=defAx1(tτ1)Bx1(t)Cx12(t).

Let u(t) be a solution of the following equation

u˙(t)=Au(tτ1)Bu(t)Cu2(t),

with u(T + τ1) = x1(T + τ1). It follows from condition (3.1) that

AB=12(b1Ler1Mτ1q1ME1Ma12Mb2Mer2Lτ2q2ME2Ma22L)>0.

From Lemma 2.2

limt+u(t)=ABC=α1>0.

Therefore, we obtain

x_1=lim inft+x1(t)α1>0.

Given ε=12α1, there exists a T1T such that

x1(t)x_1α12α112α1=α12,tT1.

Let α2 = min{x1(t) : 0 ≤ tT1} > 0 and α=min{α12,α2}>0. It follows that x1(t) ≥ α > 0 for all t ≥ 0.

Noting that above proof only use the fact

b1Ler1Mτ1q1ME1Ma12M>b2Mer2Lτ2q2ME2Ma22L.

Condition (3.3) and (3.4) all implies this inequality holds, hence, under the assumption of (3.2) or (3.3) or (3.4), the conclusion of Lemma 3.1 holds. This ends the proof of Lemma 3.1.

Our main results are the following Theorems.

Theorem 3.1

Assume that (3.2) holds. Then

m1lim inft+x1(t)lim supt+x1(t)M1.n1lim inft+y1(t)lim supt+y1(t)N1.limt+x2(t)=0,limt+y2(t)=0,

where

m1=b1Ler1Mτ1q1ME1Ma11M,n1=b1Lm1r1M(1er1Lτ1).

Theorem 3.2

Assume that (3.3) holds. Then the conclusions of Theorem 3.1 hold.

Theorem 3.3

Assume that (3.4) holds. Then the conclusions of Theorem 3.1 hold.

Noting that system (1.9) is the special case of system (1.15), (qi(t) ≡ 0, Ei(t) ≡ 0, i = 1, 2) Then as a direct corollary of Theorem 3.1, we have

Corollary 3.1

Assume that in system (1.9)

b1Ler1Mτ1b2Mer2Lτ2>max{a12Ma22L,a11Ma21L,d1Md2L}

hold. Then

m1lim inft+x1(t)lim supt+x1(t)M1.n1lim inft+y1(t)lim supt+y1(t)N1.limt+x2(t)=0,limt+y2(t)=0,

where

m1=b1Ler1Mτ1a11M,n1=b1Lm1r1M(1er1Lτ1).

Remark 3.1

Corollary 3.1 is Theorem 3.1 of Li and Chen [10], hence we generalize the main result of [10].

As a direct corollary of Theorem 3.2 and 3.3, we have

Corollary 3.2

Assume that in system (1.9)

b1Ler1Mτ1b2Mer2Lτ2>max{a12Ma22L,a11M+d1MM11a21L}

hold, where M11=b1Mer1Lτ1a11L. Then the conclusion of Corollary 3.1 holds.

Corollary 3.3

Assume that in system (1.9)

b1Ler1Mτ1b2Mer2Lτ2>max{a12M+d1MM22a22L,a11Ma21L}(3.8)

hold, where M22=b2Mer2Lτ1a22L. Then the conclusion of Corollary 3.1 holds.

Remark 3.2

As was showed in Example 1.1, though the conditions of Theorem 3.1 in [10] are not satisfied, the second species still be possible of driving to extinction. Corollary 3.2 and 3.3 are two set of new sufficient conditions which ensure the extinction of the second species, hence, Corollary 3.2 and 3.3 supplement and complement the main results of [10].

Proof of Theorem 3.1

It follows from Lemma 2.1 and 2.3 that xi(t), i = 1, 2 are bounded and positive for all t ≥ 0. Let x_1=lim inft+x1(t)andx¯2=lim supt+x2(t). For above ε1 > 0, it follows from Lemma 2.3 that

x_1<M1+ε1,x¯2<M2+ε1.(3.9)

From Lemma 3.1 we know that x1α > 0. Obviously, x2 ≥ 0. To prove limt+ x2(t) = 0, it suffices to show that x2 = 0. In order to get a contradiction, we suppose that x2 > 0. According to the Fluctuation lemma (Lemma 2.4), there exist sequences γn → +∞, σn → +∞ such that x1(γn)0,x2(σn) → 0, x1(γn) → x1 and x2(σn) → x2 as n → +∞. It follows from the first equation of system (1.15) that

x1˙(γn)=b1(γnτ1)eγnτ1γnr1(s)dsx1(γnτ1)a11(γn)x12(γn)a12(γn)x1(γn)x2(γn)d1(γn)x12(γn)x2(γn)q1(γn)E1(γn)x1(γn)b1Ler1Mτ1inftγnτ1x1(t)a11Mx12(γn)a12Mx1(γn)suptγnx2(t)d1Mx12(γn)suptγnx2(t)q1ME1Mx1(γn).(3.10)

