The ideal free distribution in temporally varying environments

This article studies the Ideal Free Distribution (IFD) in a two-patch stochastic environment under three alternative fitnesses: one density-independent and two density-dependent (hyperbolic and logistic). The hyperbolic fitness function yields the input-matching rule commonly used in experimental studies. The IFD is calculated both for a fixed total population size and, when it exists, for the stationary population distribution. The analysis shows that environmental variability across patches determines whether overmatching or undermatching occurs. Overmatching arises when variability in the patch with the higher deterministic per capita growth rate is sufficiently low relative to that in the other patch, whereas sufficiently high variability reverses this pattern and leads to undermatching.