1. Introduction
In practical world, owing to many natural and man-made factors (e.g., fire, drought, flooding, crop-dusting, deforestation, hunting, harvesting, etc.), the biological species or ecological environments usually undergo some discrete changes of relatively short duration at some fixed times. Such sudden changes can often be characterized mathematically in the form of impulses. With the development of impulsive differential equations, many experts have adequate mathematical models to investigate the dynamical behaviors of such ecosystems with impulsive effects [1] [2] [3] [4] [5]. On the other hand, the Lotka-Volterra competition systems are very important and significant mathematical models in a non-autonomous environment. Many interesting results of the competitive systems on the existence of positive periodic solutions, permanence, extinction, global stability had been studied extensively (see [6] [7] [8] [9] [10] ). For example, Wang [10] investigated the following competitive system with impulsive effects:
The author obtained sufficient conditions on the uniform persistence and extinction of the system by applying the theorem of differential equations.
In ecological environment, because of the natural enemy, severe competition, deterioration of the patch environment, spatial heterogeneity and human activities, species dispersal in two or more patches becomes one of widespread phenomena of nature. It is an important subject to study the effects of dispersion on the dynamics of species living in patchy environments. Many works on population dynamics in patch environment have been investigated [11] [12] [13] [14]. Moreover, in real ecology environment, the existed number on the history will affect indirectly the number of the species at the moment. Therefore, in order to establish more realistic models, the past history of systems should be taken into account, which has led to the introduction of time-delays in differential equations. Such biological system with infinite delay can be found in [15] [16] [17].
Motivated by above arguments, we establish an impulsive competitive system with infinite delay and diffusion as follows:
(1.1)
where and represent the population densities at time t, respectively. Let and as . Species competes with y in patch 1, while can disperse between patch 1 and patch 2, and y is confined to patch 1. denotes the diffusion coefficients of species x. and are impulsive coefficients at time , respectively.
We consider system (1.1) with the following initial conditions
(1.2)
where , for all and for . is the space of bounded function : which is continuous everywhere expect at the point and exists with and with norm .
In this paper, for any continuous function , we denote
, .
Throughout this paper, we assume that the system (1.1) satisfies ((C1), (C2) see [12] , (C3) see [17] ):
(C1) all functions are positive, continuous and bounded defined on , .
(C2) and are positive constants for all .
(C3) is a non-negative, piece-wise continuous function defined on and satisfy .
Applying some inequality techniques, comparison theorem of impulsive differential equations and Lyapunov function, we study the dynamic behaviors of an impulsive competitive system with infinite delay and diffusion, included permanence, extinction and globally attractive. This paper is organized as follows. Section 2 contains some preliminaries and presents the proof of the lemma. In Section 3, we establish some sufficient conditions which guarantee the system is permanence. In finally section, we give some conditions on the extinction of the system. In Section 4, we study the globally attractive of system (1.1).
2. Preliminaries
We consider the following impulsive non-autonomous logistic model
(2.1)
where and are bounded and continuous functions defined on , for all and impulsive coefficients are positive constants for any . Then we have the following Lemma 2.1.
Lemma 2.1. Suppose that there is a constant such that
(2.2)
(2.3)
and function
(2.4)
is bounded on and . Then we have
1) There exist constant and such that
(2.5)
for any positive solution of system (2.1).
2) If all conditions of (1) hold, further if
(2.6)
then we have for any positive solution of system (2.1).
The proof of this Lemma can be found in [18] , here we omit it.
Next we consider the following impulsive periodic single species logistic system with diffusion
(2.5)
Assume that are positive, continuous and bounded functions defined on . , for all and , for all , , then we have the following conclusions.
Lemma 2.2. Suppose that there is a positive constant such that
(2.6)
(2.7)
and function
, (2.8)
is bounded on and . Then we have
1) There are constants and such that
for any positive solution of system(2.1). .
2) If all conditions of (1) hold, further if
, (2.9)
where for all , then system (2.2) is globally attractive.
Proof: Firstly, we prove system (2.5) is permanent. Let be any solution of system (2.5). Define the function , when calculating the upper-right derivative of , we have
when , we have
Consider the following auxiliary system
(2.10)
with initial condition . Obviously, from condition (2.6), (2.8) and Lemma 2.1, there exists a constant such that . Then according to comparison theorem of impulsive differential equations, we derive
for . (2.11)
Now we prove there is a constant such that . Defined , calculating the right-lower derivative of when , similar to above conclusion, we can obtain
when , we have
by comparison theorem of impulsive differential equations, we derive for all , where is the solution of the auxiliary equation
(2.12)
with initial condition . Clearly, from condition (2.7), (2.8) and Lemma 2.1, there is a constant such that . Therefore we have
for .
