Abstract
A discrete time non‐autonomous two‐species competitive system with delays is proposed, which involves the influence of many generations on the density of species population. Sufficient conditions for permanence of the system are given. When the system is periodic, by using the continuous theorem of coincidence degree theory and constructing a suitable Lyapunov discrete function, sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions are obtained. As an application, examples and their numerical simulations are presented to illustrate the feasibility of our main results.
Highlights
In recent years, the application of theories of functional differential equations in mathematical ecology has developed rapidly
Various delayed models have been proposed in the study of population dynamics, ecology, and epidemic
The dynamic behaviors of population models governed by difference equations have been studied by many authors, see 12–18 and the references cited therein
Summary
The application of theories of functional differential equations in mathematical ecology has developed rapidly. Many authors 9–11 have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have non-overlapping generations. The dynamic behaviors of population models governed by difference equations have been studied by many authors, see 12–18 and the references cited therein. Motivated by the above work 19–24 , in this paper we will investigate the following discrete time non-autonomous two-species competitive system with delays: x1 k 1. The principle aim of this paper is to study the dynamic behaviors of system 1.1 , such as permanence, existence, and global attractivity of positive periodic solutions. A positive periodic solution { x1 k , x2 k } of system 1.1 is said to be globally attractive if each other solution { x1 k , x2 k } of system 1.1 satisfies kl→im∞|x1 k − x1 k | 0, kl→im∞|x2 k − x2 k | 0
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