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Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 249364, 9 pages doi:10.1155/2010/249364 ResearchArticleNoteonthePersistentPropertyofaDiscreteLotka-VolterraCompetitiveSystemwithDelaysandFeedback Controls Xiangzeng Kong, 1, 2 Liping Chen, 1, 2 and Wensheng Yang 1, 2 1 Key Lab of Network Security and Cryptology, Fujian Normal University, Fuzhou 350007, China 2 School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China Correspondence should be addressed to Xiangzeng Kong, xzkong@fjnu.edu.cn Received 26 June 2010; Accepted 12 September 2010 Academic Editor: P. J. Y. Wong Copyright q 2010 Xiangzeng Kong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A nonautonomous N-species discreteLotka-Volterracompetitivesystemwithdelaysandfeedback controls is considered in this work. Sufficient conditions onthe coefficients are given to guarantee that all the species are permanent. It is shown that these conditions are weaker than those of Liao et al. 2008. 1. Introduction Traditional Lotka-Volterracompetitive systems have been extensively studied by many authors 1–7.The autonomous model can be expressed as follows: u i t b i u i t ⎡ ⎣ 1 − N j1 a ij u j t ⎤ ⎦ ,i 1, ,N, 1.1 where b i > 0, a ii > 0, a ij ≥ 0 i / j, u i tdenoting the density ofthe ith species at time t.Montes de Oca and Zeeman 6 investigated the general nonautonomous N-species Lotka-Volterracompetitivesystem u i t u i t ⎡ ⎣ b i t − N j1 c ij t u j t ⎤ ⎦ ,c ij ≥ 0,i 1, ,N, 1.2 2 Advances in Difference Equations and obtained that if the coefficients are continuous and bounded above and below by positive constants, and if for each i 2, ,N,there exists an integer k i <isuch that b i c ij < b ki c k i j ,j 1, ,i, 1.3 then u i → 0 exponentially for 2 ≤ i ≤ N, and u i t → X ∗ , where X ∗ is a certain solution ofa logistic equation. Teng 8 and Ahmad and Stamova 9 also studied the coexistence ona nonautonomous Lotka-Volterracompetitive system. They obtained the necessary or sufficient conditions for the permanence andthe extinction. For more works relevant to system 1.1, one could refer to 1–9 andthe references cited therein. However, to the best ofthe authors’ knowledge, to this day, still less scholars consider the general nonautonomous discreteLotka-Volterracompetitivesystemwithdelaysandfeedback controls. Recently, in 1 Liao et al. considered the following general nonautonomous discreteLotka-Volterracompetitivesystemwithdelaysandfeedback controls: x i n 1 x i n exp ⎧ ⎨ ⎩ b i n − N j1 a ij n x j n − τ ij − d i n u i n ⎫ ⎬ ⎭ , Δu i n r i n − e i n u i n c i n x i n − σ i ,i 1, 2, ,N, x i θ φ i θ ≥ 0,θ∈ N −τ,0 : { −τ,−τ 1, ,−1, 0 } , 1.4 where x i ni 1, 2, ,N is the density ofcompetitive species; u i n is the control variable; e i n : Z → 0, 1; bounded sequences r i n, c i n, b i n, a ij n,andd i n : Z → R ; τ ij and σ i are positive integer; Z, R denote the sets of all integers and all positive real numbers, respectively; Δ is the first-order forward difference operator Δu i nu i n 1 − u i n; τ max{max 1≤i,j≤N τ ij , max 1≤i≤N σ i } > 0. In 1, Liao et al. obtained sufficient conditions for permanence ofthesystem 1.4. They obtained what follows. Lemma 1.1. Assume that min 1≤i≤N M i Δ i > 1 1.5 hold, then system 1.4 is permanent, where M i Δ i exp b u i − 1 a l ii exp −b u i τ ii · a u ii exp τ ii N j1 a u ij M j W i d u i − b l i b l i − N j1,j / i a u ij M j − d u i W i , W i r u i c u i M i e l i ,M i exp b u i − 1 a l ii exp −b u i τ ii . 