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

TL;DR

This work develops a rigorous local bifurcation theory for θ-periodic integrodifference equations on compact habitats, recasting periodic solutions as zeros of a cyclic operator G and applying Fredholm- and Morse-index arguments to obtain fold, crossing-curve, and period-doubling bifurcation criteria. The results apply to general IDEs with integral-growth and dispersal structures, and are complemented by a Nyström-based discretization framework for numerics, enabling explicit verification in examples with degenerate kernels and common ecological kernels. Key contributions include explicit conditions for transcritical, pitchfork, and sub-/supercritical folds, as well as symmetry-induced bifurcation patterns and stability exchanges along branches. The applications section demonstrates the framework on Beverton–Holt and Ricker-type models, illustrating primary bifurcations on the trivial branch and complex dynamics such as period-doubling cascades in spatially structured populations. Overall, the paper provides a versatile analytic-numeric toolkit for identifying and classifying critical transitions in periodic IDEs relevant to theoretical ecology and related fields.

Abstract

We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinite-dimensional 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 curve-type, 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.

Figures (15)

• Figure 1: Equilibrium branches $\phi^\pm(aL)$ as functions of the dispersal parameter $\alpha=aL$ (left). Four largest eigenvalues $\lambda^\pm(aL)$ along these two branches of nontrivial solutions to \ref{['noall']} (right)
• Figure 2: Subcritical ($\tfrac{g_{20}}{g_{11}}>0$) and supercritical ($\tfrac{g_{20}}{g_{11}}<0$) fold bifurcation of $\theta$-periodic solutions to \ref{['deq']} described in Thm. \ref{['thmfold']}, as well as the exchange of stability between the branches $\Gamma^+$ and $\Gamma^-$ from unstable (dashed line) to exponentially stable (solid) covered in Cor. \ref{['bifurmorse']}
• Figure 3: Transcritical bifurcation of $\theta$-periodic solutions to \ref{['deq']} from a branch $\Gamma_1$ into $\Gamma_2$ described in Prop. \ref{['proptrans']}, as well as the exchange of stability from unstable (dashed line) to exponentially stable (solid)
• Figure 4: Subcritical $(\bar{g}/g_{11}>0)$ and supercritical $(\bar{g}/g_{11}<0)$ pitchfork bifurcation of $\theta$-periodic solutions to \ref{['deq']} from a branch $\Gamma_1$ into $\Gamma_2$ described in Prop. \ref{['proppitch']}, as well as the exchange of stability from unstable (dashed line) to exponentially stable (solid)
• Figure 5: Branch of the subcritical fold bifurcation for \ref{['deq']} with right-hand side \ref{['exfold1']} and kernel \ref{['kerfinrad']} (left). Total population over $\alpha\in[0.0,0.3]$ with $a=\tfrac{1}{4}$, $L=2$ along the branch (right)

Theorems & Definitions (72)

• Theorem 2.1
• proof
• Proposition 2.2: properties of $G$
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof