Synthesizing the evolutionary invasion analysis for high-dimensional population dynamics

I will present a linear-algebraic (spectral) method for analyzing nonnegative matrices to study the dynamics of natural selection. This is a joint project with Troy Day (Queen's University, Canada).

Within adaptive dynamics theory, evolutionary invasion analysis provides a powerful framework for studying adaptive evolution. It allows us to evaluate (i) whether a new type of individuals (mutants) can successfully invade and replace the resident type, and (ii) whether recurrent substitutions converge to an equilibrium that resists further invasion (an evolutionary Nash equilibrium). A central task is to quantify the reproductive success of mutants, which corresponds to computing the spectral radius (largest eigenvalue) of a nonnegative matrix. However, the high dimensionality of population dynamics often makes the analytical treatment of eigenvalues intractable.

To address this problem, we have developed a methodology that applies to any high-dimensional adaptive dynamics system. I will first introduce the principles of adaptive dynamics and the associated eigenvalue problem. I will then present our new method, which translates the high-dimensional eigenvalue problem into another, lower-dimensional eigenvalue problem of arbitrary size, using (i) Perron–Frobenius theory and (ii) graph-theoretic arguments.