Next we consider the globally attractive of system (2.5). Construct a Lyapunov function
when , we always have
then is continuous for all . In addition, when and we have
(2.13)
Moreover, calculating the derivative of , we also have
(2.14)
where
For all , we think about under the following two cases:
Case 1: if , then
Case 2: if , then
From Case 1, Case 2 and (2.14), we can obtain
(2.15)
By (2.15) and condition (2.9), we have as . Furthermore, by (2.13), we have , that is, system (2.5) has globally attractive positive solution. This completes the proof of Lemma 2.2.
3. Permanence
Note that system (1.1) always has a positive solution for all if it has a positive initial condition. Here we state and prove the permanent of system (1.1).
Theorem 3.1. There exists a constant such that and for any positive solution of system (1.1) if there exists a constant such that
(3.1)
(3.2)
and function
, (3.3)
are bounded on and for all .
Proof. Firstly, we prove that there is a constant such that . Define the function , we have two cases when .
1) If , we have
(3.4)
2) If , we have
(3.5)
clearly, from (3.4) and (3.5), we get
for all
On the other hand, when , we have
.
Consider the following auxiliary equation
(3.6)
with initial condition . Since the condition of Lemma 2.1 holds from (3.1) and (3.3), using Lemma 2.1, we get that there exists a constant such that . Applying the comparison theorem of impulsive differential equation, we obtain for all . Finally, we have
for (3.7)
Then we prove that there is a constant such that . From the third and sixth equations of system (1.1), we obtain
considering the following subsystem
(3.8)
with initial condition . Obviously, the condition of Lemma 2.1 holds from (3.2) and (3.3), we obtain that there is a constant such that . Similar to the prove process of the bounded of species x, we have .
Let , evidently, we have
,
The proof of Theorem 3.1 is completed.
Theorem 3.2. Assume that all conditions of Theorem 3.1 are satisfied. In addition, there is a constant such that
(3.9)
(3.10)
(3.11)
and function
, (3.12)
are bounded on and , where for all .
Then the system is permanent.
Proof. Above all, we must prove that there exists a constant such that , .
Firstly, we prove . Defined and
(3.13)
When , consider the following two cases.
Case 1: If , we can choose a constant such that , then we have
Case 2: If , we have
By Case 1 and Case 2, we derive .
When , we can obtain
.
Research the following equation with impulsive
(3.14)
By condition (3.9) and (3.12), we know that condition of Lemma 2.1 is satisfied. Consequently, there exists a constant such that
for . (3.15)
Then we investigate the following system:
(3.16)
From Lemma 2.2 and condition (3.1), (3.10), (3.12) and (3.13), we can know that there are positive constants p and P such that
where is globally attractive for the system (3.16). In addition, we assume that is a positive solution of system (3.16) with initial condition . Evidently, we obtain that there exists a constant small enough such that
(3.17)
Similar to the discussion in [18] , we obtain that condition (3.10) is independent of the choice of .
From condition (3.10), there are constant small enough and large enough such that
(3.18)
for all . By (3.12), we can get a positive constant G such that
. (3.19)
From system (1.1), we consider the following subsystem,
By (3.16), (3.17) and comparison theorem of impulsive differential equations, we get that
for all . (3.20)
Next we prove there is a constant such that .
In the beginning, we prove . Suppose that the proposition is not true, we have , that is for all . Furthermore, we can choose a constant such that
(3.21)
Consequently, we have
(3.22)
For any and , we can choose an integer such that , where is a constant. Integrating (3.22) from to t, due to (3.18) and (3.19), we derive
where . Therefore, we can get as , which is contradiction with . Obviously, we have
.
Then we prove there exists a constant such that . Assume that the proposition is not true, then there exists a sequence
such that for all .
From (3.19), we have
. (3.23)
we choose an integer such that for all and any solution of system (1.1) satisfied:
1) If , then , for some ,
2) If , then , for some .
From above inequality, there exist two time sequences and such that for each , we have
and
as (3.24)
(3.25)
(3.26)
for all (3.27)
Let for each , we choose a constant such that
, (3.28)
by (3.20), for each , there is a such that
for all . (3.29)
Clearly, from (3.22), there is an such that for . Hence for any and , . By (3.27) and (3.28), we can obtain
choose an integer such that . Integrating above inequality from to , we obtain
Consequently, from (3.25) and (3.26), we have
(3.30)
For any and , we have
(3.31)
and
(3.32)
For each , there exist an and a constant such that
for all (3.33)
and
(3.34)
We can choose an integer such that , where is a constant. By (3.30), there exists a large enough such that
. (3.35)
For all , and , by (3.27) and (3.31)-(3.34), we have
Integrating above system from to , we derive
which is a contradiction. This contradiction shows that there exists a constant such that . Hence, choose a constant , then we finally have
Therefore Theorem 3.2 holds. This completes the proof.
4. Extinction
In this section, we investigate the extinction of system (1.1). We note that, under conditions of Theorem 3.2, system (1.1) is always permanent.
