Topological degree in the generalized Gause prey–predator model

Abstract

We consider a generalized Gause prey-predator model with T -periodic continuous coefficients. In the case where the Poincaré map P over time T is well defined, the result of the paper can be explained as follows: we locate a subset U of R 2 such that the topological degree d(I − P, U) equals to +1. The novelty of the paper is that the later is done under only continuity and (some) monotonicity assumptions for the coefficients of the model. A suitable integral operator is used in place of the Poincaré map to cope with possible non-uniqueness of solutions. The paper, therefore, provides a new framework for studying the generalized Gause model with functional differential perturbations and multi-valued ingredients.

Coexistence in seasonally varying predator–prey systems with Allee effect

“…We have that if λ + 2 > 1 the same is verified by the basic reproduction number associated to (N * + (t), 0) but in this case persistence is not guaranteed by this condition. The result on the existence of a periodic solution is based on degree theory following ideas in [44], see also [34,3,2,32]. In the case 1 < λ − 2 < λ + 2 , numerical simulations lead us to conjecture that the only feature is the predator extinction, but this is still an open problem that we want to address in a future work.…”