Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays

Abstract In this paper, we consider a nonautonomous two-species impulsive competitive system with infinite delays. By the impulsive comparison theorem and some mathematical analysis, we investigate the permanence, extinction and global attractivity of the system, as well as the influence of impulse perturbation on the dynamic behaviors of this system. For the logistic type impulsive equation with infinite delay, our results improve those of Xuxin Yang, Weibing Wang and Jianhua Shen [Permanence of a logistic type impulsive equation with infinite delay, Applied Mathematics Letters, 24(2011), 420-427]. For the corresponding nonautonomous two-species impulsive competitive system without delays, we discuss its permanence, extinction and global attractivity, which weaken and complement the results of Zhijun Liu and Qinglong Wang [An almost periodic competitive system subject to impulsive perturbations, Applied Mathematics and Computation, 231(2014), 377-385].


Introduction
The logistic system is considered to be one of the most important systems in mathematical ecology, and a great deal of research works have been done based on this system. Because of the seasonal uctuations in the environment and hereditary factors, many scholars have investigated the logistic system with time delays (see [1][2][3][4][5][6][7][8]). Noticing that the disturbance of environmental factors at certain time moments can give rise to instantaneous and changes of population density, many scholars have investigated the dynamic behaviors of impulsive di erential equations (see [9][10][11][12][13][14][15][16][17][18][19][20][21]). Especially, Yang [21] investigated the following logistic system with in nite delayẋ with the initial condition x(t) = ϕ(t), t ≤ , which is continuous and bounded on (−∞, ] to [ , +∞) with ϕ( ) > . Here a(t) and b(t) are continuous functions, bounded above and below by positive constants; x (t + k ) = h k x (t k ), k = , , · · · . (1.2) For any given continuous function f (t), let f L and f M denote inf ≤t<+∞ f (t) and sup ≤t<+∞ f (t), respectively. The authors discussed the permanence of system (1.2) under the following conditions: (H1) Π <t k <t h ik , i = , , are bounded above and below by positive constants for all t > ; (H2) r iL − b iM > , i = , . But the authors did not consider its competition exclusion, global attractivity and extinction. For the permanence of system (1.2), we also want to know whether conditions (H ) and (H ) can be weakened? To answer this question, we rst introduce the following example. Example 1. 1 For system (1.2), let r (t) = . t+ . t+ , r (t) = t+ t+ , a (t) = + sin √ t, a (t) = . + . sin t, b (t) = + sin t, b (t) = + . sin t, h k = . − . cos k, h k = . + . sin k and t k = k + k . Obviously, condition (H2) does not hold, but Figure 1 shows that system (1.2) is permanent. This example gives a certain answer to the above question. So it requires us to give its strict mathematical veri cation and to discuss the competition exclusion, global attractivity and extinction of (1.2). Our results improve and complement the corresponding results of Liu and Wang [32]. Motivated by the above papers, in this paper we consider the following systeṁ under an initial condition Here x (t) and x (t) are population densities of species x and x at time t respectively; r (t) > and r (t) > are the growth rates; a (t) > and a (t) > are the e ects of intra-speci c competition; r i (t) and a i (t) are continuous functions, bounded above and below by positive constants for all t > ; the continuous functions b (t) ≥ and b (t) ≥ are the rates of inter-speci c competition, which are bounded for all t > ; K i : [ , +∞) → ( , +∞) (i = , ) are continuous kernels such that +∞ K i (s)ds = ; < t < t < · · · < t k < t k+ < · · · are impulse points with lim k→+∞ t k = +∞; the impulse perturbations {h ik : k = , , · · · } (i = , ) are positive sequences bounded above and below by positive constants.

Preliminaries
In this section, we present the following de nitions and lemmas which are useful in proving our main results.

