global positive periodic solutions for periodic two species competitive systems with multiple delays and impulses

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 785653, 23 pages http://dx.doi.org/10.1155/2014/785653 Research Article Global Positive Periodic Solutions for Periodic Two-Species Competitive Systems with Multiple Delays and Impulses Zhenguo Luo,1,2 Liping Luo,1 Liu Yang,1 and Yunhui Zeng1 Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China Department of Mathematics, National University of Defense Technology, Changsha 410073, China Correspondence should be addressed to Zhenguo Luo; robert186@163.com Received 19 November 2013; Accepted 25 February 2014; Published April 2014 Academic Editor: Francisco J S Lozano Copyright © 2014 Zhenguo Luo 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 set of easily verifiable sufficient conditions are derived to guarantee the existence and the global stability of positive periodic solutions for two-species competitive systems with multiple delays and impulses, by applying some new analysis techniques This improves and extends a series of the well-known sufficiency theorems in the literature about the problems mentioned previously Introduction Throughout this paper, we make the following notation and assumptions: let 𝜔 > be a constant and 𝐶𝜔 = {𝑥 | 𝑥 ∈ 𝐶(𝑅, 𝑅), 𝑥(𝑡 + 𝜔) = 𝑥(𝑡)}, with the norm being defined by |𝑥|0 = max𝑡∈[0,𝜔] |𝑥(𝑡)|; 𝐶𝜔1 = {𝑥 | 𝑥 ∈ 𝐶1 (𝑅, 𝑅), 𝑥(𝑡 + 𝜔) = 𝑥(𝑡)}, with the norm being defined by ‖𝑥‖ = max𝑡∈[0,𝜔] {|𝑥|0 , |𝑥󸀠 |0 }; 𝑃𝐶 = {𝑥 | 𝑥 : 𝑅 → 𝑅+ , lim𝑠 → 𝑡 𝑥(𝑠) = 𝑥(𝑡), if 𝑡 ≠ 𝑡𝑘 , lim𝑡 → 𝑡𝑘− 𝑥(𝑡) = 𝑥(𝑡𝑘 ), lim𝑡 → 𝑡𝑘+ 𝑥(𝑡) exists, 𝑘 ∈ 𝑍+ }; 𝑃𝐶1 = {𝑥 | 𝑥 : 𝑅 → 𝑅+ , 𝑥󸀠 ∈ 𝑃𝐶}; then those spaces are all Banach spaces We also denote that 𝑓= 𝜔 ∫ 𝑓 (𝑡) 𝑑𝑡, 𝜔 𝑓𝐿 = 𝑓 (𝑡) , 𝑡∈[0,𝜔] 𝑓𝑀 = max 𝑓 (𝑡) , (1) 𝑡∈[0,𝜔] for any 𝑓 ∈ 𝑃𝐶𝜔 In this paper, we investigate the existence, uniqueness, and