Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays
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RESEARCH
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Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays Changjin Xu1* , Yusen Wu2 and Lin Lu1 *
Correspondence: [email protected] Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550004, P.R. China Full list of author information is available at the end of the article 1
Abstract In this paper, we deal with a discrete Lotka-Volterra predator-prey model with time-varying delays. For the general non-autonomous case, sufficient conditions which ensure the permanence and global stability of the system are obtained by using differential inequality theory. For the periodic case, sufficient conditions which guarantee the existence of a unique globally stable positive periodic solution are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions. MSC: 34K20; 34C25; 92D25 Keywords: Lotka-Volterra predator-prey model; permanence; global attractivity; delay
1 Introduction After the pioneering work of Berryman [] in , the dynamic relationship between predators and their preys has become one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Dynamic nature (including the local and global stability of the equilibrium, the persistence, permanence and extinction of species, the existence of periodic solutions and positive almost periodic solutions, bifurcation and chaos and so on) of predator-prey models has been investigated in a number of notable studies [–]. In many applications, the nature of permanence is of great interest. For example, Fan and Li [] made a theoretical discussion on the permanence of a delayed ratio-dependent predator-prey model with Holling-type functional response. Chen [] addressed the permanence of a discrete n-species delayed foodchain system. Zhao and Jiang [] focused on the permanence and extinction for a nonautonomous Lotka-Volterra system. Chen [] analyzed the permanence and global attractivity of a Lotka-Volterra competition system with feedback control. Zhao and Teng et al. [] established the permanence criteria for delayed discrete non-autonomous-species Kolmogorov systems. For more research on the permanence behavior of predator-prey models, one can see [–]. In , Lv et al. [] investigated the existence and global attractivity of a periodic solution to the following Lotka-Volterra predator-prey system: ©2014 Xu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Xu et al. Advances in Difference Equations 2014, 2014:208 http://www.advancesindifferenceequations.com/content/2014/1/208
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