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Random Leslie matrices in population dynamics

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Abstract

We generalize the concept of the population growth rate when a Leslie matrix has random elements (correlated or not), i.e., characterizing the disorder in the vital parameters. In general, we present a perturbative formalism to deal with linear non-negative random matrix difference equations, then the non-trivial effective eigenvalue of which defines the long-time asymptotic dynamics of the mean-value population vector state is presented as the effective growth rate. This effective eigenvalue is calculated from the smallest positive root of a secular polynomial. Analytical (exact and perturbative calculations) results are presented for several models of disorder. In particular, a 3 × 3 numerical example is applied to study the effective growth rate characterizing the long-time dynamics of a biological population model. The present analysis is a perturbative method for finding the effective growth rate in cases when the vital parameters may have negative covariances across populations.

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Authors and Affiliations

  1. Centro Atomico Bariloche, Instituto Balseiro, Universidad Nacional de Cuyo, CNEA, CONICET, 8400, Bariloche, Argentina

    Manuel O. Cáceres

  2. CRUB, Universidad Nacional del Comahue, 8400, Bariloche, Argentina

    Iris Cáceres-Saez

  3. Museo Argentino de Ciencias Naturales “Bernardino Rivadavia” CONICET, Av. A. Gallardo 470, Buinos Aires, Argentina

    Iris Cáceres-Saez

Authors
  1. Manuel O. Cáceres
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  2. Iris Cáceres-Saez
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Correspondence to Manuel O. Cáceres.

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Cáceres, M.O., Cáceres-Saez, I. Random Leslie matrices in population dynamics. J. Math. Biol. 63, 519–556 (2011). https://doi.org/10.1007/s00285-010-0378-0

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  • DOI: https://doi.org/10.1007/s00285-010-0378-0

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