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

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].


Introduction
Throughout this paper, for a given function g(t), we let g L and g M 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 e ect of toxic substances is another important research area, many excellent results have been obtained, see  and the references cited therein. Li et al. [13] studied the stability property of a competitive system with the e ect of toxic substances, they showed that the toxic substance have no in uence to the stability property of the system, though it has in uence on the position of the equilibrium. Their result is then generalized by Chen et al. [23] to the in nite 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] rst time proposed the following two-species competitive model with stage structurė 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 E and E , the authors obtained the following results.
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 e ects of toxic substances: where x i (t) and y i (t)(i = , ) represent the density of mature and immature species at time t > , respectively; b i (t), a ij (t), r i (t), d i (t)(i, j = , ) are all nonnegative continuous and ω-periodic functions. Li and Chen [10] obtained the following result.
Theorem E. If the coe cients of system (1.9) Then second species will be driven to extinction while the rst 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 rst 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 stagestructured 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 nd out the suitable restrictions on the coe cients 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 rst two inequalities in Theorem E hold, while the third inequality does not holds?
Above example enlighten us to revisit the dynamic behaviors of the system (1.9), and to nd out some new su cient 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 q , q are the catchability coe cients 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 di erent agencies, both the species are harvested with harvesting e orts E and E , respectively. This leads to the following modeling The authors showed that the system (1.14) may exists two saddle-node bifurcations for di erent bifurcation parameters. Now stimulated by the works of [36,37], it is natural to incorporating the harvesting e orts 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 = , ) represent the density of mature and immature species at time t > , respectively; 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 in uence 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{τ , τ }. 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 brie y discussion.

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 > .
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, a > , a ≥ and δ ≥ are constants, then: , y (t)) T be any solution of system (1.15) with initial conditions (1.16) and (1.17). where Proof. It follows from the rst or third equation of system (1.15) thaṫ Consider the following equatioṅ , 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. is, over shing will lead to the extinction of both species.

Main results
As indicated by the Remark 2.1, over shing 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 M i , i = , are de ned by (2.2).
Before we begin to prove the main results, we need several Lemmas again.
So, for all t ≥ T, from the rst equation of system (1.15), it follows thaṫ Let u(t) be a solution of the following equatioṅ with u(T + τ ) = x (T + τ ). It follows from condition (3.1) that Therefore, we obtain Given ε = α , there exists a T ≥ T such that Let α = min{x (t) : ≤ t ≤ T } > and α = min{ α , α } > . It follows that x (t) ≥ α > for all t ≥ .
Noting that above proof only use the fact Condition Our main results are the following Theorems.   ( . ) 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 (3.13) together with (3.14) leads to It follows from (3.2) that A i < , i = , , , this together with the fact x > , x > leads to which is contradiction with (3.15). Then we obtain lim t→+∞ x (t) = . Since it immediately follows that lim t→+∞ y (t) = .
Above analysis shows that for < ε < For t ≥ T + τ, from the rst equation of system (1.15), we havė Let u(t) be the solution of the equatioṅ Setting ε → , it follows that Noting that In order to get a contradiction, we suppose that x > . 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 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 B i < , i = , . This together with the fact x > , x > leads to which is contradiction with (3.23). Then we obtain lim t→+∞ x (t) = . 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 (x (t), y (t), x (t), y (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 ε > enough small, such that Let x and x be de ned as that of Lemma 3.3. For above ε > , it follows from Lemma 2.3 that From Lemma 3.1 we know that x ≥ α > . Obviously, x ≥ . To prove lim t→+∞ x (t) = , it su ces to show that x = . In order to get a contradiction, we suppose that x > . 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 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 C i < , i = , . This together with the fact x > , x > leads to which is contradiction with (3.30). Then we obtain lim t→+∞ x (t) = . 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 rst species, we have the following result.
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.

Examples
In this section we shall give two examples to illustrate the feasibility of main results in the previous section.
Example . . Consider Example 1.1 in the introduction Section. Already, we had veri ed Noting that (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. Example . . Now let's further incorporate the harvesting e ort to system (1.11), this leads to the following where τ = . , τ = . , b (t) = , r (t) = , a (t) = . + . cos(t), a (t) = + sin(t), d (t) = . , d (t) = . , b (t) = , r (t) = , a (t) = , a (t) = . + . cos(t). (1) Take q (t)E (t) = , q (t)E (t) = , in this case, b M i e −r L i τ i < q L i E L i , i = , holds, and so, from Remark 2.1, this is over shing case, and all the species will be driven to extinction. Fig 2. support this assertion.
(2) Take q (t)E (t) = . e − . , q (t)E (t) = , in this case, there are no harvest on the second species, also, the harvesting of the rst species is restrict to a limited case. b

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.
That is, inequality (3.5) holds, from Theorem 2.4, the second species will be driven to extinction. Fig. 4 also support this assertion.
(4) Take q (t)E (t) = , q (t)E (t) = , in this case, the rst species is over shing, while the second one is free of harvesting. From Remark 2.1, the rst species will be driven to extinction. Due to the extinction of the rst species, the second one will be permanent. Fig.5 also support this assertion. (5) Take q (t)E (t) = . , q (t)E (t) = . Numeric simulation (Fig. 6) shows that in this case, two species could be coexist in a stable state.

Discusion
Li and Chen [10] proposed a two species periodic competitive stage-structured Lotka-Volterra model with the e ects 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 in uence of harvesting e ect on the competition system with toxic substance. Their success motivated us to propose a two species competitive stage-structured system with the e ect of toxic substance and harvesting (system (1.15)). We rst show that due to the over shing, two of the species will be driven to extinction (Remark 2.1). After that, for the appropriate harvesting case, by applying the uctuation theorem, we are able to establish su cient 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 rst species, and if q (t)E (t) = , that is, without the capture of the rst species, the second species will be driven to extinction. Then, depending on the harvesting e ect q (t)E (t), the system may have the following dynamic behaviors: (1) the second species still be driven to extinction (case (2)); (2) the rst 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 in uence the dynamic behaviours of the system.

Declarations
Competing interests