Conjectures on secondary bifurcations for single-hump kernel perturbations

Let $\overline{\phi}\in \overline{K}(\mathbb{R})$ be an admissible kernel perturbation supported in $[-1/2,1/2]$, with $||\overline{\phi}||_1^m$ small. Consider parameters $(a,\overline{D})\in (0,1/4)\times\mathbb{R}^+$ and the positive periodic steady state $\mathcal{F}_p(x,\lambda,D;\overline{\phi})$, where $\lambda=2a+1/2$, continued from $\Omega^+(\overline{\phi})$ into $\Omega^-(\overline{\phi})$. A single-hump perturbation has a positive maximum at the centre of its support and is negative towards the ends; an inverted single-hump perturbation has a negative minimum at the centre and is positive towards the ends. Single-hump kernel perturbation conjectures. For a single-hump perturbation, the continuation provides a unique positive periodic steady state for each $a\in(0,1/4)$ and sufficiently large $\overline{D}$, with the structure described by the source's asymptotic formulas and a symmetric single hump on $[-a,a]$; as $\overline{D}$ decreases, it is expected to develop a symmetric two-hump form, whose separation induces a new localized central hump and a local secondary bifurcation as a fold in the primary branch. For an inverted single-hump perturbation, the continuation likewise provides a unique positive periodic steady state for each $a\in(0,1/4)$ and sufficiently large $\overline{D}$, retaining its symmetric single-hump form and focusing toward a central spike as $\overline{D}$ decreases, thereby inhibiting secondary bifurcations from the principal branch. These proposed extensions of the paper's analysis are not established in the source; the first mechanism is investigated numerically in the subsequent section.