Extinction in Nonautonomous Discrete Lotka-Volterra Competitive System with Pure Delays and Feedback Controls

The coexistence and global stability of population models are of the interesting subjects in mathematical biology. Many authors have argued that the discrete time models are governed by differential equations which are more appropriate than the continuous ones to describe the dynamics of population when the population has nonoverlapping generations, a lot has been done on discrete Lotka-Volterra systems. May in 1 firstly considered the following autonomous discrete two-species LotkaVolterra competitive system:


Introduction
The coexistence and global stability of population models are of the interesting subjects in mathematical biology.Many authors have argued that the discrete time models are governed by differential equations which are more appropriate than the continuous ones to describe the dynamics of population when the population has nonoverlapping generations, a lot has been done on discrete Lotka-Volterra systems.
May in 1 firstly considered the following autonomous discrete two-species Lotka-Volterra competitive system: x n 1 x n exp r 1 − a 11 x n − a 12 y n , y n 1 y n exp r 2 − a 21 x n − a 22 y n 1.1 and studied the stable points, stable cycles, and the chaos behaivor.Further, Lu and Wang 2 studied the permanence and global attractivity of this system.
Chen and Zhou in 3 considered the following periodic discrete two-species Lotka-Volterra competitive system: x n 1 x n exp r and studied the permanence and existence of a periodic solution, and further, sufficient conditions are established on the global stability of the periodic solution.
Zhang and Zhou in 4 investigated the following nonautonomous discrete twospecies Lotka-Volterra competitive system: x n 1 x n exp r 1 n − a 11 n x n − k 1 − a 12 n y n − k 2 , y n 1 y n exp r 2 n − a 21 n x n − l 1 − a 22 n y n − l 2 .

1.3
Some sufficient conditions were obtained for the permanence of the system.Wang et al. in 5 studied the following general nonautonomous discrete n-species Lotka-Volterra systems: , 2, . . ., n.

1.4
By applying the linear method and constructing the appropriate Lyapunov functionals, the author established the sufficient conditions which guarantee that any positive solution of this system is stable and attracts others, and obtained some applications of main results.Muroya in 6, 7 considered the following general nonautonomous discrete n-species Lotka-Volterra systems: and related pure delays models, that is, a i p 0 for all p ≥ 0. The author obtained the permanence and the global asymptotically stable by applying mean-value conditions and the method of constructing discrete Lyapunov functionals.
Liao et al. in 8 discussed the following general discrete nonautonomous n-species competitive system with feedback controls: Some sufficient conditions are established on the permanence and the global stability of the system.
Recently, we see that in 9, 10 the authors studied the following nonautonomous continuous Lotka-Volterra competitive system with pure delays and feedback controls:

1.7
The sufficient conditions for which a part of the n-species is driven to extinction and the surplus part of the n-species remains permanence are established.However, we see that for general discrete n-species population systems the results for which a part of the n-species is driven to extinction and the surplus part of the n-species remains the permanence, up to now, are still not obtained.Therefore, motivated by the above works, in this paper we study the following discrete nonautonomous Lotka-Volterra competitive system with pure delays and feedback controls

1.8
The main purpose is to establish a criterion for which guarantee the part species x r 1 , x r 2 , . . ., x n in system 1.8 is driven to extinction.The method used in this paper is to constructing the multiple discrete Lyapunov functions.On the permanence of the surplus species x 1 , x 2 , . . ., x r , owing to the length of this paper, we will give the discussion in another paper.
This paper is organized as follows.In next section, as preliminaries, some assumptions and useful lemmas are introduced.In Section 3, the main results of this paper on the extinction of a part of the n-species of system 1.8 are established.In Section 4, an example is presented to illustrate the feasibility of our results.

Preliminaries
Let Z denote the set of all nonnegative integers.For any bounded sequence y k , we denote y sup k∈Z {y k }, y inf k∈Z {y k }.Throughout this paper, we introduce the following assumptions.1 − e i s < 1.

2.2
Let R 0, ∞ and R n {x x 1 , x 2 , . . ., x n : x i ∈ R , i 1, 2, . . ., n}.We denote by int R n the interior of R n .For any nonnegative constants a and b with a < b, we denote by a, b Z the set of all integers in the interval a, b .For some integer m ≥ 0, we denote by D −m, 0 Z the space of all nonnegative discrete time function Let τ max{τ ij , τ i , δ i , σ ij : i, j 1, 2, . . ., n}.Motivated by the biological background of system 1.8 , in this paper we only consider all solutions of system 1.8 that satisfy the following initial conditions: , by the fundamental theory of difference equations, system 1.8 has a unique solution x s, z , s, z satisfying the initial condition 2.3 , where x s, z x 1 s, z , x 2 s, z , . . ., x n s, z and u s, z u 1 s, z , u 2 s, z , . . ., u n s, z .It is obvious that solution x s, z , s, z is positive, that is, x i s, z > 0 and u i s, z > 0 i 1, 2, . . ., n for all s ∈ Z.
We first consider the following nonautonomous difference inequality system: where α n and β n are bounded sequences defined on Z and β n ≥ 0 for all n ∈ Z.We have the following result.
Lemma 2.1 see 11 .Assume that there exists an integer λ > 0 such that then there exists a constant M > 0 such that for any nonnegative solution y n of system 2.4 with initial value y n 0 y 0 ≥ 0, where n 0 ∈ Z is some integer, Next, we consider the following nonautonomous linear difference equation: where γ n and ω n are nonnegative bounded sequences defined on Z.We have the following results.
Lemma 2.2 see 11 .Assume that there exists an integer λ > 0 such that then there exists a constant M > 0 such that for any nonnegative solution v n of system 2.7 with initial value v n 0 v 0 ≥ 0, where n 0 ∈ Z is some integer, Lemma 2.3 see 11 .Assume that the conditions of Lemma 2.2 hold, then for any constants ε > 0 and M 1 > 0 there exist positive constants δ δ ε and n n ε, M 1 such that for any n 0 ∈ Z and where v n, n 0 , v 0 is the solution of 2.7 with initial value v n 0 v 0 .
Lemma 2.4.Assume that assumptions H 1 -H 4 hold, then there exists a constant M 0 > 0 such that lim sup 12 then, for any integer k ≥ 0 and θ ≤ 0 with k θ ≥ 0, summing inequality 2.12 from k θ to k − 1, we obtain Therefore, for any integer k ≥ τ, from 2.13 and the first equation of system 1.8 we obtain

