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Mathematical Biosciences and Engineering

2013, Volume 10, Issue 5&6: 1419-1436. doi: 10.3934/mbe.2013.10.1419
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Some recent developments on linear determinacy

  • 1.
    Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287
  • 2.
    Department of Mathematics, University of Louisville, Louisville, KY 40292
  • 3.
    School of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069
  • Received: 01 October 2012 Accepted: 29 June 2018 Published: 01 August 2013

  • MSC : Primary: 92D25; Secondary: 39A10, 32K45, 34C60.

  • The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called``linear determinacy'' conjecture equates full nonlinearmodel spread rates with the spread rates computed from linearized systems with the linearization carried out around the leadingedge of the invasion. A survey that accounts forrecent developments in the identification of conditions under which linear determinacy gives the ``right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributionscovered in this survey are also discussed.

    Citation: Carlos Castillo-Chavez, Bingtuan Li, Haiyan Wang. Some recent developments on linear determinacy[J]. Mathematical Biosciences and Engineering, 2013, 10(5&6): 1419-1436. doi: 10.3934/mbe.2013.10.1419

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  • The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called``linear determinacy'' conjecture equates full nonlinearmodel spread rates with the spread rates computed from linearized systems with the linearization carried out around the leadingedge of the invasion. A survey that accounts forrecent developments in the identification of conditions under which linear determinacy gives the ``right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributionscovered in this survey are also discussed.


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