1.6 Advances in Difference Equations 3 Since exp b u i − 1 > 0,a l ii exp −b u i τ ii > 0,a u ii exp ⎧ ⎨ ⎩ τ ii ⎛ ⎝ N j1 a u ij M j W i d u i − b l i ⎞ ⎠ ⎫ ⎬ ⎭ > 0. 1.7 Hence, the above inequality 1.5 implies b l i − N j1,j / i a u ij M j − d u i W i > 0. 1.8 That is b l i > N j1,j / i a u ij M j d u i W i N j1,j / i a u ij M j d u i r u i c u i M i e l i N j1,j / i a u ij M j d u i r u i e l i d u i c u i M i e l i . 1.9 It was shown that in [1] Liao et al. considered system 1.4 where all coefficients r i n, c i n, d i n, a ij n, e i n, and b i n were assumed to satisfy conditions 1.9. In this work, we shall study system 1.4 and get the same results as 1 do under the weaker assumption that b l i > N j1,j / i a u ij M j d u i r u i e l i . 1.10 Our main results are the following Theorem 1.2. Theorem 1.2. Assume that 1.10 holds, then system 1.4 is permanent. Remark 1.3. The inequality 1.9 implies 1.10, but not conversely, for N j1,j / i a u ij M j d u i r u i e l i ≤ N j1,j / i a u ij M j d u i r u i e l i d u i c u i M i e l i . 1.11 Therefore, we have improved the permanence conditions of 1 for system 1.4. Theorem 1.2 will be proved in Section 2.InSection 3, an example will be given to illustrate that 1.10 does not imply 1.9; that is, the condition 1.10 is better than 1.9. 4 Advances in Difference Equations 2. Proof of Theorem 1.2 The following lemma can be found in 10. Lemma 2.1. Assume that A>0 and y0 > 0, and further suppose that (1) y n 1 ≤ Ay n B n ,n 1, 2, 2.1 Then for any integer k ≤ n, y n ≤ A k y n − k k−1 i0 A i B n − i − 1 . 2.2 Especially, if A<1 and B is bounded above with respect to M,then lim n →∞ sup y n ≤ M 1 − A . 2.3 2 y n 1 ≥ Ay n B n ,n 1, 2, 2.4 Then for any integer k ≤ n, y n ≥ A k y n − k k−1 i0 A i B n − i − 1 . 2.5 Especially, if A<1 and B is bounded below with respect to m ∗ ,then lim n →∞ inf y n ≥ m ∗ 1 − A . 2.6 Following comparison theorem of difference equation is Theorem 2.1 of [11 , page 241]. Lemma 2.2. Let n ∈ N n 0 {n 0 ,n 0 1, ,n 0 l, }, r ≥ 0. For any fixed n, gn, r is a nondecreasing function with respect to r, and for n ≥ n 0 , following inequalities hold: yn 1 ≤ gn, yn, un 1 ≥ gn, un. If gn 0 ≤ un 0 ,thenyn ≤ un for all n ≥ n 0 . Now let us consider the following single species discrete model: N n 1 N n exp { a n − b n N n } , 2.7 where {an} and {bn} are strictly positive sequences of real numbers defined for n ∈ N {0, 1, 2, } and 0 <a l ≤ a u ,0<b l ≤ b u . Similarly to the proof of Propositions 1 and 3 in 12, we can obtain the following. Advances in Difference Equations 5 Lemma 2.3. Any solution ofsystem 2.7 with initial condition N0 > 0 satisfies m ≤ lim n →∞ inf N n ≤ lim n →∞ sup N n ≤ M, 2.8 where M 1 b l exp { a u − 1 } ,m a l b u exp a l − b u M . 2.9 The following lemma is direct conclusion of 1. Lemma 2.4. Let xnx 1 n,x 2 n, ,x N n,u 1 n,u 2 n, ,u N n denote any positive solution ofsystem 1.4.Then there exist positive constants M i ,W i i 1, 2, ,N such that lim n →∞ sup x i n ≤ M i , lim n →∞ sup u i n ≤ W i ,i 1, 2, ,N, 2.10 where M i exp b u i − 1 a l ii exp −b u i τ ii ,W i r u i c u i M i e l i i 1, 2, ,N . 