Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances

Fengde Chen; Xiaoxing Chen; Shouying Huang

doi:10.1515/math-2016-0099

Article Open Access

Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances

Fengde Chen , Xiaoxing Chen and Shouying Huang

Published/Copyright: December 30, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

A two species non-autonomous competitive phytoplankton system with Beddington-DeAngelis functional response and the effect of toxic substances is proposed and studied in this paper. Sufficient conditions which guarantee the extinction of a species and global attractivity of the other one are obtained. The results obtained here generalize the main results of Li and Chen [Extinction in two dimensional nonautonomous Lotka-Volterra systems with the effect of toxic substances, Appl. Math. Comput. 182(2006)684-690]. Numeric simulations are carried out to show the feasibility of our results.

Keywords: Extinction; Competition; Functional response; Phytoplankton system

MSC 2010: 34D23; 92D25; 34D20; 34D40

1 Introduction

Given a function g(t), let g_L and g_M denote inf_{–∞<t<g(t)}and sup_–∞<t<∞g(t), (0, respectively.

The aim of this paper is to investigate the extinction property of the following two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances

(1)

$x˙1(t)=x1(t)r1(t)−a1(t)x1(t)−b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−c1(t)x1(t)x2(t),x˙2(t)=x2(t)r2(t)−b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t)−a2(t)x2(t)−c2(t)x1(t)x2(t),$

where r_i(t), a_i(t), b_i(t), d_i(t), i = l, 2, c_i(t) are assumed to be continuous and bounded above and below by positive constants, e_i(t), f_i(t),i = 1,2 are all non-negative continuous functions bounded above by positive constants. x₁(t), x₂(t)are population density of species x₁and x₂ 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. Here we make the following assumptions:

(1) The interspecific competition between two species takes the Beddington-DeAngelis functional response type

$b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t),b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t),$

respectively;

(2) The terms c₁(t) x₁(t) x₂(t)and c₂x₁ (t) x₂(t) denote the effect of toxic substances, each species produces a substance toxic to the other, only when the other is present.

We also consider the extinction property of the following two species non-autonomous competitive phytoplankton system with Beddington-DeAngelis functional response

(2)

$x˙1(t)=x1(t)r1(t)−a1(t)x1(t)−b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−c1(t)x1(t)x2(t),x˙2(t)=x2(t)r2(t)−b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t)−a2(t)x2(t),$

where all the coefficients have the same meaning as that of system (1). However, we assume that the second species could produce toxic, while the first one is non-toxic produce.

Traditional two species Lotka-Volterra competition model takes the form:

(3)

$x˙1(t)=x1(t)r1−a1x1(t)−b1x2(t),x˙2(t)=x2(t)r2−a2x2(t)−b2x1(t),$

where r_i ,a_i,b_i,i = 1,2 are all positive constants, and x₁(t), x₂(t)are population density of species x₁and x₂ at time t, respectively. r_i, 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. This model is the foundation stone in the study of competition model. Depending on the relationship of the coefficients, the system could have three different kinds of dynamics: (1) a unique positive equilibrium which is globally attractive; (2) bistable; the positive equilibrium is unstable, and the stability of the boundary equilibrium is dependent on the initial conditions; (3) the boundary equilibrium is globally stable, which means the extinction of the partial species.

Based on the Lotka-Volterra model (3), Chattopadhyay [2] proposed 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:

(4)

$x˙1(t)=x1(t)r1−a1x1(t)−a2x2(t)−d1x1(t)x2(t),x˙2(t)=x2(t)r2−b1x2(t)−b2x2(t)−d2x1(t)x2(t).$

By constructing some suitable Lyapunov function, he obtained sufficient conditions which ensure the global stability of the unique positive equilibrium. By using the iterative method, Li and Chen [6] showed that if the system without toxic substance admits the unique positive equilibrium, then system (4) also admits a unique positive equilibrium, in this case, the toxic substance term has no influence on the stability of the positive equilibrium.

