Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / A / 1 / iv

At low density the per-generation growth factor is approximately $\alpha /\beta$. The power $\gamma$ controls how sharply crowding reduces recruitment or survival when resources become limiting; at high density $F(u)\sim \alpha u^{1-\gamma }$. Larger $\gamma$ therefore means a stronger overcompensation in a population model.

The flip threshold says that sufficiently strong density feedback can destabilize a constant positive biological population and create alternating generations, under the usual nondegeneracy conditions. A local loss of fixed-point stability alone does not establish chaos, nor does it prove every ecological system with stronger crowding must follow the same route.