By taking the limit of the above inequality as n → +∞, we obtain the inequality

b1Ler1Mτ1q1ME1Ma11Mx_1+a12Mx¯2+d1Mx_1x¯2.(3.11)

From the third equation of system (1.15), by a similar argument as above, we obtain

b2Mer2Lτ2q2LE2La21Lx_1+a22Lx¯2+d2Lx_1x¯2.(3.12)

(3.11) is equivalent to

1a11Mb1Ler1Mτ1q1ME1Mx_1+a12Mb1Ler1Mτ1q1ME1Mx¯2+d1Mb1Ler1Mτ1q1ME1Mx_1x¯2.(3.13)

(3.12) is equivalent to

1a21Lb2Mer2Lτ2q2LE2Lx_1+a22Lb2Mer2Lτ2q2LE2Lx¯2+d2Lb2Mer2Lτ2q2LE2Lx_1x¯2.(3.14)

(3.13) together with (3.13) leads to

A1x_1+A2x¯2+A3x_1x¯20,(3.15)

where

A1=a11Mb1Ler1Mτ1q1ME1Ma21Lb2Mer2Lτ2q2LE2L,A2=a12Mb1Ler1Mτ1q1ME1Ma22Lb2Mer2Lτ2q2LE2L,A3=d1Mb1Ler1Mτ1q1ME1Md2Lb2Mer2Lτ2q2LE2L.

It follows from (3.2) that Ai < 0, i = 1, 2, 3, this together with the fact x1 > 0, x2 > 0 leads to

A1x_1+A2x¯2+A3x_1x¯2<0,(3.16)

which is contradiction with (3.15). Then we obtain limt+ x2(t) = 0. Since

y2(t)=tτ2tb2(s)x2(s)etsr2(u)duds,

it immediately follows that

limt+y2(t)=0.

Above analysis shows that for 0 < ε < (b1Ler1Mτ1q1ME1M)a12M, there exists a T1 > 0, such that for all tT1, y2(t) < ε. Lemma 2.3 had showed that

lim supt+x1(t)M1,lim supt+y1(t)N1.

To end the proof of Theorem 3.1, it’s enough to show that

lim inft+x1(t)m1,lim inft+y1(t)n1.

For tT1 + τ, from the first equation of system (1.15), we have

x˙1(t)b1Ler1Mτ1x1(tτ1)(a11M+d1Mε)x12(t)(a12Mε+q1ME1M)x1(t).(3.17)

Let u(t) be the solution of the equation

u˙=b1Ler1Mτ1u1(tτ1)(a11M+d1Mε)u12(t)(a12Mε+q1ME1M)u(t)

with u(T1 + τ) = x1(T1 + τ). It follows from Lemma 2.2 that

limt+u(t)=b1Ler1Mτ1(a12Mε+q1ME1M)a11M+d1Mε.

Therefore, we have

lim inft+x1(t)b1Ler1Mτ1(a12Mε+q1ME1M)a11M+d1Mε.

Setting ε → 0, it follows that

lim inft+x1(t)b1Ler1Mτ1q1ME1Ma11Mmdef1.

Noting that

y1(t)=tτ2tb1(s)x1(s)etsr1(u)duds.

From this, one could easily obtain

lim inft+y1(t)b1Lm1r1M(1er1Lτ1).

The proof of Theorem 3.1 is completed.

Proof of Theorem 3.2

Let (x1(t), y1(t), x2(t), y2(t))T be any solution of system (1.15) with initial conditions (1.16) and (1.17). It follows from (3.3) that there exists a ε2 > 0 enough small, such that

b1Ler1Mτ1q1ME1Mb2Mer2Lτ2q2LE2L>max{a12Ma22L,a11M+d1M(M2+ε2)a21L}.(3.18)

Let x1and x2 be defined as that of Lemma 3.3. For above ε2 > 0, it follows from Lemma 2.3 that

x_1<M1+ε,x¯2<M2+ε2.(3.19)