2.14
Since for any k ≥ τ and l ∈ 0, where r u i sup k∈Z r i k , we have from 2.14 , for any k ≥ τ, where h i exp{−τr u i }.We consider the following auxiliary equation: where l 0 b iil k , then by assumption H 3 and applying Lemma 2.1 there exists a constant N i > 0 such that lim sup for any positive solution y i k of 2.17 .Therefore, from the comparison theorem of difference equation, we finally obtain lim sup

2.19
Further form inequality 2.19 , there exists a positive constant k i ≥ τ such that

2.20
Thus, from the second equation of system 1.8 , we obtain 21 for all k ≥ k i .We consider the following auxiliary equation:

Main Results
In this section, we discuss the extinction of the part of species x r 1 , . . ., x n of system 1.8 .Define functions as follows: where i, j 1, 2, . . ., n.
Theorem 3.1.Assume that assumptions H 1 -H 4 hold and there exists an integer 1 ≤ r < n such that for any h > r there exists an integer i h < h such that then for each i r 1, . . ., n one has for any positive solution x 1 k , x 2 k , . . ., x n k , u 1 k , u 2 k , . . ., u n k of system 1.8 .

Proof.
From assumption H 2 , there exist constant η 0 > 0 and integer K 0 > 0 such that We first prove that lim k → ∞ x n k 0. Let h n and i h p. From conditions 3.2 we can find positive constants α n1 , α n2 , α n3 , α n4 , ε n and integer K n ≥ K 0 such that for all k ≥ K n and j 1, 2, . . ., n.Consequently, for all k ≥ K n and j 1, 2, . . ., n.Let x k , u k x 1 k , x 2 k , . . ., x n k , u 1 k , u 2 k , . . ., u n k be any positive solution of system 1.8 .Constructing the following discrete Lyapunov functional

3.12
By calculating, we obtain

3.14
For any k > K n , we choose an integer m ≥ 0, such that k ∈ K n mω, K n m 1 ω , then from 3.6 and 3.14 we further have

3.16
On the other hand, from assumptions H 1 , we have for all k ≥ τ, where constant M 0 is given in Lemma 2.4.Hence, there exist a positive constant

3.18
From 3.15 and 3.18 , we obtain for all k ≥ K n τ, where From 3.19 , we finally obtain

3.21
Next, we consider the second equation of system 1.8 , applying Lemma 2.3 we can easily obtain u n k → 0 as k → ∞.Now, we suppose that for any t > r, we have obtained for all i > t.We further will prove

3.31
Calculating V t k 1 /V t k , similarly to V n k , we can obtain for all k ≥ K t .From inequalities 3.26 -3.30 , we further obtain for all k ≥ K t , where Thus, from 3.22 we can obtain that there exists an integer for all k > K * t .By calculating, from 3.33 we obtain x t k < ∞.

3.39
Next, we consider the second equation of system 1.8 , applying Lemma 2.3 we can easily obtain u t k → 0 as k → ∞.
Finally, according to the induction principle, we have lim

3.40
This completes the proof of Theorem 3.1.
As consequences of Theorem 3.1, we consider the following two special cases of system 1.8 .Case 1. Nondelayed nonautonomous discrete n-species Lotka-Volterra competitive systems with feedback controls

3.43
then for each i r 1, . . ., n one has

3.45
For system 3.41 , assumption H 1 becomes into the following form.
Directly from Theorem 3.1, we have the following corollary.

Corollary 3.3. Assume that assumptions H 1 and H 3 hold and there exists an integer 1 ≤ r < n such that for any h > r there exists an integer i
for any positive solution x 1 k , x 2 k , . . ., x n k of system 3.45 .Remark 3.4.By comparison, we easily see that the results obtained in this paper are a very good extension of the corresponding results obtained in 9 on the extinction of species for nonautonomous continuous Lotka-Volterra competitive system with pure-delays and feedback controls to discrete ones.

An Example
We consider the following periodic discrete three-species Lotka-Volterra competitive system with pure-delays and feedback controls:

4.3
From these inequalities we see that all conditions 3.2 in Theorem 3.1 hold.In addition, we also see that assumptions H 1 -H 4 obviously hold.Therefore, by Theorem 3.1 we obtain that species x 3 in system 4.3 is extinct.
1, 2, ..., n.For system 3.41 , assumptions H 1 and H 3 become into the following formH 1 r i k is a bounded sequence defined on Z; a ij k , d i k and g i k i, j 1, 2, ..., nare nonnegative bounded sequences defined on Z.H 3 There exist positive integers ω and λ such that for each i 1, 2, . . ., n Assume that assumptions H 1 , H 2 , H 3 and H 4 hold and there exists a integer 1 ≤ r < n such that for any h > r there exists an integer i h < h such that Clearly, in system 4.2 , r 2, h 3, and i h 1.By calculating, we obtain