Multi-time scales in adaptive dynamics: microscopic interpretation of a trait substitution tree model

We consider a fitness-structured population model with competition and migration between nearest neighbors. Under a combination of large population and rare migration limits we are particularly interested in the asymptotic behavior of the total population partition on supporting trait sites. For the population without mutation on a finite trait space we obtain the equilibrium configuration and characterize the right time scale for fixation. For the model with mutation on an infinite trait space a jump process-trait substitution tree model is established on a rarer mutation time scale against the rare migration constrained in terms of a large population limit. Due to a change of the fitness landscape provoked by a new mutant, some temporarily unfit types can be recovered from time to time. In the end we shed light to illustrate sexual reproduction in a diploid population on the genetic level.