Reactivity and transient dynamics of discrete-time ecological systems

Most studies of ecological models focus exclusively on the asymptotic stability properties of equilibria. However, short-term transient effects can be important, and can in some cases dominate the dynamics seen in experimental or field studies. The reactivity of a stable equilibrium point measures the potential for short-term amplification of perturbations. The reactivity of a fixed point in a discrete-time system is given by the natural logarithm of the largest singular value of the Jacobian matrix of the linear approximation near the fixed point. If the reactivity is positive, the fixed point is said to be reactive. Here we examine the reactivity of discrete-time predator - prey models and density-dependent matrix population models. We find reactivity to be common ( but not universal) and sometimes extremely high. Predator - prey or food web models that include a predator whose per-capita growth rate depends on the density of its prey, but not on its own density, are a special case. Any positive equilibrium of such a model must be reactive. Reactivity of discrete-time models depends on the timing of the census relative to the timing of reproduction. Perturbation analysis of singular values can be used to calculate the sensitivity and elasticity of reactivity to changes in model parameters. We conclude that transient amplification of perturbations should be a common ecological phenomenon. The interaction of these transient effects with the asymptotic nonlinear dynamics warrants further study.