Continuation of fixed points and bifurcations from ODE to flow-kick disturbance models

Abstract

Some ODE models treat ecological disturbance as a continuous process, even disturbances such as fire that occur almost instantaneously on the timescale of system recovery. Alternatively, flow-kick models resolve disturbances as discrete impulses that change an ODE system's state periodically in time. Here we compare the dynamics of continuously disturbed ODE models to those of flow-kick models with the same average disturbance rate. In the case that kicks are small and high-frequency, we find multiple similarities between continuous and analogous discrete disturbance models. First, we prove that flow-kick maps generate an analogous vector field in the limit as the period between kicks approaches zero. Second, we present conditions under which equilibria, saddle-node bifurcations, and transcritical bifurcations continue from ODE to flow-kick systems. On the other hand, we also provide numerical evidence that similarities between continuous and discrete disturbance models can break down as the period between kick grows. We illustrate implications of these differences for climate change in a nonspatial Klausmeier model of vegetation and precipitation dynamics. We conclude that although ODEs may suffice to model high-frequency disturbances, resolving lower-frequency disturbances in time may be essential to effectively predicting their effects.