Harmless and Profitless Delays in Discrete Competitive Lotka–Volterra Systems

Abstract

Recently, time-delayed discrete population dynamical systems have received much attention. Many authors are interested in studying the effects of time-delays on the dynamical behaviors of discrete systems. Among them, Saito et al. [Y. Saito, W. Ma and T. Hara (2001). Necessary and sufficient condition for permanence of a Lotka–Volterra discrete system with delays. J. Math. Anal. Appl., 256, 162–174; Y. Saito, T. Hara and W. Ma (2002). Harmless delays for permanence and impersistence of Lotka–Volterra discrete predator–prey system. Nonlinear Analysis, 50, 703–715.], Tang and Xiao [S. Tang and Y. Xiao (2001). Permanence in Kolmogorov-Type systems of delay difference equations. J. Diff. Eqns. Appl., 7, 1–15.] have considered the two-species Lotka–Volterra discrete system with time-delays, and they conclude that time-delays therein are harmless for permanence. How will time-delays affect the dynamical behaviors of the general Lotka–Volterra discrete systems? In this article, we discuss a general n-species discrete competitive Lotka–Volterra system with delayed density dependence and delayed interspecific competition. We obtain some new results about the effect of time-delays on permanence, extinction and balancing survival. We conclude that under some conditions, the inclusion, exclusion and change of time-delays do not affect the conditions for the permanence, extinction and balancing survival of species. We also find that time-delays are harmless for both the permanence and balancing survival of species, in addition to being profitless to the extinction of species. In particular, when n = 2, the extinction and permanence of this system are corresponded to some inequalities that only involve the coefficients therein. Importantly, permanence and extinction in this two-species system are determined only by three elements: growth rate, density dependence and interspecific competition rate.