A generalized definition of reactivity for ecological systems and the problem of transient species dynamics

Perturbations to an ecosystem's steady state can trigger transient responses of great ecological relevance. Asymptotic stability determines whether a generic perturbation will fade out in the long run, but falls short of characterizing the dynamics immediately after an equilibrium has been perturbed. Reactivity, traditionally defined as the maximum instantaneous growth rate of small perturbations to a stable steady state, is a simple yet powerful measure of the short-term instability of a system as a whole. In many ecological applications, however, it could be important to focus on the reactivity properties of just some specific, problem-dependent state variables, such as the abundance of a focal species engaged in interspecific competition, either predators or preys in a trophic community, or infectious individuals in disease transmission. We propose a generalized definition of reactivity (g-reactivity) that allows to evaluate the differential contribution of the state space components to the transient behaviour of an ecological system following a perturbation. Our definition is based on the dynamic analysis of a system output, corresponding to an ecologically motivated linear transformation of the relevant state variables. We demonstrate that the g-reactivity properties of an equilibrium are determined by the dominant eigenvalue of a Hermitian matrix that can be easily obtained from the Jacobian associated with the equilibrium and the system output transformation. As a testbed for our methodological framework, we analyse the g-reactivity properties of simple spatially implicit metapopulation models of some prototypical ecological interactions, namely competition, predation and transmission of an infectious disease. We identify conditions for the temporary coexistence of an invader with a (possibly competitively superior) resident species, for transitory invasion of either prey or predator in otherwise predator- or prey-dominated ecosystems, and for transient epidemic outbreaks. Through suitable examples, we show that characterizing the transient dynamics associated with an ecosystem's steady state can be, in some cases, as important as determining its asymptotic behaviour, from both theoretical and management perspective. Because g-reactivity analysis can be performed for systems of any complexity in a relatively straightforward way, we conclude that it may represent a useful addition to the toolbox of quantitative ecologists.