Bistability conjecture for a three-class structured population model

Let $u^\star$ be the unique stationary state of the model, and suppose that the model has $N'=3$ structured classes and satisfies the reduction-to-ordinary-differential-equations assumption. Let the $\omega$-limit set describe the asymptotic states of a trajectory.

Bistability conjecture. There exists a choice of parameters such that $u^\star$ is locally asymptotically stable but not globally asymptotically stable. The $\omega$-limit set also contains a locally asymptotically stable limit cycle. In addition, the operator $\mathcal{M}_{\mathcal{B}}$ can be chosen self-adjoint.

This conjecture asserts the existence of a bistable regime in which the unique equilibrium and a stable limit cycle coexist, with limited basins of attraction. The proposed mechanism is a succession of a supercritical Hopf bifurcation and a subcritical Hopf bifurcation; the rigorous proof is identified as future work in the supplied text.