A  reaction-diffusion model for  the evolution  of dispersal  rates is
considered in  which there is both spatial  heterogeneity and temporal
periodicity.   The model is  restricted to  two phenotypes  because of
technical difficulties,  but a  wide range of  mathematical techniques
and computational effort are needed to obtain useful answers.  We find
that the  question of  selection is  a great deal  richer than  in the
autonomous  case, where  the phenotype  with the  lowest  diffusion is
selected  for.   In the  current  model  either  the lower  or  higher
diffuser  rate  may  be  selected,  or there  may  be  coexistence  of
phenotypes.  The  paper raises several open questions  and suggests in
particular  that  a  mutation-selection multi-phenotypic  model  would
repay study.