The extinction of slowly evolving dynamical systems

The time evolution of slowly evolving discrete dynamical systems x_{i + 1}= T(r_i,x_i), defined on an interval [0, L], where a parameter r_ichanges slowly with respect to i is considered. For certain transformations T, once r_i reaches a critical value the system faces a non-zero probability of extinction because some x_j [0, L]. Recent ergodic theory results of Ruelle, Pianigiani, and Lasota and Yorke are used to derive a simple expression for the probability of survival of these systems. The extinction process is illustrated with two examples. One is the quadratic map, T(r, x) = rx(1 – x), and the second is a simple model for the growth of a cellular population. The survival statistics for chronic myelogenous leukemia patients are discussed in light of these extinction processes. Two other dynamical processes of biological importance, to which our results are applicable, are mentioned.