Journal of Mathematical Biology

, Volume 10, Issue 4, pp 333–345

The extinction of slowly evolving dynamical systems

  • A. Lasota
  • Michael C. Mackey

DOI: 10.1007/BF00276093

Cite this article as:
Lasota, A. & Mackey, M.C. J. Math. Biology (1980) 10: 333. doi:10.1007/BF00276093


The time evolution of slowly evolving discrete dynamical systems xi + 1= T(ri,xi), defined on an interval [0, L], where a parameter richanges slowly with respect to i is considered. For certain transformations T, once ri reaches a critical value the system faces a non-zero probability of extinction because some xj ∋ [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.

Key words

Ergodic theoryExtinctionCell populationsLeukemia

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • A. Lasota
    • 1
  • Michael C. Mackey
    • 2
  1. 1.Institute of MathematicsSilesian UniversityKatowicePoland
  2. 2.Department of PhysiologyMcGill UniversityMontrealCanada