global stability of the positive periodic solution for two corresponding periodic Lotka-Volterra competitive systems involving multiple delays and impulses: 𝑥1󸀠 (𝑡) = 𝑥1 (𝑡) [𝑟1 (𝑡) − 𝑎1 (𝑡) 𝑥1 (𝑡) 𝑛 𝑃𝐶𝜔 = {𝑥 | 𝑥 ∈ 𝑃𝐶, 𝑥(𝑡 + 𝜔) = 𝑥(𝑡)}, with the norm being defined by |𝑥|0 = max𝑡∈[0,𝜔] |𝑥(𝑡)|; + ∑ 𝑏1𝑖 (𝑡) 𝑥1 (𝑡 − 𝜏𝑖 (𝑡)) 𝑃𝐶𝜔1 = {𝑥 | 𝑥 ∈ 𝑃𝐶1 , 𝑥(𝑡 + 𝜔) = 𝑥(𝑡)}, with the norm being defined by ‖𝑥‖ = max𝑡∈[0,𝜔] {|𝑥|0 , |𝑥󸀠 |0 }; − ∑ 𝑐1𝑗 (𝑡) 𝑥2 (𝑡 − 𝛿𝑗 (𝑡))] , 𝑗=1 ] 𝑖=1 𝑚 Abstract and Applied Analysis They had assumed that the net birth 𝑟(𝑡), the selfinhibition rate 𝑎(𝑡), and the delay 𝜏(𝑡) are continuously differentiable 𝜔-periodic functions, and 𝑟(𝑡) > 0, 𝑎(𝑡) > 0, 𝑏(𝑡) ≥ 0, and 𝜏(𝑡) ≥ for 𝑡 ∈ 𝑅 The positive feedback term 𝑏(𝑡)𝑦(𝑡 − 𝜏(𝑡)) in the average growth rate of species has a positive time delay (the sign of the time delay term is positive), which is a delay due to gestation (see [1, 2]) They had established sufficient conditions which guarantee that system (5) has a positive periodic solution which is globally asymptotically stable In [3], Fan and Wang investigated the following periodic single-species population growth models with periodic delay: 𝑥2󸀠 (𝑡) = 𝑥2 (𝑡) [𝑟2 (𝑡) − 𝑎2 (𝑡) 𝑥2 (𝑡) 𝑚 + ∑ 𝑏2𝑗 (𝑡) 𝑥2 (𝑡 − 𝜂𝑗 (𝑡)) 𝑗=1 𝑛 −∑ 𝑐2𝑖 (𝑡) 𝑥1 (𝑡 − 𝜎𝑖 (𝑡))] , 𝑖=1 𝑡 ≠ 𝑡𝑘 , Δ𝑥𝑙 (𝑡) = 𝑥𝑙 (𝑡+ ) − 𝑥𝑙 (𝑡) = 𝜃𝑙𝑘 𝑥𝑙 (𝑡) , 𝑙 = 1, 2, 𝑘 = 1, 2, , 𝑡 = 𝑡𝑘 , (2) 𝑥1󸀠 (𝑡) = 𝑥1 (𝑡) [𝑟1 (𝑡) − 𝑎1 (𝑡) 𝑥1 (𝑡) 𝑛 − ∑ 𝑏1𝑖 (𝑡) 𝑥1 (𝑡 − 𝜏𝑖 (𝑡)) 𝑖=1 𝑚 − ∑ 𝑐1𝑗 (𝑡) 𝑥2 (𝑡 − 𝛿𝑗 (𝑡))] , 𝑗=1 ] 𝑥2󸀠 (𝑡) = 𝑥2 (𝑡) [𝑟2 (𝑡) − 𝑎2 (𝑡) 𝑥2 (𝑡) (3) 𝑚 − ∑ 𝑏2𝑗 (𝑡) 𝑥2 (𝑡 − 𝜂𝑗 (𝑡)) 𝑗=1 𝑛 −∑ 𝑐2𝑖 (𝑡) 𝑥1 (𝑡 − 𝜎𝑖 (𝑡))] , 𝑖=1 𝑘 = 1, 2, , 𝑦2󸀠 (𝑡) = 𝑦2 (𝑡) [𝑟2 (𝑡) − 𝑎2 (𝑡) 𝑦2 (𝑡) − 𝑐2 (𝑡) 𝑦1 (𝑡)] 𝑡 = 𝑡𝑘 , with initial conditions 𝑥𝑙 (𝜉) = 𝜙𝑙 (𝜉) , 𝜉 ∈ [−𝜏, 0] , 𝑥𝑙󸀠 (𝜉) = 𝜙𝑙󸀠 (𝜉) , 𝜙𝑙 (0) > 0, 𝜙𝑙 ∈ 𝐶 ([−𝜏, 0) , 𝑅+ ) ⋂ 𝐶1 ([−𝜏, 0) , 𝑅+ ) , (4) (7) They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (7) by