2.11 Proposition 2.5. Suppose assumption 1.10 holds, then there exist positive constant m i and w i such that lim n →∞ inf x i n ≥ m i , lim n →∞ inf u i n ≥ w i . 2.12 Proof. We first prove lim n →∞ inf x i n ≥ m i . By Lemma 2.4 and by the first equation ofsystem 1.4, we have x i n 1 x i n exp ⎧ ⎨ ⎩ b i n − N j1 a ij n x j n − τ ij − d i n u i n ⎫ ⎬ ⎭ ≥ x i n exp ⎧ ⎨ ⎩ b i n − N j1 a ij M j ε − d i n W i ε ⎫ ⎬ ⎭ 2.13 for n sufficiently large, then n−1 sn−τ ii x i s 1 x i s ≥ exp ⎧ ⎨ ⎩ n−1 sn−τ ii ⎛ ⎝ b i s − N j1 a ij s M j ε − d i s W i ε ⎞ ⎠ ⎫ ⎬ ⎭ . 2.14 6 Advances in Difference Equations Thus x i n − τ ii ≤ x i n exp n−1 sn−τ ii D i s , 2.15 where D i s N j1 a ij s M j ε d i s W i ε − b i s . 2.16 From the second equation ofsystem 1.4, we have u i n 1 − e i n u i n c i n x i n − σ i r i n ≤ 1 − e l i u i n c i n x i n − σ i r i n : A i u i n B i n . 2.17 Then, Lemma 2.1 implies that for any k ≤ n − τ ii , u i n ≤ A k i u i n − k k−1 j0 A j i B i n − j − 1 A k i u i n − k k−1 j0 A j i r i n − j − 1 c i n − j − 1 x i n − j − 1 − σ i ≤ A k i u i n − k k−1 j0 A j i r i n − j − 1 c u i exp j 1 σ i D u i x i n ≤ A k i u i n − k k−1 j0 A j i r u i k−1 j0 A j i c u i c u i exp j 1 σ i D u i x i n ≤ A k i W i 1 − A k i 1 − A i r u i H i x i n , 2.18 where H i ⎡ ⎣ k−1 j0 A j i c u i c u i exp j 1 σ i D u i ⎤ ⎦ u . 2.19 For any small positive constant ε>0, there exists a K>0 such that d u i W i − r u i d u i 1 − A i A k i <ε ∀k>K. 2.20 Advances in Difference Equations 7 From the first equation ofsystem 1.4, 2.18,and2.20, we have x i n 1 ≥ x i n exp ⎧ ⎨ ⎩ b i n − N j1,j / i a ij n M j − a u ii exp τ ii D u i x i n −d u i W i A k i − 1 − A k i 1 − A i r u i d u i − d u i H i x i n ⎫ ⎬ ⎭ x i n exp ⎧ ⎨ ⎩ b i n − N j1,j / i a ij n M j − r u i d u i 1 − A i − d u i W i − r u i d u i 1 − A i A k i − a u ii exp τ ii D u i d u i H i x i n ⎫ ⎬ ⎭ ≥ x i n exp ⎧ ⎨ ⎩ b i n − N j1,j / i a ij n M j − r u i d u i 1 − A i − ε − a u ii exp τ ii D u i d u i H i x i n ⎫ ⎬ ⎭ . 2.21 By Lemmas 2.2 and 2.3, we have lim n →∞ inf x i n ≥ b l i − N j1,j / i a u ij M j − r u i d u i /e l i − ε a u ii exp τ ii D u i d u i H i · exp ⎧ ⎨ ⎩ b l i − N j1,j / i a u ij M j − r u i d u i e l i − ε − a u ii exp τ ii D u i d u i H i M i ⎫ ⎬ ⎭ . 2.22 Setting ε → 0in2.22 leads to lim n →∞ inf x i n ≥ b l i − N j1,j / i a u ij M j − r u i d u i /e l i a u ii exp τ ii D u i d u i H i · exp ⎧ ⎨ ⎩ b l i − N j1,j / i a u ij M j − r u i d u i e l i − a u ii exp τ ii D u i d u i H i M i ⎫ ⎬ ⎭ . 2.23 Thus, lim n →∞ inf x i n ≥ m i , 2.24 8 Advances in Difference Equations where m i b l i − N j1,j / i a u ij M j − r u i d u i /e l i a u ii exp τ ii D u i d u i H i · exp ⎧ ⎨ ⎩ b l i − N j1,j / i a u ij M j − r u i d u i e l i − a u ii exp τ ii D u i d u i H i M i ⎫ ⎬ ⎭ . 2.25 Second, we prove lim n →∞ inf u i n ≥ w i . For enough small ε>0, from the second equation ofsystem 1.4, we have u i n 1 1 − e i n u i n r i n c i n x i n − σ i ≥ r l i c l i m i − ε 1 − e u i u i n 2.26 for sufficient large n. Hence u i n ≥ 1 − e u i n u i 0 1 − 1 − e u i e u i r l i c l i m i − ε . 2.27 Thus, we obtain lim n →∞ inf u i n ≥ w i . 