Some recent developments on linear determinacy

  • 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

    Related Papers:

  • 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.


    加载中
    [1] Landscape Ecol., 4 (1990), 177-188.
    [2] in "Evolution of Insect Pests: The Pattern of Variations" (ed. K. C. Kim), Wiley, New York, (1993), 219-242.
    [3] in "Partial Differential Equations and Related Topics" (ed. J. A. Goldstein), Lecture Notes in Mathematics, 446, Springer-Verlag, Berlin, 1975, 5-49.
    [4] Adv. Math., 30 (1978), 33-76.
    [5] Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.
    [6] Second edition, Texts in Applied Mathematics, 40, Springer, New York, 2012.
    [7] J. Math. Biol., 13 (1981), 173-184.
    [8] BioScience, 60 (2010), 488-489.
    [9] J. Math. Biol., 6 (1978), 109-130.
    [10] Trans. Amer. Math. Soc., 286 (1984), 557-594.
    [11] MIT Press, 1996.
    [12] Princeton University Press, 2007.
    [13] Nature, 429 (2004), 180-184.
    [14] J. Dynamics and Differential Equations, 21 (2009), 663-680.
    [15] Ann. of Eugenics, 7 (1937), 355-369.
    [16] Nature, 437 (2005), 209-214.
    [17] Nature, 442 (2006), 448-452.
    [18] SIAM J. Appl. Math., 44 (1984), 56-79.
    [19] Proc Natl. Acad. Sci. USA, 103 (2006), 5935-5940.
    [20] J. Math. Bio., 2 (1975), 251-263.
    [21] Proc. Edinb. Math. Soc., 31 (1988), 89-97.
    [22] in "Mathematics Inspired by Biology" (eds. V. Capasso and O. Diekmann), Lecture Notes in Mathematics, 1714, Springer, Berlin, 1999.
    [23] Can. Appl. Math. Q., 10 (2002), 473-499.
    [24] Theoretical Population Biology, 9 (1976), 202-221.
    [25] Ecology Letters, 8 (2005), 91-101.
    [26] Chapman and Hall, London, New York, 1989.
    [27] Ecology, 84 (2003), 1945-1956.
    [28] Nonlinear World, 1 (1994), 277-290.
    [29] Math. Models Methods Appl. Sci., 5 (1995), 935-966.
    [30] SIAM J. Math. Anal., 40 (2008), 776-789.
    [31] J. Math. Biol., 46 (2003), 132-152.
    [32] J. Diff. Equations, 251 (2011), 1549-1561.
    [33] J. Dyn. Diff. Equat., 24 (2012), 633-644.
    [34] Bull. Math. Biol., 60 (1998), 435-448.
    [35] J. Mar. Res., 12 (1953), 141-147
    [36] Proc. Natl. Acad. Sci., 97 (2000), 9840-9843.
    [37] J. Chem. Phys., 116 (2002), 10083-10091.
    [38] Int. J. Infect. Dis., 11 (2007), 98-108.
    [39] Mathematical Biosciences, 107 (1991), 255-287.
    [40] Bull. Moscow Univ. Math. Mech., 1 (1937), 1-26.
    [41] J. of Math. Biol., 30 (1992), 413-436.
    [42] Ecology, 77 (1996), 2027-2042.
    [43] Report from "Toward a Science of Sustainability Conference Airlie Center," March, Warrenton, Virginia, National Science Foundation, (2009), 4-10.
    [44] Proc. Nat. Acad. Sci. USA, 71 (1974), 2744-2747.
    [45] Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980.
    [46] Journal of Mathematical Biology, 45 (2002), 219-233.
    [47] Forma, 11 (1996), 1-25.
    [48] Journal of Differential Equations, 252 (2012), 4842-4861.
    [49] Nonlinearity, 24 (2011), 1759-1776.
    [50] Math. Biosciences, 196 (2005), 82-98.
    [51] Journal of Mathematical Biology, 58 (2009), 323-338.
    [52] Chaos Solitons Fractals, 3 (2008), 476-486.
    [53] J. Dynam. Di . Equ., 23 (2011), 903-921.
    [54] Commun. Pure Appl. Math., 60 (2007), 1-40.
    [55] Bull. Math. Biol., 63 (2001), 655-684.
    [56] Springer-Verlag, New York, 2002.
    [57] Math. Biosciences, 93 (1989), 269-295.
    [58] Ecology, 8 (2000), 1613-1628.
    [59] J. Math. Biol., 28 (1990), 529-565.
    [60] Biometrika, 38 (1951), 196-218.
    [61] in "Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges" (eds. A. Gumel, C. Castillo-Chavez, D. P. Clemence and R. E. Mickens), American Mathematical Society, Vol. 410, (2006), 297-310.
    [62] J. of Math. Biol., 8 (1979), 173-187.
    [63] IMA. J. Appl. Math., 72 (2007), 801-816.
    [64] $32^nd$ International Conference on Distributed Computing Systems Workshops (ICDCSW), (2012), 133-139.
    [65] Journal of Differential Equations, 247 (2009), 887-905.
    [66] Discrete and Continuous Dynamical Systems B, 17 (2012), 2243-2266.
    [67] J. of Nonlinear Sciences, 21 (2011), 747-783.
    [68] Biosciences, 171 (2001), 83-97.
    [69] J. Math. Biol., 44(2002), 150-168.
    [70] Discrete and Continuous Dynamical Systems A, 32 (2012), 3303-3324.
    [71] J. Math. Biol., 45 (2002), 183-218.
    [72] J. Math. Biol., 55 (2007), 207-222.
    [73] SIAM J. Math. Anal., 13 (1982), 353-396.
    [74] in "Nonlinear Partial Differential Equations and Applications" (ed. J. M. Chadam), Lecture Notes in Mathematics, Vol. 648, Springer, New York, 47-98.
    [75] in "Nonlinear Partial Differential Equations and Applications" (ed. J. M. Chadam), Lecture Notes in Mathematics, Vol. 648, Springer-Verlag, Berlin, (1978), 47-96.
    [76] Discrete and Continuous Dynamical Systems, 23 (2009), 1087-1098.
    [77] Math. Biosci. Eng., 7 (2010), 187-194.
    [78] J. Dyn. Diff. Eqs., 13 (2001), 651-686.
  • Reader Comments
  • © 2013 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1400) PDF downloads(451) Cited by(7)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog