Bifurcation structure at the reproduction threshold

Characterize the exact bifurcation structure at \(\mathcal R_0=1\) for the reduced nonlinear stationary problem.

Background

The stationary-state result proves existence of at least one nontrivial state when the reduced reproduction number exceeds one under a parametrized compact-operator hypothesis. It does not establish uniqueness, stability, or bifurcation behavior. The precise nature of the transition at the critical value R0=1\mathcal R_0=1 is therefore explicitly identified as unresolved.

References

Other open problems include the proof of uniform global persistence for the nonlinear semiflow, the exact bifurcation structure at \mathcal R_0=1, and the derivation of first-order necessary optimality conditions via a rigorous backward adjoint system.

— Rigorous Analysis of a Nonlocal Transport--Renewal System for Physiologically Structured Populations  (2609.00735 - Yu et al., 1 Sep 2026) in Section Discussion

The case $\beta_1\rho(A)=\mu$ is not decided by the linearization.

— Higher-Order Co-Infection on Hypergraphs  (2610.26072 - Liu et al., 19 Oct 2026) in Proposition 1, Section 3.1, subsection “Disease-free linearization under a mean-hazard closure”