Li and Chen [4] argued that with the change of the circumstance, the coefficients of the system should be time-varying, and they studied the nonautonomous case of system (4), i.e.,

(5)

$x˙1(t)=x1(t)r1(t)−a1(t)x1(t)−b2(t)x2(t)−c1(t)x1(t)x2(t),x˙2(t)=x2(t)r2(t)−b2(t)x2(t)−a2(t)x2(t)−c2(t)x1(t)x2(t).$

By applying the fluctuation theorem, they obtained a set of sufficient conditions which guarantee the extinction of the second species and the globally attractive of the first species.

Solé et al [16] and Bandyopadhyay [14] considered a Lotka-Volterra type of model for two interacting phytoplankton species, where one species could produce toxic, while the other one is non-toxic produce. The model takes the form

(6)

$x˙1(t)=x1(t)r1−a1x1(t)−a2x2(t)−d1x1(t)x2(t),x˙2(t)=x2(t)r2−b1x2(t)−b2x2(t).$

By constructing some suitable Lyapunov function, Bandyopadhyay [14] obtained a set of sufficient conditions which ensure the global attractivity of the positive equilibrium. For more work on competitive system with toxic substance, one could refer to [1-20, 35-38] and the references cited therein.

On the other hand, based on the traditional Lotka-Volterra competition model, some scholars argued that the more appropriate competition model should with nonlinear inter-inhibition terms. Wang, Liu and Li [23] proposed the following two species competition model,

(7)

$x˙1(t)=x1(t)r1(t)−a1(t)x1(t)−b1(t)x2(t)1+x2(t),x˙2(t)=x2(t)r2(t)−b2(t)x1(t)1+x1(t)−a2(t)x2(t).$

In this system, the inter-inhibition terms take the form $b1tx2t1+x2tandb2tx1t1+x1t,$ respectively, which is of Holling II type. By using 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 of system (7).

Corresponding to system (7), several scholars [24, 25] investigated the dynamic behaviors of the discrete type two species competition system with nonlinear inter-inhibition terms.

(8)

$x1(k+1)=x1(k)expr1(k)−a1(k)x1(k)−b1(k)x2(k)1+x2(k),x2(k+1)=x2(k)expr2(k)−b2(k)x1(k)1+x1(k)−a2(k)x2(k).$

Wang and Liu [24] studied the almost-periodic solution of the system (8) and Yu [25] further incorporated the feedback control variables to the system (8) and investigated the persistent property of the system.

Recently, combining with the effect of toxic substance and the nonlinear inter-inhibition term, Yue [1] proposed the following two species discrete competitive system

(9)

$x1(k+1)=x1(k)expr1(k)−a1(k)x1(k)−b1(k)x2(k)1+x2(k)−c1(k)x1(k)x2(k),x2(k+1)=x2(k)expr2(k)−b2(k)x1(k)1+x1(k)−a2(k)x2(k).$

By constructing some suitable Lyapunov type extinction function, 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 the functional response plays important role in the predator-prey model, and during the past two decades, the Beddington-DeAngelis functional response, which is a combination of the famous Holling II functional response and ratio-dependent functional response, having overcome the defect of the both functional response, is studied by many scholars, see [26-30] and the references therein.

The success of [26-30] motivated us to propose the competition system with Beddington-DeAngelis functional response, also, if we further assume that each species produces a substance toxic to the other only when the other is present, or assume that one species is toxic produce while the other one is non-toxic producing, then, we could establish the model (1) and (2), respectively. It is, to the best of the knowledge of the authors, the first time such kind of model proposed. During the last decade, many scholars ([3-5], [8], [11-13], [31-38]) investigated the extinction property of the competition system. In this paper, we still focus our attention to the extinction property of the system (1) and (2).

The aim of this paper is, by developing the analysis technique of [1, 8, 9], to investigate the extinction property of the system (1) and (2). The remaining part of this paper is organized as follows. In Section 2, we state several useful Lemmas and we state the main results in Section 3. These results are then proved in Section 4. Some examples together with their numerical simulations are presented in Section 5 to show the feasibility of our results. We give a brief discussion in the last section.

