Overcompensation in a population model

Overcompensation occurs when a larger present population produces a smaller total population in the next generation over part of the relevant range. It is stronger than a decline of per-capita growth alone. For $F(u)=\alpha u/(\beta +u^{\gamma })$ with $\gamma >1$, the large-abundance response $F(u)\sim \alpha u^{1-\gamma }$ decreases with abundance. Stronger feedback can drive a fixed-point multiplier through minus one, but that local threshold alone does not establish chaotic behaviour.