Evolutionary branching points in multi-dimensional trait spaces

Ecological interactions play a crucial role in driving speciation and adaptive radiation. According to the adaptive dynamics theory of evolution in one-dimensional trait spaces, such evolutionary diversifications can occur when evolutionary branching points (convergence stable and evolutionarily unstable singular points) exist in the trait spaces. However, these evolutionary diversifications may occur across multiple traits at once, as exemplified by various within-species ecotypic diversifications involving multiple traits or genes. This paper attempt to extend this branching point condition to evolution in multi-dimensional trait spaces. Two criteria, termed branching possibility and branching inevitability, are developed to evaluate the likelihood of evolutionary branching in multi-dimensional trait spaces. Branching possibility guarantees the existence of evolutionary pathways that lead to branching (where deterministic parts of these pathways are described with the canonical equation of adaptive dynamics theory). Branching inevitability guarantees a steady progression of evolutionary branching (where the progression degree is described with a time-increasing function related to the local Lyapunov function, the potential function in an evolutionary potential game, and Fisher's fundamental theorem). Numerically simulated evolution in multi-dimensional trait spaces show that points having branching possibility or inevitability can induce evolutionary branching not only under asexual reproduction but also under sexual reproduction.