2 Lemmas

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

Lemma 2.1

If a > 0, b > 0 and ẋ ≥x(b — ax), when t≥ 0 and x(0) > 0, we have

$liminft→+∞⁡x(t)≥ba.$

If a > 0, b > 0 andẋ≤x(b — ax), when t≥ 0 and x(0) > 0, we have

$limsupt→+∞⁡x(t)≤ba.$

Lemma 2.2

Let x(t) = (x₁(t), x₂(t))^T be any solution of system (1) (system (2)) with x_i(t₀) > 0, i = 1, 2, then x_i (t) > 0, t ≥ t₀and there exists toa positive constant M₀ such that

$limsupt→+∞xi(t)≤M0,i=1,2,$

i.e, any positive solution of system (1) are ultimately bounded above by some positive constant.

Proof

Let x(t) = (x₁(t), x₂(t))^T be any solution of system (1) with x_i(t₀) > 0, i = 1,2, then

(10)

$x1(t)=x1(t0)exp∫t0tΔ1(s)ds>0.x2(t)=x2(t0)exp∫t0tΔ2(s)ds>0.$

where

$Δ1(s)=r1(s)−a1(s)x1(s)−b1(s)x2(s)d1(s)+e1(s)x1(s)+f1(s)x2(s)−c1(s)x1(s)x2(s),Δ2(s)=r2(s)−b2(s)x1(s)d2(s)+e2(s)x1(s)+f2(s)x2(s)−a2(s)x2(s)−c2(s)x1(s)x2(s).$

From the first equation of system (1), we have

(11)

$x˙1(t)≤x1(t)r1(t)−a1(t)x1(t)≤x1(t)r1M−a1Lx1(t).$

By applying Lemma 2.1 to differential inequality (11), it follows that

(12)

$limsupt→+∞x1(t)≤r1Ma1L=defM1.$

Similarly to the analysis of (11) and (12), from the second equation of system (1), we have

(13)

$limsupt→+∞x2(t)≤r2Ma2L=defM2.$

Set M₀ = max{M₁, M₂}, then the conclusion of Lemma 2.2 follows.

The proof for system (2) is similar to the above proof, with some minor revision, we omit the detail here. This ends the proof of Lemma 2.2.

Lemma 2.3

([21], Fluctuation lemma). Let x(t) be a bounded differentiable function on (α, ∞), then there exist sequences τ_n→ ∞, σ_n → ∞ such that

$(a)x˙(τn)→0andx(τn)→limsupt→+∞x(t)=x asn→,(b)x˙(σn)→0andx(σn)→liminft→+∞x(t)=x_asn→∞.$

For the Logistic equation

(14)

$x˙1(t)=x1(t)r1(t)−a1(t)x1(t).$

From Lemma 2.1 of Zhao and Chen [22], we have

Lemma 2.4

Suppose that r₁(t) and a₁(t) are continuous functions bounded above and below by positive constants, then any positive solutions of Eq. (14) are defined on [0, +∞), bounded above and below by positive constants and globally attractive.

3 Main results

Our main results are the following Theorem 3.1-3.9.

Theorem 3.1

Assume that

(15)

$r1Lr2M>maxa1M(d2M+e2MM1+f2MM2)b2L,c1Mc2L,b1Md1La2L$

holds, then the species x₂ will be driven to extinction, that is, for any positive solution (x₁(t), x₂(t))^T of system (1), x₂(t)) → 0 as t → +∞.

Remark 3.2

The main result of Li and Chen [4] is the special case of Theorem 3.1, If we take d_i (t) = 1, e_i (t) = f_i (t) = 0, i = 1, 2 in system (1), then system (1) is degenerate to system (5), and Theorem 3.1 is degenerate to the main result in [4]. Hence we generalize the main result of [4].

Theorem 3.3

Assume that

(16)

$a1M(d2M+e2MM1+f2MM2)b2L<r1L−c1MM1M2r2M$

and

(17)

$b1Md1La2L<r1L−c1MM1M2r2M$

hold, then the species x₂ will be driven to extinction, that is, for any positive solution (x₁(t), x₂(t))^T of system (1), x₂(t)) → 0 as t → +∞.

Theorem 3.4

Assume that

(18)

$(a1M+c1MM2)(d2M+e2MM1+f2MM2)b2L<r1Lr2M$

and

(19)

$b1Md1La2L<r1Lr2M$

hold, then the species x₂ will be driven to extinction, that is, for any positive solution (x₁(t), x₂(t))^T of system (1), x₂(t)) → 0 as t → +∞.

