Abstract
A nonautonomous discrete two-species Lotka-Volterra competition system with infinite delays and single feedback control is considered in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained. It is shown that if the the discrete Lotka-Volterra competitive system with infinite delays and without feedback control is permanent, then, by choosing some suitable feedback control variable, the permanent species will be driven to extinction. That is, the feedback control variable, which represents the biological control or some harvesting procedure, is the unstable factor of the system. Such a finding overturns the previous scholars’ recognition on feedback control variables.
Highlights
Wu and Zhang [19] proposed a discrete autonomous Lotka-Volterra competition system with infinite delays and feedback controls; by using the iterative method, sufficient conditions which ensure the global attractivity of the system were obtained
One of the purposes of this paper is to find out the influence of feedback control variable on the persistent property of the system
(1) Li et al [3] consider a continuous and autonomous Lotka-Volterra competitive system with infinite delays and feedback controls; if the Lotka-Volterra competitive system is globally stable, they showed that the feedback controls only change the position of the unique positive equilibrium and retain the stable property
Summary
The study of the dynamic behaviors of discrete time models governed by difference equation has become one of the most important topics in mathematics biology; many interesting results concerned with permanence, extinction, and existence of positive periodic solution (almost periodic solution) and so forth have been extensively studied by many scholars; see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] and the references cited therein. Chen [29] and Liao et al [30] proposed a discrete time periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments; some sufficient conditions which ensure the existence of unique globally asymptotically stable periodic solution were obtained. Wu and Zhang [19] proposed a discrete autonomous Lotka-Volterra competition system with infinite delays and feedback controls; by using the iterative method, sufficient conditions which ensure the global attractivity of the system were obtained. We mention here that this is the first time such kind of model is proposed and studied, and, as far as system (5) is concerned, whether the single feedback control variable has influence on the persistent property of the system or not is an interesting problem.
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