Variation in risk in single-species discrete-time models.

Simple, discrete-time, population models typically exhibit complex dynamics, like cyclic oscillations and chaos, when the net reproductive rate, R, is large. These traditional models generally do not incorporate variability in juvenile "risk,'' defined to be a measure of a juvenile's vulnerability to density-dependent mortality. For a broad class of discrete-time models we show that variability in risk across juveniles tends to stabilize the equilibrium. We consider both density-independent and density-dependent risk, and for each, we identify appropriate shapes of the distribution of risk that will stabilize the equilibrium for all values of R. In both cases, it is the shape of the distribution of risk and not the amount of variation in risk that is crucial for stability.