Theorem 3.5

Assume that

(20)

$a1M(d2M+e2MM1+f2MM2)b2L<r1Lr2M$

and

(21)

$b1Md1L+c1MM1a2L<r1Lr2M$

hold, then the species x₂ will be driven to extinction, that is, for any positive solution (x₁(t), x₂(t))^T of system (1), x₂(t) → 0 as t → +∞.

Remark 3.6

From the proof of Theorem 3.3-3.5 in Section 4, one could easily see that under the assumption of Theorem 3.3-3.5, the conclusion also holds for system (2). i.e., under the assumption of Theorem 3.3, 3.4 or 3.5, the species x₂ in system (2) will be driven to extinction.

Remark 3.7

Another interesting thing is to investigate the extinction property of species x₁ in system (1). One could easily establish some parallel results as that of Theorem 3.1-3.5 for the extinction of species x₁, and we omit the detail here.

Theorem 3.8

Assume that the conditions of Theorem 3.1 or 3.3 or 3.4 or 3.5 hold, let x(t) = (x₁(t), x₂(t))^T be any positive solution of system (1), then the species x₂ will be driven to extinction, that is, x₂(t)→ 0 as t → +∞, and $x1t→x1∗tast→+∞,wherex1∗t$ is any positive solution of the system

$x˙1(t)=x1(t)r1(t)−a1(t)x1(t).$

Theorem 3.9

Assume that

(22)

$r1Mr2L<b1La2M(d1M+e1MM1+f1MM2)$

and

(23)

$r1Mr2L<a1Ld2Lb2M$

hold, then the species x₁ will be driven to extinction, that is, for any positive solution (x₁(t), x₂(t))^T of system (1), x₁(t) → 0 as t → +∞ $x2t→x2∗tast→+∞,wherex2∗t$ is any positive solution of system ẋ₂(t) = x₂(t)(r₂(t) – b₂(t)x₂(t)).

4 Proof of the main results

Proof of Theorem 3.1. It follows from (15) that one could choose enough small positive constant ε₁ > 0 such that

(24)

$r1Lr2M>maxa1M(d2M+e2M(M1+ε1)+f2M(M2+ε1))b2L,c1Mc2L,b1Md1La2L.$

(24) is equivalent to

(25)

$a1Mb2Ld2M+e2M(M1+ε1)+f2M(M2+ε1)<r1Lr2M,b1Md1La2L<r1Lr2M,c1Mc2L<r1Lr2M.$

Therefore, there exist two constants α, β such that

(26)

$a1Mb2Ld2M+e2M(M1+ε1)+f2M(M2+ε1)<βα<r1Lr2M,b1Md1La2L<βα<r1Lr2M,c1Mc2L<βα<r1Lr2M.$

That is

(27)

$αc1M−βc2L<0,αa1M−βb2Ld2M+e2M(M1+ε1)+f2M(M2+ε1)<0,αb1Md1L−βa2L<0,−αr1L+βr2M=def−δ1<0.$

Let x(t) = (x₁(t), x₂(t))^T be a solution of system (1) with x_i(0) > 0, i = 1,2. For above ε₁ > 0, from Lemma 2.2 there exists T₁ large enough such that

(28)

$x1(t)<M1+ε1,x2(t)<M2+ε1for allt≥T1.$

From(l) we have

(29)

$x˙1(t)x1(t)=r1(t)−a1(t)x1(t)−b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−c1(t)x1(t)x2(t),x˙2(t)x2(t)=r2(t)−b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t)−a2(t)x2(t)−c2(t)x1(t)x2(t).$

Let

$V(t)=x1−α(t)x2β(t).$

From (27)-(29), for t ≥Τ₁, it follows that

$V˙(t)=V(t)−α(r1(t)−a1(t)x1(t)−b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−c1(t)x1(t)x2(t))+β(r2(t)−a2(t)x2(t)−b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t)−c2(t)x1(t)x2(t))=V(t)[(−αr1(t)+βr2(t))+(αc1(t)−βc2(t))x1(t)x2(t)+(αa1(t)−βb2(t)d2(t)+e2(t)x1(t)+f2(t)x2(t))x1(t)+(αb1(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−βa2(t))x2(t)]≤V(t)[(−αr1L+βr2M)+(αc1M−βc2L)x1(t)x2(t)+(αa1M−βb2Ld2M+e2M(M1+ε1)+f2M(M2+ε1))x1(t)+(αb1Md1L−βa2L)x2(t)]≤−δ1V(t),t≥T1.$

