In a seminal publication Hassell and May demonstrated that sufficiently uneven spatial distributions can stabilize predator-prey systems. In this article we investigate whether such spatial distributions (of either predators or prey) can be caused by behavior that is favored by natural selection. If selection operates on predators only, evolutionarily stable patch selection strategies (ESSs) will lead to predator aggregation, provided the prey are unevenly distributed. However, to render the ecological equilibrium stable, prey aggregation needs to be very strong. If selection operates at both trophic levels, then simultaneous ESSs will exist for predator and prey. Where patches are of equal quality (as is implicitly assumed in Hassell and May's model), the distributions of both predators and prey will be homogeneous, and ecological stability will vanish. Where patches differ, for example, in prey reproduction or survival, aggregated distributions of prey and predators will result. A stable ecological equilibrium is then possible, but only if there are many patches of marginal quality. This article shows that the combination of both evolutionary and ecological stability criteria not only allows one to test whether ecological theories are compatible with the theory of natural selection but may also lead to new insights, such as why low-quality patches may constitute a partial refuge for the prey.