Bifurcations in periodic integrodifference equations in <inline-formula><tex-math id="M1">\begin{document}$ C(\Omega) $\end{document}</tex-math></inline-formula> Ⅱ: Discrete torus bifurcations

Abstract

We provide a convenient Neimark-Sacker bifurcation result for time-periodic difference equations in arbitrary Banach spaces. It ensures the bifurcation of "discrete invariant tori" caused by a pair of complex-conjugated Floquet multipliers crossing the complex unit circle. This criterion is made explicit for integrodifference equations, which are infinite-dimensional discrete dynamical systems popular in theoretical ecology, and are used to describe the temporal evolution and spatial dispersal of populations with nonoverlapping generations. As an application, we combine analytical and numerical tools for a detailed bifurcation analysis of a spatial predator-prey model. Since such realistic models can frequently only be studied numerically, we formulate our assumptions in such a fashion as to allow for numerically stable verification.

Multiparameter bifurcation in periodic integrodifference equations

“…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 $$$ {\phi}&#x0005E;{\ast }&#x0003D;{\left({\phi}_t&#x0005E;{\ast}\right)}_{t\in \mathbb{Z}} $$$ for some critical parameters.…”

“…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 (1) has a…”