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Entropy and convergence in dynamics and demography

S Tuljapurkar1

  • 1Stanford University, CA 94305.

Journal of Mathematical Biology
|January 1, 1993
PubMed
Summary

Population entropy, a measure of convergence rate in Markov chains, offers insights into demographic dynamics. This study explores its relationship with eigenvalues of the Leslie matrix, clarifying its demographic meaning.

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Response.

Science (New York, N.Y.)·1995

Area of Science:

  • Demography
  • Mathematical Biology
  • Dynamical Systems

Background:

  • Demographic dynamics can be modeled as Markov chains.
  • Population entropy (Kolmogorov-Sinai entropy) relates to convergence rate to demographic equilibrium.
  • Previous work by Tuljapurkar (1982) established entropy's role in convergence analysis.

Purpose of the Study:

  • To provide elementary proofs for the relationship between convergence rate and entropy in finite state Markov chains.
  • To detail the applications and constraints of using entropy as a convergence metric.
  • To explore the demographic interpretation of population entropy and its connection to Leslie matrix eigenvalues.

Main Methods:

  • Analysis of finite state Markov chains.
  • Qualitative and quantitative arguments for demographic interpretation.
  • Derivation of an exact relationship between population entropy and Leslie matrix eigenvalues.

Main Results:

  • Elementary proofs demonstrating the link between convergence rate and entropy.
  • Discussion on the utility and limitations of entropy as a convergence measure.
  • An exact formula connecting population entropy to the eigenvalues of the Leslie matrix characteristic equation.

Conclusions:

  • Population entropy is a valuable tool for understanding demographic convergence rates.
  • The study clarifies the demographic significance of entropy, especially in relation to Leslie matrix dynamics.
  • The findings are applicable to both Markovian dynamical systems and demographic modeling.

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