Invasion analysis for structured populations: analytical principles and numerical workflows

All real populations show some form of structure—differences among individuals, such as life stage or physiological condition, that shape a species' ability to persist. Yet it remains challenging to incorporate population structure into models of species interactions. We provide a general framework for studying competition in structured populations by quantifying species' invasion growth rates. The crux of our approach is separation of timescales: the initial dynamics of a structured population depend on how exactly it is introduced, but these transient differences do not affect long-term invasion success. Depending on the research question, the relevant long-term growth rate when rare may equivalently be derived using analytical separation of timescales, matrix population theory, or numerical simulations. Using case studies spanning developmental stage, environmental state, and infection status, we illustrate how our methodology derives simple, ecologically meaningful invasion conditions for a variety of systems. Finally, we extend the approach to fluctuating resident dynamics, showing how to use simulations or a general mathematical framework known as Floquet theory to correctly quantify invasion growth. By unifying the quantitative methods of population biology and community ecology, our framework will allow ecologists to study species persistence and coexistence across a greater variety of demographically complex systems.