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A population is said to have an ideal free distribution in a spatiallyheterogeneous but temporally constant environment if each of its members havechosen a fixed spatial location in a way that optimizes its individual fitness,allowing for the effects of crowding. In this paper, we extend the idea ofindividual fitness associated with a specific location in space to account forthe full path that an individual organism takes in space and time over aperiodic cycle, and extend the mathematical formulation of an ideal freedistribution to general time periodic environments. We find that, as in manyother cases, populations using dispersal strategies that can produce ageneralized ideal free distribution have a competitive advantage relative topopulations using strategies that do not produce an ideal free distribution. Asharp criterion on the environmental functions is found to be necessary andsufficient for such ideal free distribution to be feasible. In the case thecriterion is met, we showed that there exist dispersal strategies that can beidentified as producing a time-periodic version of an ideal free distribution,and such strategies are evolutionarily steady and are neighborhood invadersfrom the viewpoint of adaptive dynamics. Our results extend previous works inwhich the environments are either temporally constant, or temporally periodicbut the total carrying capacity is temporally constant.

Ideal Free Dispersal under General Spatial heterogeneity and Time Periodicity