Theorem 4.1. Assume that there is a constant such that
(4.1)
(4.2)
and function
(4.3)
(4.4)
are bounded function on and , where , then we can obtain
(4.5)
for any positive solution of system (1.1).
Proof. Firstly, we prove the extinction of species x. Define . When , calculating the right-upper derivative of , we have
when , we get
By the comparison theory of impulsive differential equations, we have for all , where is a solution of the auxiliary equation
(4.6)
with the initial condition . Since system (4.6) satisfies all conditions of Lemma 2.1 from conditions (4.1) and (4.3), we obtain for any positive solution of system (4.6). Then we get
(4.7)
In the following, we prove the extinction of species y. From third equation and sixth equation of system (1.1), we have
(4.8)
By Theorem 3.2, there exist and such that for all . Obviously, when , for any , we obtain
Then we consider the following auxiliary system
(4.9)
with the initial condition . Since all condition of Lemma 2.1 holds from condition (4.2) and (4.4), we can obtain for any positive solution of system (4.9). Clearly, we have
(4.10)
From (4.7) and (4.10), we finally obtain (4.5) holds. This completes the proof of Theorem 4.1.
5. Globally Attractive
In this section, by constructing appropriate Lyapunov function, we establish the sufficient conditions on the globally attractive of system (1.1).
Theorem 5.1. Assume that all conditions of Theorem3.2 hold, further, there exists a constant such that
, (5.1)
where
and . Then system (1.1) is globally attractive, that is, for any two positive solutions and of system (1.1), the following limit hold.
(5.2)
Proof. For any two positive solutions , by Theorem 3.2, we obtain that there exist constants such that
(5.3)
Then we have for any and ,
(5.4)
(5.5)
Define a Lyapunov function
for any impulsive time , we have
is continuous for all . For any and , calculating the derivative of , then we get
(5.6)
Let
for , we consider the following cases:
1) If for all , then
2) If for all , then
3) If for all , similar to the arguments above, we can get the same conclusion as (1) and (2). From (1), (2) and (3), we have
(5.7)
Due to (5.6) and (5.7), we can obtain
(5.8)
Moreover, we define
Obviously, and are continuous for all and . Calculating the upper right derivative, we derive that
(5.9)
Define , then we can follows from (5.8) and (5.9) that
Integrating above inequality, we further have for all . Then from (5.1) we have as . Thus, we have as . Finally, from (5.4) and (5.5) we know (5.2) holds. This completes the proof of Theorem 5.1.
6. Conclusions
In this paper, we investigated an impulsive competitive system with infinite delay and diffusion, in which can disperse between patch 1 and patch 2, but competitor y is confined to patch 1. We also gave some sufficient conditions on permanence, extinction and global attractivity of system (1.1). From Theorem 3.1-Theorem 5.1, we can see that the impulse and dispersal have an influence on permanence, extinction and global attractivity. Moreover, we note that the infinite delay is harmless for the extinction, but it affects the permanence and global attractivity of system (1.1).
Further, we can observe that impulsive perturbations play an important role in the permanence and extinction from Theorem 3.1-Theorem 4.1. In ecological environment, many natural and man-made factors which can be described impulse in mathematical always lead to rapid decrease or increase of the population number. So we consider the following two cases.
Theorem 3.1 shows that if the density-coefficients are greater than zero and the impulsive coefficients are bounded, the species x and y are always ultimately bounded. In following discussion, we also assume that satisfies this condition.
Discuss 1 On condition that the impulses lead to decrease of the number of species (such as fire, drought, hunting, harvesting, flooding deforestation), then the impulsive coefficients satisfy and for all .
1) Theorem 3.2 shows that if the impulsive perturbations are relatively small compared to the intrinsic growth rate of x, the species x can keep permanence; if the impulsive perturbations are relatively small, in addition, the delay, competition coefficients of y and dispersal coefficients of x relatively small make the intrinsic growth rate of y to increase, then the species y keeps permanence.
2) Theorem 4.1 shows that if the impulsive perturbations are relatively large compared to the intrinsic growth rate and dispersal coefficient of x, then the species x tends to extinction; if the impulsive perturbations are relatively large and the intrinsic growth rate of y is relatively small, the species y tends to extinction.
Discuss 2 On condition that impulses lead to increase of the number of species (such as feed, replenishment, input or other protective measures from human), that is the impulsive coefficients satisfy for all .
1) Theorem 3.2 shows that the species x always keep permanence; if the delay, competition coefficients of y and dispersal coefficients of x are relatively small making the intrinsic growth rate of y to increase; regardless of impulsive influence which is large or small, the species y keeps permanence.
2) Theorem 4.1 shows that the species x never tends to extinction; if the impulsive perturbations and the intrinsic growth rate are relatively small; besides, the competition coefficient is relatively large, then the species y tends to extinction.
Acknowledgements
This paper is supported by Natural Science Foundation of Guangxi (2016GXNSFAA380194).
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