Periodicity in a generalized semi-ratio-dependent predator–prey system with time delays and impulses

Abstract

With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of positive periodic solutions in a generalized semi-ratio-dependent predator-prey system with time delays and impulses, which covers many models appeared in the literature. When the results reduce to the semi-ratio-dependent predator-prey system without impulses, they generalize and improve some known ones.

Positive Almost Periodic Solutions for a Delayed Predator–Prey Model with Hassell-Varley Type Functional Response

“…In System (32), | sin √ 2t| + | sin √ 3t| and cos 2 ( √ 2t) + cos 2 ( √ 3t) are almost periodic functions, which are not periodic functions. Similar to the argument as given in Example 2, it is easy to prove that System (32) gives at least one positive almost periodic solution (see Figures 3 and 4).…”

“…where a, b, c, d, r, δ, τ and σ are nonnegative almost periodic functions and m is a nonnegative constant. It is well known that Mawhin's continuation theorem of coincidence degree theory is an important method to investigate the existence of positive periodic solutions to some kinds of non-linear ecosystems (see [11][12][13][26][27][28][29][30][31][32][33][34]). However, it is difficult to use it to investigate the existence of positive almost periodic solutions of non-linear ecosystems.…”

Dynamic analysis of an impulsive differential equation with time-varying delays

“…However, in real world, delay is not always constant. It is often time-varying [2], [12]. Then how a time-varying delay affects the dynamical behavior of the system?…”

Multiplicity of periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response and harvesting terms

“…The main method which usually used to gain the existence of periodic solutions for periodic systems is the coincidence degree theory developed by Gaines and Mawhin [10], we refer to [11][12][13][14][15][16][17][18][19][20] and references cited therein. While in this paper, we will use another method: the generalized continuation theorem.…”