Integrating this inequality from Τ₁ to t (≥ Τ₁ ), it follows

(30)

$V(t)≤V(T1)exp⁡(−δ1(t−T1)).$

By Lemma 2.2 we know that there exists M > M₀ > 0 such that

(31)

$xi(t)<Mfor alli=1,2andt≥T1.$

Therefore, (30) implies that

(32)

$x2(t)<Cexp−δ1β(t−T1),$

where

(33)

$C=Mα/β(x1(T1))−α/βx2(T1)>0.$

Consequently, we have x₂(t) → 0 exponentially as t→ +∞. This ends the proof of Theorem 3.1.

Proof of Theorem 3.3. It follows from (16) and (17) that one could choose a positive constant ε₂ > 0 small enough such that

(34)

$a1Mb2Ld2M+e2M(M1+ε2)+f2M(M2+ε2)<r1L−c1MM1+ε2M2+ε2r2M$

and

(35)

$b1Md1La2L<r1L−c1MM1+ε2M2+ε2r2M$

hold. Therefore, there exist two constants a, β such that

(36)

$a1Mb2Ld2M+e2M(M1+ε2)+f2M(M2+ε2)<βα<r1L−c1MM1+ε2M2+ε2r2M$

and

(37)

$b1Md1La2L<βα<r1L−c1M(M1+ε2)(M2+ε2)r2M$

hold. That is

(38)

$αa1M−βb2Ld2M+e2M(M1+ε2)+f2M(M2+ε2)<0,αb1Md1L−βa2L<0,−αr1L+βr2M+αc1M(M1+ε2)(M2+ε2)=def−δ2<0.$

Let x(t) = (x₁(t), x₂(t))^T be a solution of system (1) with x_i (0) > 0, i = 1, 2. For above ε₂ > 0, from Lemma 2.2 there exists T₂ large enough such that

(39)

$x1(t)<M1+ε2,x2(t)<M2+ε2for allt≥T2.$

Let

$V(t)=x1−α(t)x2β(t).$

From (29), (38) and (39), for t ≥T₂, it follows that

$V˙(t)=V(t)−α(r1(t)−a1(t)x1(t)−b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−c1(t)x1(t)x2(t))+β(r2(t)−a2(t)x2(t)−b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t)−c2(t)x1(t)x2(t))≤V(t)(−αr1(t)+βr2(t))+αc1(t)x1(t)x2(t)+(αa1(t)−βb2(t)d2(t)+e2(t)x1(t)+f2(t)x2(t))x1(t)+(αb1(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−βa2(t))x2(t)≤V(t)−αr1L+βr2M+αc1M(M1+ε2)(M2+ε2)+(αa1M−βb2Ld2M+e2M(M1+ε2)+f2M(M2+ε2))x1(t)+(αb1Md1L−βa2L)x2(t)≤−δ2V(t),t≥T2.$

Integrating this inequality from T₂ to t (≥ T₂), it follows

(40)

$V(t)≤V(T2)exp⁡(−δ2(t−T2)).$

From (40), similarly to the analysis of (30)-(33), we have x₂(t) → 0 exponentially as t → +∞. This ends the proof of Theorem 3.3.

Proof of Theorem 3.4. It follows from (18) and (19) that one could choose a positive constant ε₃ > 0 small enough such that

(41)

$a1M+c1MM2+ε3b2Ld2M+e2M(M1+ε3)+f2M(M2+ε3)<r1Lr2M$

and

(42)

$b1Md1La2L<r1Lr2M.$

hold. Therefore, there exist two constants α, β such that

(43)

$a1M+c1MM2+ε3b2Ld2M+e2M(M1+ε3)+f2M(M2+ε3)<βα<r1Lr2M$

and

(44)

$b1Md1La2L<βα<r1Lr2M$

hold. That is

(45)

