Alanna Haslam-Hyde

Title: A New Framework for Harnessing Reactivity

Abstract. In modeling biological systems and other applications, an important recurring question is whether those systems maintain healthy regimes not just at dynamic attractors but also during transient excursions away from those attractors. For ODE models, these excursions are not due only to nonlinearities; they can also occur in the linearization. Since transient excursions in linearized systems are not preserved by diagonalization, we develop a new framework for analyzing linear systems through the lens of reactivity (the maximum instantaneous rate of radial amplification). We shift attention to a radial and tangential decomposition of the vector field and conjugate by rotation to generate equivalent matrices that preserve reactivity. In particular, we use our new framework to pick four standard matrices that give more direct access to reactivity and highlight interesting parallels between reactivity and eigenvectors. We finish by exploring applications of this framework to examples in ecology.