Lyapunov Exponents and the Mathematics of Invasion in Oscillatory or Chaotic Populations

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93 Citations (Scopus)

Abstract

This paper concisely reviews the mathematical properties of the dominant Lyapunov exponent of a matrix sequence in the context of population biology. The concept of Lyapunov exponent provides a valuable tool for investigating processes of invasion in ecology or genetics, which are crucial in shaping community diversity, determining the spread of epidemics or the fixation of a new mutation. The appeal of the invasibility criterion based on the dominant Lyapunov exponent lies in the opportunity it offers to deal with population structure, complex life cycles, and complex population dynamics resulting from the model nonlinearities (oscillations, chaos), as well as random fluctuations arising from a stochastic environment. We put emphasis on the issues of the existence, numerical approximation, and regularity of the dominant Lyapunov exponent. Our presentation is aimed at showing that, despite our inability to compute the exponent analytically, which adds to its high intrinsic instability, important biological insights can nevertheless be achieved at the cost of fairly mild assumptions on the features of the models considered.

Original languageEnglish (US)
Pages (from-to)126-171
Number of pages46
JournalTheoretical Population Biology
Volume48
Issue number2
DOIs
StatePublished - Oct 1995

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Mathematics
mathematics
invasibility
Population Dynamics
chaotic dynamics
Ecology
Life Cycle Stages
nonlinearity
Population
fixation
population structure
oscillation
mutation
life cycle (organisms)
population dynamics
life cycle
ecology
Biological Sciences
Mutation
matrix

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics
  • Agricultural and Biological Sciences(all)

Cite this

Lyapunov Exponents and the Mathematics of Invasion in Oscillatory or Chaotic Populations. / Ferriere, Regis H J; Gatto, M.

In: Theoretical Population Biology, Vol. 48, No. 2, 10.1995, p. 126-171.

Research output: Contribution to journalArticle

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