$αa1M+αc1M(M2+ε3)−βb2Ld2M+e2M(M1+ε3)+f2M(M2+ε3)<0,αb1Md1L−βa2L<0,−αr1L+βr2M=def−δ3<0.$

Let x(t) = (x₁(t), x₂(t))^T be a solution of system (1) with x_i(0) > 0, i = 1,2. For above ε₃ > 0, from Lemma 2.2 there exists T₃ large enough such that

(46)

$x1(t)<M1+ε3,x2(t)<M2+ε3for allt≥T3.$

Let

$V(t)=x1−α(t)x2β(t).$

From (29), (45) and (46), for t ≥ T₃, it follows that

$V˙(t)=V(t)−α(r1(t)−a1(t)x1(t)−b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−c1(t)x1(t)x2(t))+β(r2(t)−a2(t)x2(t)−b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t)−c2(t)x1(t)x2(t))≤V(t)(−αr1(t)+βr2(t))+(αa1(t)+αc1(t)x2(t)−βb2(t)d2(t)+e2(t)x1(t)+f2(t)x2(t))x1(t)+(αb1(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−βa2(t))x2(t)$

$≤V(t)−αr1L+βr2M+(αa1M+αc1M(M2+ε3)−βb2Ld2M+e2M(M1+ε3)+f2M(M2+ε3))x1(t)+(αb1Md1L−βa2L)x2(t)≤−δ3V(t),t≥T3.$

Integrating this inequality from T₃ to t (≥ T₃), it follows

(47)

$V(t)≤V(T3)exp⁡(−δ3(t−T3)).$

From (47), similarly to the analysis of (30)-(33), we have x₂(t) → 0 exponentially as t → +∞. This ends the proof of Theorem 3.4.

Proof of Theorem 3.5. It follows from (20) and (21) that one could choose a positive constant ε₄ > 0 small enough such that

(48)

$a1Mb2Ld2M+e2M(M1+ε4)+f2M(M2+ε4)<r1Lr2M$

and

(49)

$b1Md1L+c1M(M1+ε4)d1La2L<r1Lr2M.$

hold. Therefore, there exist two constants a, β such that

(50)

$a1Mb2Ld2M+e2M(M1+ε4)+f2M(M2+ε4)<βα<r1Lr2M$

and

(51)

$b1Md1L+c1M(M1+ε4)d1La2L<βα<r1Lr2M$

hold. That is

(52)

$αa1M−βb2Ld2M+e2M(M1+ε4)+f2M(M2+ε4)<0,αb1Md1L+αc1M(M2+ε4)−βa2L<0,−αr1L+βr2M=def−δ3<0.$

Let x(t) = (x₁(t), x₂(t))^T be a solution of system (1) with x_i (0) > 0, i = 1,2. For above ε4 > 0, from Lemma 2.2 there exists T₄large enough such that

(53)

$x1(t)<M1+ε4,x2(t)<M2+ε4for allt≥T4.$

Let

$V(t)=x1−α(t)x2β(t).$

From (29), (52) and (53), for t ≥T₄, it follows that

$V˙(t)=V(t)−α(r1(t)−a1(t)x1(t)−b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−c1(t)x1(t)x2(t))+β(r2(t)−a2(t)x2(t)−b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t)−c2(t)x1(t)x2(t))≤V(t)(−αr1(t)+βr2(t))+(αc1(t)x1(t)−βa2(t)+αb1(t)d1(t)+e1(t)x1(t)+f1(t)x2(t))x2(t)+(αa1(t)−βb2(t)d2(t)+e2(t)x1(t)+f2(t)x2(t))x1(t)≤V(t)[(−αr1L+βr2M)+(αa1M−βb2Ld2M+e2M(M1+ε4)+f2M(M2+ε4))x1(t)+(αb1Md1L+αc1M(M1+ε4)−βa2L)x2(t)≤−δ4V(t),t≥T4.$

Integrating this inequality from T₄to t (≥ T₄), it follows

(54)

$V(t)≤V(T4)exp⁡(−δ4(t−T4)).$

From (54), similarly to the analysis of (30)-(33), we have x₂(t) → 0 exponentially as t → +∞. This ends the proof of Theorem 3.5.

Proof of Theorem 3.8. By applying Lemma 2.3 and 2.4, the proof of Theorem 3.8 is similar to that of the proof of Theorem in [4]. We omit the detail here.

