Sharpness of saturation in harvesting and predation

Harvesting and predation occur through contact processes in which the rate at which the managed (prey) population can be found depends on the population size, usually saturating at high densities. Many models incorporate saturation in this process without considering the effects of the particular function used to describe it. We show that the sharpness with which this saturation occurs has an important effect upon the resulting population dynamics, with bistability (sometimes involving a stable equilibrium and a stable limit cycle) occurring for saturation that is any sharper than the commonly used Michaelis-Menten (Holling type II) functional response. This sharpness threshold occurs across a wide range of model types, from simple harvesting to density-dependent and ratio-dependent predation.