Iteration of the Cycle II: Extension to the General Multilevel Hierarchy

Abstract

In the previous section we have followed phases 0–4 for up to 3 fitness levels and 2 spatial levels, i.e. we treated the transition $E_{0} \rightarrow E_{1}$ and then $E_{1} \rightarrow E_{2}$. The latter case of 3 fitness levels leading to fixation and neutral equilibrium on E ₂ includes all the complexity of higher level systems, namely, the necessity to deal treating the McKean–Vlasov dynamic on types in E ₂ the perturbations arising with the McKean–Vlasov dynamics at the next lower lower level in smaller space-time scales. The program can otherwise then be continued in essentially the same way to describe the emergence, fixation and neutral equilibrium at levels $E_{3},E_{4},E_{5},\ldots$ in finite spatial systems with finitely many fitness levels. In the next subsection we briefly indicate how the general transition $E_{j-1} \rightarrow E_{j}$ is carried out. In the second subsection we then explain how these transitions for the finite (spatial and number of fitness levels) systems is used to obtain the transitions for the original infinite systems.