Evolutionary branching points in multi-dimensional trait spaces

Ecological interaction can induce evolutionary diversification of a biological population into distinct multiple populations. Such a process is called evolutionary branching in adaptive dynamics theory. In one-dimensional trait spaces, the existence of an evolutionary branching point ensures evolutionary branching of an asexual monomorphic population in its neighborhood under rare and small mutations. An evolutionary branching point is a convergence stable point (i.e., a point attractor for a monomorphic population through directional selection) that is not an ESS (evolutionarily stable strategy). For analysis of arbitrarily higher-dimensional trait spaces with respect to evolutionary branching induced by convergence stable non-ESSes, this study develops two kinds of criteria. The two criteria, referred to as branching possibility and branching inevitability, are based respectively on the canonical equation of adaptive dynamics theory and on the local Lyapunov function. The branching possibility ensures the existence of evolutionary paths achieving evolutionary branching. The branching inevitability, which corresponds to Fisher’s fundamental theorem of natural selection, ensures steady progress of evolutionary branching through repeated invasions by arbitrary mutants having positive invasion fitnesses, even when the timescale separation between population dynamics and evolutionary dynamics is incomplete. All strongly convergence stable non-ESSes have the branching possibility. All absolutely convergence stable non-ESSes have both the branching possibility and branching inevitability.