  We consider a population living in a patchy environment that varies
stochastically in space and time. The population is composed of two morphs
(that is, individuals of the same species with different genotypes). In terms
of survival and reproductive success, the associated phenotypes differ only in
their habitat selection strategies. We compute invasion rates corresponding to
the rates at which the abundance of an initially rare morph increases in the
presence of the other morph established at equilibrium. If both morphs have
positive invasion rates when rare, then there is an equilibrium distribution
such that the two morphs coexist; that is, there is a protected polymorphism
for habitat selection. Alternatively, if one morph has a negative invasion rate
when rare, then it is asymptotically displaced by the other morph under all
initial conditions where both morphs are present. We refine the
characterization of an evolutionary stable strategy for habitat selection from
[Schreiber, 2012] in a mathematically rigorous manner. We provide a necessary
and sufficient condition for the existence of an ESS that uses all patches and
determine when using a single patch is an ESS. We also provide an explicit
formula for the ESS when there are two habitat types. We show that adding
environmental stochasticity results in an ESS that, when compared to the ESS
for the corresponding model without stochasticity, spends less time in patches
with larger carrying capacities and possibly makes use of sink patches, thereby
practicing a spatial form of bet hedging.
