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

Theorem A

E₁ is globally asymptotically stable provided

Theorem B

E₂ is globally asymptotically stable provided

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

where i = 1, 2, …, n, τ_i are nonnegative constants. b_i(t), a_ij(t), d_i(t) (i, j = 1, 2, …, n) are all nonnegative continuous and ω-periodic functions. b_i(t), a_ii(t), d_i(t) > 0 for all t ∈ [0, ω]. Set

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

Theorem C

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

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

Theorem D

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

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:

where x_i(t) and y_i(t) (i = 1, 2) represent the density of mature and immature species at time t > 0, respectively; b_i(t), a_ij(t), r_i(t), d_i(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

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

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

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

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

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:

where x_i(t) and y_i(t) (i = 1, 2) represent the density of mature and immature species at time t > 0, respectively; b_i(t), a_ij(t), r_i(t), d_i(t), q_i(t), E_i(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

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

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.

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:

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

1. If b ≥ a₁, then limt→+∞x(t)=b−a1a2;

2. If b ≤ a₁, 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

where

Proof

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

Consider the following equation

with u_i(t) = x_i(t)(–τ ≤ t ≤ 0), i = 1, 2. By Lemma 2.2, limt→+∞ui(t)=biMe−riLτi−qiLEiLaiiL, and so,

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 biMe−riLτi≤qiLEiL,i=1,2, then limt→+∞ x_i(t) = 0, and consequently, limt→+∞ y_i(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 τ_n → ∞, σ_n → ∞ such that

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)≤b2Me−r2Lτ1−q2LE2La22L. Given ε=12(b1Le−r1Mτ1−q1ME1Ma12M−b2Me−r2Lτ2−q2ME2Ma22L), there exists a T ≥ 0 such that for all t ≥ T

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

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

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

From Lemma 2.2

Therefore, we obtain

Given ε=12α1, there exists a T₁ ≥ T such that

Let α₂ = min{x₁(t) : 0 ≤ t ≤ T₁} > 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

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

where

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.

Corollary 3.1

Assume that in system (1.9)

hold. Then

where

Remark 3.1

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

Corollary 3.2

Assume that in system (1.9)

hold, where M11=b1Me−r1Lτ1a11L . Then the conclusion of Corollary 3.1 holds.

Corollary 3.3

Assume that in system (1.9)

hold, where M22=b2Me−r2Lτ1a22L . 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 x_i(t), i = 1, 2 are bounded and positive for all t ≥ 0. Let x_1=lim inft→+∞x1(t)  and  x¯2=lim supt→+∞x2(t) . For above ε₁ > 0, it follows from Lemma 2.3 that

From Lemma 3.1 we know that x₁ ≥ α > 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 γ_n → +∞, σ_n → +∞ such that x1′(γn)→0,x2′(σn) → 0, x₁(γ_n) → x₁ and x₂(σ_n) → x₂ as n → +∞. It follows from the first equation of system (1.15) that

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

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

(3.11) is equivalent to

(3.12) is equivalent to

(3.13) together with (3.13) leads to

where

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

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