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The extinction of slowly evolving dynamical systems

A Lasota, M C Mackey

    Journal of Mathematical Biology
    |December 1, 1980
    PubMed
    Summary

    This study introduces a formula for the survival probability of slowly evolving discrete dynamical systems. It explains extinction risks in systems like the quadratic map and cellular growth models.

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    Area of Science:

    • Dynamical Systems Theory
    • Ergodic Theory
    • Mathematical Biology

    Background:

    • Slowly evolving discrete dynamical systems exhibit complex behavior as parameters change.
    • Extinction events can occur in these systems when parameters reach critical values.

    Purpose of the Study:

    • To derive a simple expression for the probability of survival in slowly evolving discrete dynamical systems.
    • To illustrate extinction processes using mathematical models and discuss their biological relevance.

    Main Methods:

    • Utilized recent ergodic theory results from Ruelle, Pianigiani, and Lasota and Yorke.
    • Analyzed the time evolution of systems defined on an interval [0, L] with slowly changing parameters.
    • Applied the derived survival probability expression to specific examples.

    Main Results:

    • A straightforward formula for system survival probability was derived.
    • The quadratic map and a cellular growth model were used to illustrate extinction dynamics.
    • The findings are discussed in the context of chronic myelogenous leukemia patient survival statistics.

    Conclusions:

    • The derived survival probability expression provides a valuable tool for analyzing extinction risks in dynamical systems.
    • The study highlights the applicability of these mathematical concepts to biological processes, including disease progression.

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