From Lemma 3.1 we know that x1α > 0. Obviously, x2 ≥ 0. To prove limt+ x2(t) = 0, it suffices to show that x2 = 0. In order to get a contradiction, we suppose that x2 > 0. Already, by using the Fluctuation lemma, we had established the inequalities (3.11) and (3.12). Now, from (3.11) and (3.19), we have

b1Ler1Mτ1q1ME1M(a11M+d1M(M2+ε))x_1+a12Mx¯2,(3.20)

which is equivalent to

1a11M+d1M(M2+ε)b1Ler1Mτ1q1ME1Mx_1+a12Mb1Ler1Mτ1q1ME1Mx¯2.(3.21)

Also, it follows from (3.12) that

1a21Lb2Mer2Lτ2q2LE2Lx_1+a22Lb2Mer2Lτ2q2LE2Lx¯2.(3.22)

(3.21) combine with (3.22) leads to

B1x_1+B2x¯20,(3.23)

where

B1=a11M+d1M(M2+ε)b1Ler1Mτ1q1ME1Ma21Lb2Mer2Lτ2q2LE2L.

B2=a12Mb1Ler1Mτ1q1ME1Ma22Lb2Mer2Lτ2q2LE2L.

Condition (3.18) implies that Bi < 0, i = 1, 2. This together with the fact x1 > 0, x2 > 0 leads to

B1x_1+B2x¯2<0,(3.24)

which is contradiction with (3.23). Then we obtain limt+ x2(t) = 0. The rest of the proof is similar to that of the proof of Theorem 3.1, and we omit the detail here.

Proof of Theorem 3.3

Let (x1(t), y1(t), x2(t), y2(t))T be any solution of system (1.15) with initial conditions (1.16) and (1.17). It follows from (3.4) that there exists a ε3 > 0 enough small, such that

b1Ler1Mτ1q1ME1Mb2Mer2Lτ2q2LE2L>max{a12M+d1M(M1+ε3)a22L,a11Ma21L}.(3.25)

Let x1 and x2 be defined as that of Lemma 3.3. For above ε3 > 0, it follows from Lemma 2.3 that

x_1<M1+ε3,x¯2<M2+ε3.(3.26)

From Lemma 3.1 we know that x1α > 0. Obviously, x2 ≥ 0. To prove limt+ x2(t) = 0, it suffices to show that x2 = 0. In order to get a contradiction, we suppose that x2 > 0. Already, by using the Fluctuation lemma, we had established the inequalities (3.11) and (3.12). Now, from (3.11) and (3.26), we have

b1Ler1Mτ1q1ME1Ma11Mx_1+(a12M+d1M(M1+ε3))x¯2,(3.27)

which is equivalent to

1a11Mb1Ler1Mτ1q1ME1Mx_1+a12M+d1M(M1+ε3)b1Ler1Mτ1q1ME1Mx¯2.(3.28)

Also, it follows from (3.12) that

1a21Lb2Mer2Lτ2q2LE2Lx_1+a22Lb2Mer2Lτ2q2LE2Lx¯2.(3.29)

(3.28) combine with (3.29) leads to

C1x_1+C2x¯20,(3.30)

where

C1=a11Mb1Ler1Mτ1q1ME1Ma21Lb2Mer2Lτ2q2LE2L.C2=a12M+d1M(M1+ε3)b1Ler1Mτ1q1ME1Ma22Lb2Mer2Lτ2q2LE2L.

Condition (3.25) implies that Ci < 0, i = 1, 2. This together with the fact x1 > 0, x2 > 0 leads to

C1x_1+C2x¯2<0,(3.31)

which is contradiction with (3.30). Then we obtain limt+ x2(t) = 0. The rest of the proof is similar to that of the proof of Theorem 3.1, and we omit the detail here.

Concerned with the extinction of the first species, we have the following result.

Theorem 3.4

Assume that (3.5) or (3.6) or (3.7) hold. Then

m2lim inft+x2(t)lim supt+x2(t)M2.n2lim inft+y2(t)lim supt+y2(t)N2.limt+x1(t)=0,limt+y1(t)=0,

where

m2=b2Ler2Mτ2q2ME2Ma22M,n2=b2Lm2r2M(1er2Lτ2).

Since the proof of Theorem 3.4 is similar to that of Theorems 3.1-3.3, we omit the detail here.

As a direct corollary of Theorem 2.4, we have

Corollary 3.4

Assume that in system (1.9), one of the following three inequalities holds.

b2Ler2Mτ2b1Mer1Lτ1>max{a21Ma11L,a22Ma12L,d2Md1L},b2Ler2Mτ2b1Mer1Lτ1>max{a21Ma11L,a22M+d2MM22a12L},b12Ler2Mτ2b1Mer1Lτ1>max{a21M+d2MM11a11L,a22Ma12L},

where Mii=biMeriLτiaiiL, i = 1, 2. Then

m2lim inft+x2(t)lim supt+x2(t)M2.n2lim inft+y2(t)lim supt+y2(t)N2.limt+x1(t)=0,limt+y1(t)=0,

where

m2=b2Ler2Mτ2a22M,n2=b2Lm2r2M(1er2Lτ2).