2.28 This completes the proof. 3. An Example In this section, we give an example to illustrate that 1.10 does not imply 1.9. Consider the two-species systemwithdelaysandfeedback controls for t ∈ −∞, ∞ x 1 n 1 x 1 n exp 1 2 − 2x 1 n − 1 − 1 2 x 2 n − 3 − 1 2 u 1 n , x 2 n 1 x 2 n exp 1 2 − 1 2 x 1 n − 3 − 2x 2 n − 1 − 1 2 u 2 n , Δu 1 n 1 1 8 − 1 2 u 1 n x 1 n − 4 , Δu 2 n 1 1 8 − 1 2 u 2 n x 2 n − 8 . 3.1 We have b l 1 b l 2 1 2 ,M 1 M 2 1 2 ,a u 12 M 2 d u 1 r u 1 e l 1 3 8 ,a u 21 M 1 d u 2 r u 2 e l 2 3 8 . 3.2 Advances in Difference Equations 9 So b l 1 >a u 12 M 2 d u 1 r u 1 e l 1 ,b l 2 >a u 21 M 1 d u 2 r u 2 e l 2 . 3.3 Therefore 1.10 holds. But 1 2 b l 1 <a u 12 M 2 d u 1 r u 1 c u 1 M 1 e l 1 7 8 , 1 2 b l 2 <a u 21 M 1 d u 2 r u 2 c u 2 M 2 e l 2 7 8 . 3.4 Thus 1.9 does not hold. References 1 X. Liao, Z. Ouyang, and S. Zhou, “Permanence of species in nonautonomous discrete Lotka- Volterra competitivesystemwithdelaysandfeedback controls,” Journal of Computational and Applied Mathematics, vol. 211, no. 1, pp. 1–10, 2008. 2 S. Ahmad, “On the nonautonomous Volterra-Lotka competition equations,” Proceedings ofthe American Mathematical Society, vol. 117, no. 1, pp. 199–204, 1993. 3 S. Ahmad and A. C. Lazer, “On the nonautonomous N-competing species problems,” Applicable Analysis, vol. 57, no. 3-4, pp. 309–323, 1995. 4 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. 5 M. Kaykobad, “Positive solutions of positive linear systems,” Linear Algebra and Its Applications, vol. 64, pp. 133–140, 1985. 6 F. Montes de Oca and M. L. Zeeman, “Extinction in nonautonomous competitiveLotka-Volterra systems,” Proceedings ofthe American Mathematical Society, vol. 124, no. 12, pp. 3677–3687, 1996. 7 M. L. Zeeman, “Extinction in competitiveLotka-Volterra systems,” Proceedings ofthe American Mathematical Society, vol. 123, no. 1, pp. 87–96, 1995. 8 Z. D. Teng, “Permanence and extinction in nonautonomous Lotka-Volterracompetitive systems with delays,” Acta Mathematica Sinica, vol. 44, no. 2, pp. 293–306, 2001. 9 S. Ahmad and I. M. Stamova, “Almost necessary and sufficient conditions for survival of species,” Nonlinear Analysis. Real World Applications, vol. 5, no. 1, pp. 219–229, 2004. 10 Y H. Fan and L L. Wang, “Permanence for adiscrete model withfeedback control and delay,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 945109, 8 pages, 2008. 11 L. Wang and M. Q. Wang, Ordinary Difference Equation, Xinjiang University Press, 1991. 12 F. Chen, “Permanence and global attractivity ofadiscrete multispecies Lotka-Volterra competition predator-prey systems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 3–12, 2006. . Liao, Z. Ouyang, and S. Zhou, “Permanence of species in nonautonomous discrete Lotka- Volterra competitive system with delays and feedback controls,” Journal of Computational and Applied Mathematics,. N, and u i t → X ∗ , where X ∗ is a certain solution of a logistic equation. Teng 8 and Ahmad and Stamova 9 also studied the coexistence on a nonautonomous Lotka-Volterra competitive system. . “Permanence and extinction in nonautonomous Lotka-Volterra competitive systems with delays, ” Acta Mathematica Sinica, vol. 44, no. 2, pp. 293–306, 2001. 9 S. Ahmad and I. M. Stamova, “Almost