Invariant Manifolds in a Class-Structured Model From Adaptive Dynamics

Abstract

We consider a family of structured population models from adaptive dynamics in which cells transition through a number of growth states, or classes, before division. We prove the existence and global asymptotic stability of invariant (‘resident') manifolds in that family; furthermore, we re-derive conditions under which scarce mutants can invade established resident populations, and we show the existence of corresponding ‘invasion’ manifolds that are obtained as critical manifolds under the additional assumption that resident has attained quasi-steady state, which induces a separation of scales. Our analysis is based on standard phase space techniques for ordinary differential equations, in combination with the geometric singular perturbation theory due to Fenichel.