Hopf bifurcation behavior for intermediate nonlocal weights

Investigate the occurrence and type of Hopf bifurcation for the intermediate choices of the nonlocal weight function c_\varepsilon arising between the two explicitly constructed choices that produce supercritical and subcritical Hopf bifurcations, respectively.

Background

The paper constructs a one-dimensional nonlocal population equation with a three-peak effective growth profile and uses a finite-dimensional spectral reduction to obtain Hopf bifurcations. With the local coefficients a and b fixed, two different choices of the nonlocal weight c_\varepsilon near the second and third maximum points yield, respectively, supercritical and subcritical Hopf bifurcations.

The construction imposes an affine relation between the values c_\varepsilon(x_2) and c_\varepsilon(x_3). The authors analyze only the endpoint choices corresponding to the two bifurcation types and explicitly leave unresolved what happens for the intermediate admissible choices of c_\varepsilon.

References

The two constructions use the same q, a_{\varepsilon,\eta}, and b_{\varepsilon,\eta}. Two different choices of c_\varepsilon near x_2 and x_3 yield supercritical and subcritical Hopf bifurcations, while c_\varepsilon remains unchanged in a fixed neighborhood of x_1; see Figure~\ref{fig:three-peak-coeff-profiles}. In this construction, for each fixed sufficiently small \varepsilon, the values c_\varepsilon(x_2) and c_\varepsilon(x_3) satisfy an affine relation; see the proof of Lemma~\ref{lem:positive-profiles}. But we do not investigate the occurrence or type of Hopf bifurcation for the intermediate choices of c_\varepsilon.

— Stability and Instability of the Unique Positive Steady State for a Nonlocal Population Equation  (2610.02620 - Lou et al., 2 Oct 2026) in Section 2, paragraph immediately following Theorem 2.3 (Theorem \ref{thm:three-peak-hopf})