Proof of Theorem 3.9. Conditions (22) and (23) imply that there exist two constants α, β and a positive constant ε₅ small enough, such that

(55)

$r1Mr2L<βα<b1Ld1M+e1M(M1+ε5)+f1M(M2+ε5)a2M,r1Mr2L<βα<a1Ld2Lb2L.$

That is

(56)

$−αa1L+βb2Md2L<0,αr1M−βr2L=def−δ4<0,−αb1Ld1M+e1M(M1+ε5)+f1M(M2+ε5)+βa2M<0.$

For above ε₅ > 0, from Lemma 2.2 there exists T₅large enough such that

(57)

$x2(t)<M2+ε5for allt≥T5.$

Let

$V1(t)=x1α(t)x2−β(t).$

It follows from (56) and (57) that

$V˙1(t)=V1(t)α(r1(t)−a1(t)x1(t)−b1(t)x2(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)−c1(t)x1(t)x2(t))−β(r2(t)−b2(t)x1(t)d2(t)+e2(t)x1(t)+f2(t)x2(t)−a2(t)x2(t))=V1(t)(αr1(t)−βr2(t)+(−αa1(t)+βb2(t)d2(t)+e2(t)x1(t)+f2(t)x2(t))x1(t)+(−αb1(t)d1(t)+e1(t)x1(t)+f1(t)x2(t)+βa2(t))x2(t)≤V1(t)(αr1M−βr2L)+(−αa1L+βb2Md2L)x1(t)+(−αb1Ld1M+e1M(M1+ε5)+f1M(M2+ε5)+βa2M)x2(t)≤−δ5V1(t).$

Integrating this inequality from T₅ to t (≥ T₅), it follows

(58)

$V1t≤V1T5exp−δ5t−T5.$

From this, similarly to the analysis of (30)-(33), we have x₁(t) → 0 exponentially as t → +∞. The rest of the proof of Theorem 3.9 is similar to that of the proof of Theorem in [4]. We omit the detail here.

5 Numeric examples

Now let us consider the following examples.

Example 5.1

(59)

$x˙(t)=x10−(2.3+0.3sin⁡(4t))x−(3+12cos⁡(4t))y1+0.1y+0.1x−6xy,y˙(t)=y2−(6+sin⁡(4t))x1+0.1x+0.1y−2y−(4+cos⁡(4t))xy.$

Corresponding to system (1), one has

$r1(t)=10,a1(t)=2.3+0.3sin⁡(2t),b1(t)=3+12cos⁡(2t),c1(t)=6,r2(t)=2,a2(t)=2,b2(t)=6+sin⁡(2t),c2(t)=4+cos⁡(2t).d2(t)=d1(t)=1,e1(t)=e2(t)=f1(t)=f2(t)=0.1.$

And so,

(60)

$M1=r1Ma1L=5,M2=r2Ma2L=1.$

Consequently

$r1Lr2M=5,a1Md2M+e2MM1+f2MM2b2L=104125,c1Mc2L=2,b1Md1La2L=74.$

Since

$5>max104125,2,74,$

it follows from Theorem 3.1 that the first species of the system (59) is globally attractive, and the second species will be driven to extinction. Numeric simulations (Fig. 1, 2) also support this findings.

$Fig. 1 Dynamic behavior of the first component x(t) of the solution (x(t), y(t)) of system (59) with the initial conditions (x(0), y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.$

Fig. 1

Dynamic behavior of the first component x(t) of the solution (x(t), y(t)) of system (59) with the initial conditions (x(0), y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.

$Fig. 2 Dynamic behavior of the second component y(t) of the solution (x(t), y(t)) of system (59) with the initial conditions (x(0),y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.$

Fig. 2

Dynamic behavior of the second component y(t) of the solution (x(t), y(t)) of system (59) with the initial conditions (x(0),y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.

