  We adapt a fitness function from evolutionary game theory as a mechanism for
aggregation and dispersal in a partial differential equation (PDE) model of two
interacting populations, described by density functions $u$ and $v$. We
consider a spatial model where individuals migrate up local fitness gradients,
seeking out locations where their given traits are more advantageous. The
resulting system of fitness gradient equations is a degenerate system having
spatially structured, smooth, steady state solutions characterized by constant
fitness throughout the domain. When populations are viewed as predator and
prey, our model captures prey aggregation behavior consistent with Hamilton's
selfish herd hypothesis. We also present weak steady state solutions in 1d that
are continuous but in general not smooth everywhere, with an associated fitness
that is discontinuous, piecewise constant. We give numerical examples of
solutions that evolve toward such weak steady states. We also give an example
of a spatial Lotka--Volterra model, where a fitness gradient flux creates
instabilities that lead to spatially structured steady states. Our results also
suggest that when fitness has some dependence on local interactions, a
fitness-based dispersal mechanism may act to create spatial variation across a
habitat.