1)
where g ∈ C[R + × R + , R], ϕ k ∈ C[R, R] and ϕ k (u) is nondecreasing in u for each k = , , · · · . Let r(t) be the maximal solution of the scalar impulsive di erential equatioṅ existing on [t , +∞), then m(t + ) ≤ u implies m(t) ≤ r(t), t ≥ t . Remark 2.1 (see [10]) In Lemma 2.1, assume inequalities (2.1) reverse. Let p(t) be the minimal solution of (2.1) existing on [t , +∞), then p(t + ) ≥ u implies p(t) ≥ r(t), t ≥ t . Consider the following impulsive systeṁ where a and b are positive constants. Proof. Let z(t) = /y(t), then system (2.3) is transformed intȯ According to [9], for any T > , we can obtain First we consider aη + ln h M = , that is e a h /η M = and h M < . According to [17], we obtain Next consider aη + ln h M < , that is e a h /η M < and h M < , then because of bηh M aη + ln h M < . Therefore, it follows from the positivity of y(t) and the relationship between z(t) and y(t) that lim t→+∞ y(t) = . This completes the proof of Lemma 2.3.
where ≤ j ≤ , i ≠ j, which completes the proof of Lemma 2.4.

Lemma 2.5 For any y
The proof is similar to that of Lemma 3 in [24], so we omit it.

Main results
In this section, we present the main results of this paper. First we study the coexistence of system (1.3).
Proof. From (1.3), we can obtain for i = , thaṫ Then according to Lemma 2.1, we have For i = , , substituting this into the ith equation of (1.3), we obtaiṅ (2) If h iM < , we haveẋ All the above analysis show that Therefore for any given ε > satisfying there exists a T > such that for t > T, Substituting this into system (1.3), it follows from Lemma 2.5 that, for ≤ i, j ≤ and i ≠ j We can easily obtain that Substituting this into the ith equation of system (1.3) gives rise tȯ Next we prove lim inf By setting ε → , it follows from Lemma 2.2 that (4) If h iL < , we obtaiṅ By setting ε → , it follows from Lemma 2.2 that This proves the permanence of (1.3). and where Proof. Let (x (t), x (t)) T and (y (t), y (t)) T be any two solutions of system (1.3) with (1.4). From Theorem 3.1, for any ε > satisfying < ε < min{m , m }, there exist δ > such that and T > such that for t > T , De ne a Lyapunov function as follows For t > T and t ≠ t k , k = , , · · · , calculating the upper right derivatives of V i (t) with ≤ i, j ≤ and i ≠ j, we have where ξ j (t) lies between x j (t) and y j (t), j = , .
For i = , , de ne For t > T and t ≠ t k , k = , , · · · , calculating the upper right derivatives of V i (t), it follows that Therefore, for t > T and t ≠ t k , k = , , · · · , where ξ ij (t) ( ≤ i, j ≤ ; i ≠ j) lies between x i (t) and y i (t), i = , . For t = t k , we can easily verify that V(t + k ) = V(t k ). Integrating both sides of the above inequality from T to t, we obtain Similarly to the analysis of [17], it is obvious that

Therefore, V(t) is bounded on [T , +∞) and there is
This completes the proof of Theorem 3.2.
Next, we consider the competition exclusion of system (1.3). Then for any ε > satisfying r L − b M ε + ε θ + ln h L > , there exists a T > such that for t > T , Substituting this into system (1.3), it follows from Lemma 2.5 thaṫ Similarly we havė This completes the proof of the theorem. Consider the following impulsive systeṁ Proof. Let (x (t), x (t)) T be any positive solution of system (1.3), and x(t) be any positive solution of system (3.12). From the condition of Theorem 3.6, there exists a δ > such that a L M − a M σ ≥ δ . According to Theorem 3.5, for any < ε <m small enough, there exists a T > such that for t > T , De ne a Lyapunov function as follows Similarly to the analysis of Theorem 3.2, for t > T and t ≠ t k , k = , , · · · , calculating the upper right derivatives ofV (t), we can obtain De neV For t > T and t ≠ t k , k = , , · · · , calculating the upper right derivatives ofV (t) and denotingV(t) = V (t) +V (t), it follows that By the boundedness of x (t) and x(t) and setting ε → , we educe that For t = t k , we can easily verify thatV(t + k ) =V(t k ). Integrating both sides of the above inequality from T to t, we obtainV Similarly to the analysis of [17], it is obvious that This completes the proof of Theorem 3.4. Now we discuss the extinction of system (1.3).

