Density dependence

A population model has density dependence when its per-capita recruitment, survival or growth depends on population abundance relative to habitat or resource scale. Negative density dependence reduces per-capita growth as crowding increases. In a discrete-generation model, a strong enough decline can make the next total population decrease as the present total increases, producing overcompensation and possible alternating generations. The ecological mechanism and the mathematical stability calculation should be distinguished.

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.