4 Examples

In this section we shall give two examples to illustrate the feasibility of main results in the previous section.

Example 4.1

Consider Example 1.1 in the introduction Section. Already, we had verified

b1Ler1Mτ1b2Mer2Lτ2=32>1=a12Ma22L.(4.1)

Noting that

M11=b1Mer1Lτ1a11L=3e0.21.

Thus,

a11M+d1MM1a21L=2+0.2×3e0.22<2+0.62<32=b1Ler1Mτ1b2Mer2Lτ2.(4.2)

(4.1) together with (4.2) shows that all the conditions of Corollary 3.2 are hold, and so, the second species will be driven to extinction.

Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.1) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (1.2, 0.4, 0.8, 0.2)T, (1.8, 0.6, 0.5, 0.2) and (0.8, 1.4, 1.5, 2)T, θ ∈ [−0.2, 0], respectively.
Figure 1

Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.1) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (1.2, 0.4, 0.8, 0.2)T, (1.8, 0.6, 0.5, 0.2) and (0.8, 1.4, 1.5, 2)T, θ ∈ [−0.2, 0], respectively.

Example 4.2

Now let’s further incorporate the harvesting effort to system (1.11), this leads to the following system

x˙1(t)=3e0.2x1(t0.2)(1.5+0.5cos(t))x12(t)(2+sin(t))x1(t)x2(t)0.2x12(t)x2(t)q1(t)E1(t)x1,y˙1(t)=3x1(t)y1(t)3e0.2x1(t0.2),x˙2(t)=2e0.2x2(t0.2)(3.5+0.5cos(t))x22(t)2x1(t)x2(t)0.1x1(t)(x2(t))2q2(t)E2(t)x2,y˙2(t)=2x2(t)y2(t)2e0.2x2(t0.2),(4.3)

where τ1 = 0.2, τ2 = 0.2, b1(t) = 4, r1(t) = 1, a11(t) = 1.5 + 0.5 cos(t), a12(t) = 2 + sin(t), d1(t) = 0.2, d2(t) = 0.1, b2(t) = 2, r2(t) = 1, a21(t) = 2, a22(t) = 3.5 + 0.5 cos(t).

  1. Take q1(t)E1(t) = 3, q2(t)E2(t) = 2, in this case, biMeriLτi<qiLEiL,i = 1, 2 holds, and so, from Remark 2.1, this is overfishing case, and all the species will be driven to extinction. Fig 2. support this assertion.

    Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (1.2, 0.4, 0.8, 0.2)T, θ ∈ [−0.2, 0], where q1E1 = 3, q2E2 = 2. respectively.
    Figure 2

    Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (1.2, 0.4, 0.8, 0.2)T, θ ∈ [−0.2, 0], where q1E1 = 3, q2E2 = 2. respectively.

  2. Take q1(t)E1(t) = 0.2e−0.2, q2(t)E2(t) = 0, in this case, there are no harvest on the second species, also, the harvesting of the first species is restrict to a limited case. b1Ler1Mτ1q1ME1Mb2Mer2Lτ2=2.82,

    M11=b1Mer1Lτ1q1ME1Ma11L=2.8e0.21.

    Thus,

    a11M+d1MM1a21L=2+0.2×2.8e0.22<2+0.62<2.82=b1Ler1Mτ1q1ME1Mb2Mer2Lτ2.(4.4)

    a12Ma22L=1<2.82=b1Ler1Mτ1q1ME1Mb2Mer2Lτ2.(4.5)

    (4.4) and (4.5) show that all the conditions of Corollary 3.2 are hold, then second species will be driven to extinction. Fig. 3 also support this assertion.

    Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (1.2, 0.4, 0.8, 0.2)T, (1.8, 0.6, 0.5, 0.2) and (0.8, 1.4, 1.5, 2)T, θ ∈ [−0.2, 0], respectively. Here we take q1E1 = 0.2e−0.2, q2E2 = 0.
    Figure 3

    Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (1.2, 0.4, 0.8, 0.2)T, (1.8, 0.6, 0.5, 0.2) and (0.8, 1.4, 1.5, 2)T, θ ∈ [−0.2, 0], respectively. Here we take q1E1 = 0.2e−0.2, q2E2 = 0.

