Bifurcations in periodic integrodifference equations in C(\Omega) I: Analytical results and applications

Abstract

We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinitedimensional discrete dynamical systems arise in theoretical ecology as models to describe the spatial dispersal of species having nonoverlapping generations. Our explicit criteria allow us to identify branchings of fold-and crossing curvetype, which include the classical transcritical-, pitchfork-and flip-scenario as special cases. Indeed, not only tools to detect qualitative changes in models from e.g. spatial ecology and related simulations are provided, but these critical transitions are also classified. In addition, the bifurcation behavior of various time-periodic integrodifference equations is investigated and illustrated. This requires a combination of analytical methods and numerical tools based on Nyström discretization of the integral operators involved.

Bifurcations in periodic integrodifference equations in $C(Ω)$ I: Analytical results and applications

“…Second, this paper addressed simple Floquet multipliers crossing the stability boundary S 1 at ±1 under parameter variation. The complementary situation of complex-conjugated pairs, leading to Neimark-Sacker bifurcations, is featured in the companion paper [1], where we provide conditions for bifurcations of discrete tori.…”

“…apply to systems of IDEs like for instance predator-prey models [33,1]. Another field of applications is the analysis of models capturing an age-or sizestructure of a population [10,11,17,18], which are additionally equipped with a spatial component (cf.…”

Multiparameter bifurcation in periodic integrodifference equations

“…Thereto, for the linear space of continuous functions , it is handy to abbreviate and to write $$$$ {\left\Vert u\right\Vert}_{\infty}:= \underset{x\in \Omega}{\max}\left|u(x)\right| $$$$ for the norm on in what follows. Furthermore, it was established in Aarset and Pötzsche 3, Lemma 3.1 that the bilinear or sesquilinear form yields a bounded dual system .…”

“…Proof In case $$$ A\subset \mathbb{R} $$$, a proof is given in Aarset and Pötzsche, 3, Prop. 2.2 and our more general situation follows along these lines.…”

“…Here, due to the periodicity of (), it is natural to look for periodic solutions rather than fixed points as bifurcating objects. The situation of 1‐parameter bifurcations was extensively studied in previous studies 3,4 . For the sake of bifurcations, it is necessary to assume that () has a nonhyperbolic solution $$$ {\phi}^{\ast }={\left({\phi}_t^{\ast}\right)}_{t\in \mathbb{Z}} $$$ for some critical parameters.…”