Theorem 3.5 Let (x (t), x (t)) T be any positive solution of system (1.3). Assume that
Proof. The proof of the theorem is similar to the corresponding part of Theorem 3.3, so we omit the detail. In the following part of this section, based on the above theorems, we gives some corresponding results for systems (1.1) and (1.2) respectively. First for system (1.1), similarly to the analysis of Theorems 3.1 and 3.2, we can easy obtain the following theorem.
Remark 3.1 In Corollary 3.1, we prove the global attractivity of (1.1), but under some weaker conditions than those in Yang [21]; especially, our result does not require the following unreasonable condition: < inf k≥ h k ≤ h k ≤ (k = , , · · · ) and inf k≥ (h k − h k− ) > .
Next for system (1.2), similarly to the proof of Theorem 3.1, we can easily prove the following theorem.

Theorem 3.7
Let (x (t), x (t)) T be any solution of system (1.2) with x i ( ) > , i = , . Assume that  Proof. Let (x (t), x (t)) T and (y (t), y (t)) T be any two positive solutions of system (1.2). From Theorem 3.7, for any ε > small enough, there exist δ > satisfying m i − ε > and T > such that for t > T De ne a Lyapunov function as followsṼ For t > T and t ≠ t k , k = , , · · · , calculating the upper right derivatives ofṼ(t), for j = , and j ≠ i, we have where ζ j (t) lies between x j (t) and y j (t).
For t = t k , we can easily verify thatṼ(t + k ) =Ṽ(t k ). Integrating both sides of the above inequality from T to t, we obtainṼ Similarly to the analysis of [17], it is obvious that This completes the proof of Theorem 3.8.
Consider the following impulsive systeṁ x(t + k ) = h k x(t k ), k = , , · · · , (3.14) Similarly to the analysis of Theorems 3.7 and 3.8, we can easily prove the following theorem.

Theorem 3.9
Let (x (t), x (t)) T be any positive solution of system (1.2), x(t) be any positive solution of system (3.14). Assume that r L θ + ln h L > and r M η + ln h M ≤ Then the species x is permanent and globally attractive but the species x is extinct, that is where M is de ned in Theorem 3.7 and m = min Proof By impulsive comparison theorem and Lemma 2.3, these results can be easily obtained, so we omit the detail. We can easily verify that Thus all the conditions of Theorem 3.2 are satis ed. Therefore both species x and x are permanent and globally attractive, which is shown in Figure 2.  Figure   . − . cos k . + . sin k Permanence Extinction Figure   . − . cos k . + . sin k Extinction Permanence Figure   . − . cos k . + . sin k Extinction Extinction Figure   Furthermore, we keep the growth rates, the intra-speci c competition and the kernel functions of all species unchanged in Table 1, but adjust the values of the impulse perturbations given in Table 2, then simulations (see  show that the permanence and extinction of the species are signi cantly changed, which are in accordance with the results of Theorems 3.4 and 3.5, here we can verify the corresponding conditions similarly to those in Table 1.

Conclusion
In this paper, we are devoted to obtaining the major factors that a ect the coexistence, competition exclusion and extinction of system (1.3). Table 1 Table 2, there is a signi cant variation of the survival of each species. When choosing the impulse perturbations h ik < small enough and keeping the value of the growth rate unchanged, it is hard to maintain the permanence of the species x i . Moreover, this can result in the extinction of both species, which is di erent from the continuous system. The impulse perturbation plays an important role in the survival of the species and can deduce more situations of real ecosystems. Furthermore, for the logistic type impulsive equation with in nite delay, our results improve those of [21] and remove its unreasonable condition. For the corresponding nonautonomous two-species impulsive competitive system without delays, our results weaken and complement the results of [32].