Evolutionary invasion analysis for structured populations: a synthesis

Natural populations exhibit complex class structures that profoundly shape evolutionary trajectories. While evolutionary demography provides a formal framework to predict adaptation using invasion fitness, the high mathematical dimensionality of these models often precludes analytical solutions, obscuring biological interpretation and hindering the analysis of long-term evolutionary outcomes. Because current reduction techniques remain fragmented, a unifying theoretical foundation is critically needed. Here, we introduce "structural evolutionary invasion analysis," a systematic framework that integrates two complementary tools to simplify complex life cycles. First, we formulate the "invasion determinant," an algebraic method that yields a direct scalar condition for mutant invasion. Second, we develop the Projected Next-Generation Matrix (PNGM), which structurally compresses life-cycle graphs by eliminating secondary classes. We demonstrate that this reduction is mathematically equivalent to separating dynamical timescales, explicitly preserving Fishers reproductive values for the retained focal classes. Crucially, under the standard assumption of weak selection, our synthesized framework guarantees that all properties of evolutionary singularities--including their location, convergence stability, and evolutionary stability--are strictly identical to those derived from the full, unreduced model. Illustrated with diverse ecological examples, this framework provides modellers with a rigorous and tractable toolkit for decoding state-dependent selection in high-dimensional populations.