  3. Take q1(t)E1(t) = 2.6e−0.2, q2(t)E2(t) = 0, in this case

    b2Ler2Mτ2b1Mer1Lτ1q1LE1L=2e0.23e0.22.6e0.2=5.a21Ma11L=2,a22Ma12L=4,d2Md1L=12.

    Hence

    b2Ler2Mτ2b1Mer1Lτ1q1LE1L>max{a21Ma11L,a22Ma12L,d2Md1L}.

    That is, inequality (3.5) holds, from Theorem 2.4, the second species will be driven to extinction. Fig. 4 also support this assertion.

    Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (0.8, 0.4, 0.5, 0.4), θ ∈ [−0.2, 0], respectively. Here we take q1E1 = 2.6e−0.2, q2E2 = 0.
    Figure 4

    Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (0.8, 0.4, 0.5, 0.4), θ ∈ [−0.2, 0], respectively. Here we take q1E1 = 2.6e−0.2, q2E2 = 0.

  4. Take q1(t)E1(t) = 3, q2(t)E2(t) = 0, in this case, the first species is overfishing, while the second one is free of harvesting. From Remark 2.1, the first species will be driven to extinction. Due to the extinction of the first species, the second one will be permanent. Fig.5 also support this assertion.

    Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (1.8, 0.6, 0.5, 0.2) and (0.8, 1.4, 1.5, 2)T, θ ∈ [−0.2, 0], respectively. Here we take q1E1 = 3, q2E2 = 0.
    Figure 5

    Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (1.8, 0.6, 0.5, 0.2) and (0.8, 1.4, 1.5, 2)T, θ ∈ [−0.2, 0], respectively. Here we take q1E1 = 3, q2E2 = 0.

  5. Take q1(t)E1(t) = 1.5, q2(t)E2(t) = 0. Numeric simulation (Fig. 6) shows that in this case, two species could be coexist in a stable state.

    small Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (0.2, 0.6, 0.5, 0.2) and (0.8, 0.4, 0.5, 0.4)T, θ ∈ [−0.2, 0], respectively. Here we take q1E1 = 1.5, q2E2 = 0.
    Figure 6

    small Dynamics behaviors of the solution (x1(t), y1(t), x2(t), y2(t))T of system (4.3) with initial condition (φ1(θ), φ2(θ), ψ1(θ), ψ2(θ))T = (0.2, 0.6, 0.5, 0.2) and (0.8, 0.4, 0.5, 0.4)T, θ ∈ [−0.2, 0], respectively. Here we take q1E1 = 1.5, q2E2 = 0.

5 Discusion

Li and Chen [10] proposed a two species periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances, they studied the extinction property of the system. It is naturally to investigate the dynamic behaviours of system (1.9) if the conditions in [10] no longer hold, Example 1.1 in the introduction Section shows that some of the species still could be driven to extinction, this motivated us to revisit the extinction property of the system (1.9). On the other hand, Kar and Chaudhuri [36] and Gupta, Banerjee and Chandra [37] studied the influence of harvesting effect on the competition system with toxic substance. Their success motivated us to propose a two species competitive stage-structured system with the effect of toxic substance and harvesting (system (1.15)). We first show that due to the overfishing, two of the species will be driven to extinction (Remark 2.1). After that, for the appropriate harvesting case, by applying the fluctuation theorem, we are able to establish sufficient conditions which ensure one of the components be driven to extinction.

Theorem 3.1 can be seen as the generalization of Theorem 3.1 in [10], thus, we generalize the main result of [10] to the harvesting case. Theorem 3.2-3.4 are new results, which supplement and complement the main results of [6] and [10].

To show the feasibility of our main results, we study a numeric example (Example 4.2), here we make an assumption that we only harvest the first species, and if q1(t)E1(t) = 0, that is, without the capture of the first species, the second species will be driven to extinction. Then, depending on the harvesting effect q1(t)E1(t), the system may have the following dynamic behaviors: (1) the second species still be driven to extinction (case (2)); (2) the first species will be driven to extinction (cases (3) and (4)); (3) two species could be coexist in a stable state (case (5)).

Our results and numeric examples show that harvesting is one of the most important factors to influence the dynamic behaviours of the system.

Acknowledgment

The author would like to thank Dr. Rongyu Han for bringing our attention to the paper of Kant and Kumar.

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About the article

Received: 2019-02-06

Accepted: 2019-03-03

Published Online: 2019-07-31


Competing interestsThe authors declare that there is no conflict of interests.

FundingThe research was supported by the National Natural Science Foundation of China under Grant(11601085).

Authors’ ContributionsAll authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.


Citation Information: Open Mathematics, Volume 17, Issue 1, Pages 856–873, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2019-0067.

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© 2019 Huang et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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