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

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.

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) The interspecific competition between two species takes the Beddington-DeAngelis functional response type respectively; (2) The terms c 1 .t /x 1 .t /x 2 .t / and c 2 x 1 .t /x 2 .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 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: where r i ; a i ; b i ; i D 1; 2 are all positive constants, and x 1 .t /; x 2 .t / are population density of species x 1 and x 2 at time t , respectively. r i ; i D 1; 2 are the intrinsic growth rates of species; a i ; i D 1; 2 are the rates of intraspecific competition of the first and second species, respectively; b i ; i D 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: 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 timevarying, and they studied the nonautonomous case of system (4), i.e., 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.
Sol K e 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 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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][35][36][37][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, In this system, the inter-inhibition terms take the form b 1 .t/x 2 .t/ 1Cx 2 .t/ and b 2 .t/x 1 .t/ 1Cx 1 .t/ , 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.
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 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][27][28][29][30] and the references therein.
The success of [26][27][28][29][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][4][5], [8], [11][12][13], [31][32][33][34][35][36][37][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.

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 P x x.b ax/; when t 0 and x.0/ > 0, we have If a > 0; b > 0 and P x Ä x.b ax/; when t 0 and x.0/ > 0, we have i.e, any positive solution of system (1) are ultimately bounded above by some positive constant. where From the first equation of system (1), we have By applying Lemma 2.1 to differential inequality (11), it follows that Similarly to the analysis of (11) and (12), from the second equation of system (1), we have Set M 0 D maxfM 1 ; M 2 g; 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.
From Lemma 2.1 of Zhao and Chen [22], we have

Main results
Our main results are the following Theorem 3.1-3.9.
holds, then the species x 2 will be driven to extinction, that is, for any positive solution .
The main result of Li and Chen [4] is the special case of Theorem 3.1, If we take (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].
hold, then the species x 2 will be driven to extinction, that is, for any positive solution .x 1 .t /; x 2 .t // T of system (1), x 2 .t/ ! 0 as t ! C1: hold, then the species x 2 will be driven to extinction, that is, for any positive solution .x 1 .t /; hold, then the species x 2 will be driven to extinction, that is, for any positive solution .x 1 .t /; x 2 .t // T of system (1), 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 x 2 in system (2) will be driven to extinction.
Remark 3.7. Another interesting thing is to investigate the extinction property of species x 1 in system (1). One could easily establish some parallel results as that of Theorem 3.1-3.5 for the extinction of species x 1 , and we omit the detail here.  (1), then the species x 2 will be driven to extinction, that is, ; where x 1 .t / is any positive solution of the system hold, then the species x 1 will be driven to extinction, that is, for any positive solution .x 1 .t /;

Proof of the main results
Proof of Theorem 3.1. It follows from (15) that one could choose enough small positive constant " 1 > 0 such that (24) is equivalent to Therefore, there exist two constants˛;ˇsuch that That is˛c Let x.t/ D .x 1 .t/; x 2 .t // T be a solution of system (1) with x i .0/ > 0; i D 1; 2. For above " 1 > 0, from Lemma 2.2 there exists T 1 large enough such that From (1) we have Let From (27)- (29), for t T 1 , it follows that Integrating this inequality from T 1 to t . T 1 /, it follows Therefore, (30) implies that where C D M˛=ˇ.x 1 .T 1 // ˛=ˇx 2 .T 1 / > 0: Consequently, we have x 2 .t / ! 0 exponentially as t ! C1: This ends the proof of Theorem 3.1. (16) and (17) that one could choose a positive constant " 2 > 0 small enough such that

Proof of Theorem 3.3. It follows from
hold. Therefore, there exist two constants˛;ˇsuch that For above " 2 > 0, from Lemma 2.2 there exists T 2 large enough such that From (29), (38) and (39), for t T 2 , it follows that Integrating this inequality from T 2 to t . T 2 /, it follows From (40), similarly to the analysis of (30)-(33), we have x 2 .t / ! 0 exponentially as t ! C1: 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 " 3 > 0 small enough such that hold. Therefore, there exist two constants˛;ˇsuch that From (29), (45) and (46), for t T 3 , it follows that Integrating this inequality from T 3 to t . T 3 /, it follows From (47), similarly to the analysis of (30)-(33), we have x 2 .t / ! 0 exponentially as t ! C1: This ends the proof of Theorem 3.4.
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˛;ˇand a positive constant " 5 small enough, such that That is For above " 5 > 0, from Lemma 2.2 there exists T 5 large enough such that It follows from (56) and (57) that Integrating this inequality from T 5 to t . T 5 /, it follows From this, similarly to the analysis of (30)-(33), we have x 1 .t / ! 0 exponentially as t ! C1: 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.

Numeric examples
Now let us consider the following examples. (59) Corresponding to system (1), one has And so, Consequently  o ; 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.

P
x.t / D x 10 2:3 C 0:3 sin.4t / x .3 C 1 2 cos.4t //y 1 C 0:1y C 0:1x 1xy Á ; P y.t / D y 2 .6 C sin.4t //x 1 C 0:1x C 0:1y 2y 0:1xy Á : (61) In system (61), we let c 1 .t / D 1; c 2 .t / D 0:1, and all the other coefficients are the same as that of system (59). In this case, since the conditions of Theorem 3.1 could not satisfied, however, (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.  And so, Consequently 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.

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][4][5][6][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.