Positive periodic solutions in delayed Gause-type predator–prey systems

Abstract

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions in delayed Gause-type predator-prey systems. Some known results are shown to be special cases of the presented paper.

Topological degree in the generalized Gause prey–predator model

“…The fundamental assumptions in Ding-Su-Hao [3] and Ding-Jiang [4] are comparable with ours, however these authors assume x → b(t, x) sub-linear for all x ≥ 0 and we need the later at x = 0 only. The condition (X) plays a similar role as the requirements for time-integrals of the coefficients of (1), that literally all of the papers [12,3,26,21,16,17,4,24] assume (paper [21] doesn't impose any conditions for time-integrals because it deals with nearly constant T -periodic solutions only). Detailed comparison of (X) with the respective assumptions in these papers is outside of the scope of this introduction.…”

“…In combination with the continuity of the topological degree, formula (2) allows to incorporate delays (see Krasnosel'ski [14, Appendix II, §3], Krasnosel'ski-Zabreyko [15, §41.5]) and other functionals (see [15]) into Gause model (1), with potential bearings towards complementing the results in [12,3,16,4] (see introduction). Formula (2) also allows incorporating time-periodic impulses that can be viewed as perturbations of the integral operator Φ.…”

“…A Gause model of similar to (1) (but with a particular form of a(t, x)) is considered in Moghadas-Alexander [21] and Liu-Lou [16] where b x (t, x) > 0 and b xx (t, x) < 0 for all x > 0, t ∈ R. The paper Luo [17] considers a more general form of (1), but requires that a x (t, x) ≤ 0 for all x ≥ 0, t ∈ R and assumes boundedness of b(t, x) and c(t, x) when applied to (1). The fundamental assumptions in Ding-Su-Hao [3] and Ding-Jiang [4] are comparable with ours, however these authors assume x → b(t, x) sub-linear for all x ≥ 0 and we need the later at x = 0 only. The condition (X) plays a similar role as the requirements for time-integrals of the coefficients of (1), that literally all of the papers [12,3,26,21,16,17,4,24] assume (paper [21] doesn't impose any conditions for time-integrals because it deals with nearly constant T -periodic solutions only).…”

“…We understand that the assumptions of some of these papers can be relaxed (in particular, the proofs in Ding-Su-Hao [3] and Ding-Jiang [4] obtained for more complex versions of (1) can possibly be adjusted to our settings). Our introduction doesn't aim to document that we got stronger results, but rather wants to emphasize that our new technique leads to the assumptions, which are different from those used in the relevant literature.…”

“…Our introduction doesn't aim to document that we got stronger results, but rather wants to emphasize that our new technique leads to the assumptions, which are different from those used in the relevant literature. Moreover, our technique may appear simpler (for some readers) than those used in papers [12,3,26,21,16,17,4,24].…”