Bounds on Transient Instability For Complex Ecosystems

Stability is a desirable property of complex ecosystems. If a community of interacting species is at a stable equilibrium point then it is able to withstand small perturbations without any adverse effect. In ecology, the Jacobian matrix evalufated at an equilibrium point is known as the community matrix, which represents the population dynamics of interacting species. The system’s asymptotic short- and long-term behaviour can be determined from eigenvalues derived from the community matrix. Here we use results from the theory of pseudospectra to describe intermediate, transient dynamics. We show that the transition from stable to unstable dynamics includes a region of transient instability, where the effect of a small perturbation is amplified before ultimately decaying. The shift from stability to transient instability depends on the magnitude of a perturbation, and we show how to determine lower and upper bounds to the maximum amplitude of perturbations. Of five different types of community matrix, we find that amplification is least severe with predatorprey interactions. This analysis is relevant to other systems whose dynamics can be expressed in terms of the Jacobian matrix. Through understanding transient instability, we can learn under what conditions multiple perturbations—multiple external shocks—will irrecoverably break stability.