On the Construction of Optimal Paths to Extinction

Abstract : One of the major problems in dealing with interacting finite populations of agents, such as molecules in chemistry or people in populations, is that there always exists a probability of one species or state going extinct. Predicting the probability of extinction requires a knowledge of how the dynamics progresses towards extinction. The path that optimizes the probability to extinction is defined to be the optimal path. Here we present an algorithm for constructing, or growing, the optimal path to extinction in systems of arbitrary dimensions. The algorithm relies on the calculation of finite-time Lyapunov exponents (FTLE), which provide a quantitative measure of how sensitively a system's behavior depends on initial conditions. We also present an efficient method for approximating the FTLE field of a dynamical system.