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Mathematical Biosciences and Engineering

2015, Volume 12, Issue 4: 661-686. doi: 10.3934/mbe.2015.12.661
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Stability and persistence in ODE modelsfor populations with many stages

  • 1.
    Department of Mathematics and Philosophy, Columbus State University, Columbus, Georgia 31907
  • 2.
    Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
  • 3.
    Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804
  • 4.
    Mathematics and Statistics, York University, and Centre for Disease Modelling, York Institute of Health Research, Toronto, Ontario
  • Received: 01 February 2014 Accepted: 29 June 2018 Published: 01 April 2015

  • MSC : Primary: 92D25; Secondary: 34D20, 34D23, 37B25, 93D30.

  • A model of ordinary differential equations is formulated for populationswhich are structured by many stages. The model is motivated by tickswhich are vectors of infectious diseases, but is general enough to apply to many other species.Our analysis identifies a basic reproduction numberthat acts as a threshold between population extinction and persistence.We establish conditions for the existence and uniqueness of nonzeroequilibria and show that their local stability cannot be expected ingeneral. Boundedness of solutions remains an open problem though wegive some sufficient conditions.

    Citation: Guihong Fan, Yijun Lou, Horst R. Thieme, Jianhong Wu. Stability and persistence in ODE modelsfor populations with many stages[J]. Mathematical Biosciences and Engineering, 2015, 12(4): 661-686. doi: 10.3934/mbe.2015.12.661

    Related Papers:

  • A model of ordinary differential equations is formulated for populationswhich are structured by many stages. The model is motivated by tickswhich are vectors of infectious diseases, but is general enough to apply to many other species.Our analysis identifies a basic reproduction numberthat acts as a threshold between population extinction and persistence.We establish conditions for the existence and uniqueness of nonzeroequilibria and show that their local stability cannot be expected ingeneral. Boundedness of solutions remains an open problem though wegive some sufficient conditions.


    [1] J. Differential Equations, 217 (2005), 431-455.
    [2] SIAM J. Appl. Math., 66 (2006), 1339-1365.
    [3] Sinauer Associates Inc, Sunderland, MA, 2001.
    [4] J. Differential Equations, 247 (2009), 956-1000.
    [5] Discrete Contin. Dyn. Syst., 33 (2013), 4891-4921.
    [6] CBMS-NSF regional conference series in applied mathematics, 71, SIAM, 1998.
    [7] Academic Press, 2003.
    [8] Springer, Berlin Heidelberg, 1985.
    [9] J. Math. Biol., 61 (2010), 277-318.
    [10] Wiley, New York, 2000.
    [11] J. Math. Anal. Appl., 242 (2000), 255-270.
    [12] Math. Model. Nat. Phenom., 2 (2007), 69-100.
    [13] J. Math. Biol., (to appear).
    [14] Math. Biosci. Eng., 8 (2011), 503-513.
    [15] Vol. Two, Chelsea Publishing Company, New York, 1989.
    [16] Structured population models in biology and epidemiology, Lecture Notes in Math., Springer, Berlin, 1936 (2008), 165-208.
    [17] Theor. Popul. Biol., 28 (1985), 150-180.
    [18] The American Nat., 171 (2008), 743-754.
    [19] Comm. Pure Appl. Math., 38 (1985), 733-753.
    [20] Math. Biosci., 155 (1999), 77-109.
    [21] Can. Appl. Math. Quart., 19 (2011), 65-77.
    [22] Springer Verlag, Berlin Heidelberg, 2008.
    [23] SIAM J. Math. Anal., 37 (2005), 251-275.
    [24] Springer Verlag, Berlin Heidelberg, 1986.
    [25] Statistical Methods in Medical Research, 9 (2000), 279-301.
    [26] Int. J. Parasit., 35 (2005), 375-389.
    [27] Journal of Theoretical Biology, 224 (2003), 359-376.
    [28] Mathematical Biosciences, 208 (2007), 216-240.
    [29] Computers and Mathematics with Applications, 15 (1988), 565-594.
    [30] SIAM J. Appl. Math., 52 (1992), 541-576.
    [31] American Mathematical Society, Providence, RI, 1995.
    [32] Springer New York, 2011.
    [33] Cambridge University Press, Cambridge, 1995.
    [34] Amer. Math. Soc., Providence, 2011.
    [35] J. Math. Biology, 26 (1988), 299-317.
    [36] Princeton University Press, Princeton, NJ, 2003.
    [37] SIAM J. Appl. Math., 70 (2009), 188-211.
    [38] SIAM J. Appl. Math., 48 (1988), 549-591.
    [39] Springer, 1997.
    [40] Math. Biosc., 180 (2002), 29-48.
    [41] Structured population models in biology and epidemiology, Lecture Notes in Math., Springer, Berlin, 1936 (2008), 1-49.
    [42] Parasitology Today, 15 (1999), 258-262.
    [43] 2010 International Congress on Environmental Modelling and Software Modelling for Environment's Sake (D.A. Swayne, W. Yang, A. A. Voinov, A. Rizzoli, T. Filatova, eds.), 2010.
    [44] Journal of Theoretical Biology, 319 (2013), 50-61.
    [45] Springer, New York, 2003.

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