using differential inequalities and topological degree, respectively In fact, in many practical situations the time delay occurs so often A more realistic model should include some of the past states of the system Therefore, in [10], Liu et al considered two corresponding periodic Lotka-Volterra competitive systems involving multiple delays: 𝑙 = 1, 2, 𝑛 where 𝑎1 (𝑡), 𝑎2 (𝑡), 𝑏1𝑖 (𝑡), 𝑏2𝑗 (𝑡), 𝑐1𝑗 (𝑡), and 𝑐2𝑖 (𝑡) are all in 𝑃𝐶𝜔 Also 𝜏𝑖 (𝑡), 𝛿𝑗 (𝑡), 𝜂𝑗 (𝑡), and 𝜎𝑖 (𝑡) are all in 𝑃𝐶𝜔1 with 𝜏𝑖 (𝑡) > 0, 𝛿𝑗 (𝑡) > 0, 𝜂𝑗 (𝑡) > 0, 𝜎𝑖 (𝑡) > 0, 𝑡 ∈ [0, 𝜔], 𝜏 = max{𝜏𝑖 (𝑡), 𝛿𝑗 (𝑡), 𝜂𝑗 (𝑡), 𝜎𝑖 (𝑡)}, 𝜏𝑖󸀠 (𝑡) < 1, 𝛿𝑗󸀠 (𝑡) < 1, 𝜂𝑗󸀠 (𝑡) < 1, 𝜎𝑖󸀠 (𝑡) < (𝑖 = 1, 2, , 𝑛, 𝑗 = 1, 2, , 𝑚) Furthermore, the 𝜔 intrinsic growth rates 𝑟1 (𝑡), 𝑟2 (𝑡) ∈ 𝑃𝐶𝜔 are with ∫0 𝑟𝑙 (𝑡)𝑑𝑡 > 0, (𝑙 = 1, 2) For the ecological justification of (2) and (3) and similar types refer to [1–10] In [1], Freedman and Wu proposed the following periodic single-species population growth models with periodic delay: 𝑦󸀠 (𝑡) = 𝑦 (𝑡) [𝑟 (𝑡) − 𝑎 (𝑡) 𝑦 (𝑡) + 𝑏 (𝑡) 𝑦 (𝑡 − 𝜏 (𝑡))] (6) They had assumed that the net birth 𝑟(𝑡), the selfinhibition rate 𝑎(𝑡), and the delay 𝜏(𝑡) are continuously differentiable 𝜔-periodic functions, and 𝑟(𝑡) > 0, 𝑎(𝑡) > 0, 𝑏(𝑡) ≥ 0, and 𝜏(𝑡) ≥ for 𝑡 ∈ 𝑅 The negative feedback term −𝑏(𝑡)𝑦(𝑡 − 𝜏(𝑡)) in the average growth rate of species has a negative time delay (the sign of the time delay term is negative), which can be regarded as the deleterious effect of time delay on a species growth rate (see [4–6]) They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (6) But the discussion of global attractivity is only confined to the special case when the periodic delay is constant Alvarez and Lazer [7] and Ahmad [8] have studied the following two-species competitive system without delay: 𝑦1󸀠 (𝑡) = 𝑦1 (𝑡) [𝑟1 (𝑡) − 𝑎1 (𝑡) 𝑦1 (𝑡) − 𝑐1 (𝑡) 𝑦2 (𝑡)] , 𝑡 ≠ 𝑡𝑘 , Δ𝑥𝑙 (𝑡) = 𝑥𝑙 (𝑡+ ) − 𝑥𝑙 (𝑡) = 𝜃𝑙𝑘 𝑥𝑙 (𝑡) , 𝑙 = 1, 2, 𝑦󸀠 (𝑡) = 𝑦 (𝑡) [𝑟 (𝑡) − 𝑎 (𝑡) 𝑦 (𝑡) − 𝑏 (𝑡) 𝑦 (𝑡 − 𝜏 (𝑡))] (5) 𝑦1󸀠 (𝑡) = 𝑦1 (𝑡) [𝑟1 (𝑡) − 𝑎1 (𝑡) 𝑦1 (𝑡) + ∑ 𝑏1𝑖 (𝑡) 𝑦1 (𝑡 − 𝜏𝑖 (𝑡)) 𝑖=1 𝑚 − ∑ 𝑐1𝑗 (𝑡) 𝑦2 (𝑡 − 𝜌𝑗 (𝑡))] , 𝑗=1 ] 𝑚 𝑦2󸀠 (𝑡) = 𝑦2 (𝑡) [𝑟2 (𝑡) − 𝑎2 (𝑡) 𝑦2 (𝑡) + ∑ 𝑏2𝑗 (𝑡) 𝑦2 (𝑡 − 𝜂𝑗 (𝑡)) 𝑗=1 [ 𝑛 −∑ 𝑐2𝑖 (𝑡) 𝑦1 (𝑡 − 𝜎𝑖 (𝑡))] , 𝑖=1 (8) Abstract and Applied Analysis 𝑛 (𝐻3 ) {𝜃𝑙𝑘 } is a real sequence such that 𝜃𝑙𝑘 + > 0, ∏0 0, 𝛿𝑗 (𝑡) > 0, 𝜂𝑗 (𝑡) > 0, 𝜎𝑖 (𝑡) > 0, 𝑡 ∈ [0, 𝜔], 𝜏 = max{𝜏𝑖 (𝑡), 𝛿𝑗 (𝑡), 𝜂𝑗 (𝑡), 𝜎𝑖 (𝑡)}, 𝜏𝑖󸀠 (𝑡) < 1, 𝛿𝑗󸀠 (𝑡) < 1, 𝜂𝑗󸀠 (𝑡) < 1, and 𝜎𝑖󸀠 (𝑡) < (𝑖 = 1, 2, , 𝑛; 𝑗 = 1, 2, , 𝑚) (𝐻2 ) [𝑡𝑘 ]𝑘∈𝑁 satisfies < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑘 < ⋅ ⋅ ⋅ , lim𝑘 → ∞ 𝑡𝑘 = +∞, 𝜃𝑙𝑘 (𝑖 = 1, 2) are constants, and there exists a positive integer 𝑞 > such that 𝑡𝑘+𝑞 = 𝑡𝑘 + 𝜔, 𝜃𝑙(𝑘+𝑞) = 𝜃𝑙𝑘 Without loss of generality, we can assume that 𝑡𝑘 ≠ and [0, 𝜔]∩{𝑡𝑘 } = 𝑡1 , 𝑡2 , , 𝑡𝑚 , and then 𝑞 = 𝑚 (a) 𝑥𝑙 (𝑡) is absolutely continuous on each (𝑡𝑘 , 𝑡𝑘+1 ); (b) for each 𝑘 ∈ 𝑍+ , 𝑥𝑙 (𝑡𝑘+ ) and 𝑥𝑙 (𝑡𝑘− ) exist, and 𝑥𝑙 (𝑡𝑘− ) = 𝑥𝑙 (𝑡𝑘 ); (c) 𝑥𝑙 (𝑡) satisfies the first equation of (2) and (3) for almost everywhere (for short a.e.) in [0, ∞] \ {𝑡𝑘 } and satisfies 𝑥𝑙 (𝑡𝑘+ ) = (1 + 𝜃𝑙𝑘 )𝑥𝑙 (𝑡𝑘 ) for 𝑡 = 𝑡𝑘 , 𝑘 ∈ 𝑍+ = {1, 2, } Under the above hypotheses (𝐻1 )–(𝐻3 ), we consider the following nonimpulsive delay differential equation: 𝑦1󸀠 (𝑡) = 𝑦1 (𝑡) [𝑟1 (𝑡) − 𝐴 (𝑡) 𝑦1 (𝑡) 𝑛 + ∑ 𝐵1𝑖 (𝑡) 𝑦1 (𝑡 − 𝜏𝑖 (𝑡)) 𝑖=1 𝑚 − ∑ 𝐶1𝑗 (𝑡) 𝑦2 (𝑡 − 𝛿𝑗 (𝑡))] , 𝑗=1 ] 𝑦2󸀠 (10) (𝑡) = 𝑥2 (𝑡) [𝑟2 (𝑡) − 𝐴 (𝑡) 𝑦2 (𝑡) 𝑚 + ∑ 𝐵2𝑗 (𝑡) 𝑦2 (𝑡 − 𝜂𝑗 (𝑡)) 𝑗=1 𝑛 −∑ 𝐶2𝑖 (𝑡) 𝑦1 (𝑡 − 𝜎𝑖 (𝑡))] , 𝑖=1 𝑦1󸀠 (𝑡) = 𝑦1 (𝑡) [𝑟1 (𝑡) − 𝐴 (𝑡) 𝑦1 (𝑡) 𝑛 − ∑ 𝐵1𝑖 (𝑡) 𝑦1 (𝑡 − 𝜏𝑖 (𝑡)) 𝑖=1 𝑚 − ∑ 𝐶1𝑗 (𝑡) 𝑦2 (𝑡 − 𝛿𝑗 (𝑡))] , 𝑗=1 ] 𝑦2󸀠 (𝑡) = 𝑥2 (𝑡) [𝑟2 (𝑡) − 𝐴 (𝑡) 𝑦2 (𝑡) 𝑚 − ∑ 𝐵2𝑗 (𝑡) 𝑦2 (𝑡 − 𝜂𝑗 (𝑡)) 𝑗=1 𝑛 −∑ 𝐶2𝑖 (𝑡) 𝑦1 (𝑡 − 𝜎𝑖 (𝑡))] , 𝑖=1 (11) Abstract and Applied Analysis = ∏ (1 + 𝜃1𝑘 ) 𝑦1󸀠 (𝑡) − ∏ (1 + 𝜃1𝑘 ) 𝑦1 (𝑡) with the initial conditions 0