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.

Keywords:

• Permanence
• Extinction
• Balancing survival
• Discrete Lotka–Volterra systems
• Time-delays

Acknowledgments

The authors would like to extend their appreciation to Prof. Lin He and Dr.Yi–jie Liu for their help and comments on the first draft of this article. We also thank Dr. Sanyi Tang, Dr. Wanbiao Ma, Prof.Wendi Wang, Prof. M.J. Keeling, Prof. L.R. Ginzbur and Prof. John E. Franke for sending their reprints/preprints to us. This work was supported by the Chinese Postdoctoral Science Foundation, Academy of Finland and by the National Natural Science Foundation of China (No. 10171106).

Notes

^‡E-mail: [email protected]

^†E-mail: [email protected]

^†E-mail: [email protected]

Additional information

Notes on contributors

Ravi P. Agarwal ‡

‡E-mail: [email protected] †E-mail: [email protected]