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].
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
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)
System (1.2) admits three non-negative equilibria.
Concerned with the stability property of E1 and E2, the authors obtained the following results.
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, 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
Then, Liu et al. [7] obtained the following results.
Theorem C
For system (1.6) in the case n = 2, assume
Then
Theorem D
For system (1.6) in the case n = 2, assume
Then
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 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
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 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
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.
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:
and assume that b, a2 > 0, a1 ≥ 0 and δ ≥ 0 are constants, then:
If b ≥ a1, then
If b ≤ a1, then
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 ui(t) = xi(t)(–τ ≤ t ≤ 0), i = 1, 2. By Lemma 2.2,
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
Lemma 2.4
[32] (Fluctuation lemma) Let x(t) be a bounded differentiable function on (α, ∞), then there exist sequences τn → ∞, σn → ∞ such that
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:
Before stating the main results of this section, we introduce a set of conditions
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
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
Let α2 = min{x1(t) : 0 ≤ t ≤ T1} > 0 and
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.
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)
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].
As a direct corollary of Theorem 3.2 and 3.3, we have
Corollary 3.2
Assume that in system (1.9)
hold, where
Corollary 3.3
Assume that in system (1.9)
hold, where
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
From Lemma 3.1 we know that x1 ≥ α > 0. Obviously, x2 ≥ 0. To prove
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 Ai < 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
it immediately follows that
Above analysis shows that for 0 < ε <
To end the proof of Theorem 3.1, it’s enough to show that
For t ≥ T1 + τ, from the first equation of system (1.15), we have
Let u(t) be the solution of the equation
with u(T1 + τ) = x1(T1 + τ). It follows from Lemma 2.2 that
Therefore, we have
Setting ε → 0, it follows that
Noting that
From this, one could easily obtain
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
Let x1and x2 be defined as that of Lemma 3.3. For above ε2 > 0, it follows from Lemma 2.3 that
From Lemma 3.1 we know that x1 ≥ α > 0. Obviously, x2 ≥ 0. To prove
which is equivalent to
Also, it follows from (3.12) that
(3.21) combine with (3.22) leads to
where
Condition (3.18) implies that Bi < 0, i = 1, 2. This together with the fact x1 > 0, x2 > 0 leads to
which is contradiction with (3.23). Then we obtain
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
Let x1 and x2 be defined as that of Lemma 3.3. For above ε3 > 0, it follows from Lemma 2.3 that
From Lemma 3.1 we know that x1 ≥ α > 0. Obviously, x2 ≥ 0. To prove
which is equivalent to
Also, it follows from (3.12) that
(3.28) combine with (3.29) leads to
where
Condition (3.25) implies that Ci < 0, i = 1, 2. This together with the fact x1 > 0, x2 > 0 leads to
which is contradiction with (3.30). Then we obtain
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
where
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.
where
where
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
Noting that
Thus,
(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.
![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.](/document/doi/10.1515/math-2019-0067/asset/graphic/j_math-2019-0067_fig_001.jpg)
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
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).
Take q1(t)E1(t) = 3, q2(t)E2(t) = 2, in this case,
![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.](/document/doi/10.1515/math-2019-0067/asset/graphic/j_math-2019-0067_fig_002.jpg)
Figure 2Dynamics 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.
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.
Thus,
(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.
![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.](/document/doi/10.1515/math-2019-0067/asset/graphic/j_math-2019-0067_fig_003.jpg)
Figure 3Dynamics 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.
Take q1(t)E1(t) = 2.6e−0.2, q2(t)E2(t) = 0, in this case
Hence
That is, inequality (3.5) holds, from Theorem 2.4, the second species will be driven to extinction. Fig. 4 also support this assertion.
![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.](/document/doi/10.1515/math-2019-0067/asset/graphic/j_math-2019-0067_fig_004.jpg)
Figure 4Dynamics 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.
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.
![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.](/document/doi/10.1515/math-2019-0067/asset/graphic/j_math-2019-0067_fig_005.jpg)
Figure 5Dynamics 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.
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.
![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.](/document/doi/10.1515/math-2019-0067/asset/graphic/j_math-2019-0067_fig_006.jpg)
Figure 6small 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.
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Competing interests
The authors declare that there is no conflict of interests.
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Funding
The research was supported by the National Natural Science Foundation of China under Grant(11601085).
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Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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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