We study the resident-invader dynamics for a class of models of spatial population with a one-dimensional trait, or strategy. We prove the existence of a one-dimensional invariance manifold connecting the two exclusion states via the graph transform method, as well as various global dynamical results on coexistence and exclusion, based on local invasibility criterions including the notions of evolutionary stability and convergence stability in adaptive dynamics. Applications of our abstract results include reactiondiffusion-advection models, patch models and nonlocal dispersal models. In particular, we derive a novel conclusion that a recently established evolutionarily stable dispersal strategy in [Lam-Lou, J. Math. Biol. (2013)] is a neighborhood invader strategy. This is joint work with R.S. Cantrell (Miami) and C. Cosner (Miami).