Example 5.2

(61)

$x˙(t)=x10−2.3+0.3sin4tx−3+12cos4ty1+0.1y+0.1x−1xy,y˙(t)=y2−6+sin4tx1+0.1x+0.1y−2y−0.1xy.$

In system (61), we let c₁(t)= 1, C₂(t)= 0.1, and all the other coefficients are the same as that of system (59). In this case, since

$r1Lr2M=5<10=c1Mc2L,$

the conditions of Theorem 3.1 could not satisfied, however,

(62)

$a1M(d2M+e2MM1+f2MM2)b2L=104125<52=r1L−c1MM1M2r2M$

and

(63)

$b1Md1La2L=74<52=r1L−c1MM1M2r2M.$

(62) and (63) show that all the conditions of Theorem 3.3 are satisfied, and so, the first species in system (61) is globally attractive, and the second species will be driven to extinction. Numeric simulations (Fig. 3, 4) also support this findings.

$Fig. 3 Dynamic behavior of the first component x(t) of the solution (x(t), y(t)) of system (61) with the initial conditions (x(0), y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.$

Fig. 3

Dynamic behavior of the first component x(t) of the solution (x(t), y(t)) of system (61) with the initial conditions (x(0), y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.

$Fig. 4 Dynamic behavior of the second component y(t) of the solution (x(t), y(t)) of system (61) with the initial conditions (x(0), y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.$

Fig. 4

Dynamic behavior of the second component y(t) of the solution (x(t), y(t)) of system (61) with the initial conditions (x(0), y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.

Example 5.3

(64)

$x˙(t)=x2−(6+sin⁡(4t))y1+0.1x+0.1y−2x−0.1xy,y˙(t)=y10−(2.3+0.3sin⁡(4t))y−(3+12cos⁡(4t))x1+0.1y+0.1x.$

Corresponding to system (1), one has

$r2(t)=10,a2(t)=2.3+0.3sin⁡(2t),b2(t)=3+12cos⁡(2t),r1(t)=2,a1(t)=2,b1(t)=6+sin⁡(2t),c1(t)=0.1.d2(t)=d1(t)=1,e1(t)=e2(t)=f1(t)=f2(t)=0.1.$

And so,

(65)

$M1=r1Ma1L=1,M2=r2Ma2L=5.$

Consequently

$r1Mr2L=15<125104=b1La2M(d1M+e1MM1+f1MM2),r1Mr2L=15<a1Ld2Lb2M=47.$

Hence, all the conditions of Theorem 3.9 hold. It follows from Theorem 3.9 that the first species of the system (64) will be driven to extinction, and the second species is globally attractive, numeric simulations (Fig. 5, 6) also support this findings.

$Fig. 5 Dynamic behavior of the first component x(t) of the solution (x(t), y(t)) of system (64) with the initial conditions (x(0), y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.$

Fig. 5

Dynamic behavior of the first component x(t) of the solution (x(t), y(t)) of system (64) with the initial conditions (x(0), y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.

$Fig. 6 Dynamic behavior of the second component y(t) of the solution (x(t), y(t)) of system (64) with the initial conditions (x(0),y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.$

Fig. 6

Dynamic behavior of the second component y(t) of the solution (x(t), y(t)) of system (64) with the initial conditions (x(0),y(0)) = (1,1), (2,2), (0.2,3) and (7,4), respectively.

6 Discussion

During the last decade, many scholars paid attention to the extinction property of the competition system, in their series work, Li and Chen [3-7], Chen et al [8, 9] studied the extinction property of the competition system with toxic substance. He et al [31], Chen et al [34, 35] studied the extinction property of the Gilpin-Ayala competition model. Recently, Yue [1] proposed a competitive system with both toxic substance and nonlinear inter-inhibition terms, i.e., system (9), she also investigated the extinction property of the system. Noting that the functional response used in [1] is of Holling II type, in this paper, we consider a more plausible one, i.e., the Beddington-DeAngelis functional response. By constructing some suitable Lyapunov type extinction function, several set of sufficient conditions which ensure the extinction of a species are obtained. Our results generalize the main result of Li and Chen [6].

We mention here that in system (1) and (2), we did not consider the influence of delay, we leave this for future investigation.

Competing interests

The authors declare that there is no conflict of interests.

Authors’ contributions

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

Acknowledgement

The research was supported by the Natural Science Foundation of Fujian Province (2015J01012).

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Received: 2016-9-16

Accepted: 2016-11-1

Published Online: 2016-12-30

Published in Print: 2016-1-1

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