Abstract
This paper is devoted to the existence of periodic solutions for a semi‐ratio‐dependent predator‐prey system with time delays on time scales. With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of periodic solutions. Our results show that for the most monotonic prey growth such as the logistic, the Gilpin, and the Smith growth, and the most celebrated functional responses such as the Holling type, the sigmoidal type, the Ivlev type, the Monod‐Haldane type, and the Beddington‐DeAngelis type, the system always has at least one periodic solution. Some known results are shown to be special cases of the present paper.
Highlights
In the past decades, many authors have investigated the existence of periodic solutions for population models governed by the differential and difference equations 1–7
In order to unify differential and difference equations, people have done a lot of research about dynamic equations on time scales
It is troublesome to study the existence of periodic solutions for continuous and discrete systems, respectively
Summary
Many authors have investigated the existence of periodic solutions for population models governed by the differential and difference equations 1–7. The existence of periodic solutions for semi-ratio-dependent predator-prey systems has been studied extensively in the literature and seen great progress 8–16. It is troublesome to study the existence of periodic solutions for continuous and discrete systems, respectively. We consider the following periodic semi-ratio-dependent predator-prey system with time delays on a time scale T: uΔ1 t g t, eu[1] t−τ1 t − h t, eu[1] t , eu[2] t eu[2] t−τ2 t −u1 t , 1.1. If T R, system 1.1 reduces to the standard semi-ratio-dependent predator-prey system governed by the ordinary differential equations: xtxtgt, x t − τ1 t − h t, x t , y t y t − τ2 t , yt yt c t. Our result generalizes some theorems in 8, 9, 11, 12, 15, 16, 21 , improves and generalizes some theorems in 10, 13, 14, 19, 26
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