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Mathematical Biosciences and Engineering

2014, Volume 11, Issue 6: 1247-1274. doi: 10.3934/mbe.2014.11.1247
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Spatiotemporal complexity in a predator--prey model with weak Allee effects

  • 1.
    Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275
  • 2.
    Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh
  • 3.
    Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212
  • 4.
    College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035
  • Published: 01 September 2014

  • MSC : Primary: 35B36, 45M10; Secondary: 92C15.

  • In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.

    Citation: Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang. Spatiotemporal complexity in a predator--prey model with weak Allee effects[J]. Mathematical Biosciences and Engineering, 2014, 11(6): 1247-1274. doi: 10.3934/mbe.2014.11.1247

    Related Papers:

  • In this article, we study the rich dynamics of a diffusive predator-prey system with Allee effects in the prey growth. Our model assumes a prey-dependent Holling type-II functional response and a density dependent death rate for predator. We investigate the dissipation and persistence property, the stability of nonnegative and positive constant steady state of the model, as well as the existence of Hopf bifurcation at the positive constant solution. In addition, we provide results on the existence and non-existence of positive non-constant solutions of the model. We also demonstrate the Turing instability under some conditions, and find that our model exhibits a diffusion-controlled formation growth of spots, stripes, and holes pattern replication via numerical simulations. One of the most interesting findings is that Turing instability in the model is induced by the density dependent death rate in predator.


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    [1] Ecology, 75 (1994), 1842-1850.
    [2] Trends in Ecology & Evolution, 15 (2000), 337-341.
    [3] Nonlinear Analysis: Real World Applications, 10 (2009), 1401-1416.
    [4] SIAM Journal on Applied Mathematics, 69 (2009), 1244-1262.
    [5] Ecology, 76 (1995), 995-1004.
    [6] Journal of Differential Equations, 33 (1979), 201-225.
    [7] University of Chicago Press, Chicago, USA, 1931.
    [8] Ecology, 83 (2002), 28-34.
    [9] Journal of Theoretical Biology, 139 (1989), 311-326.
    [10] Journal of Mathematical Biology, 33 (1995), 816-828.
    [11] Mathematical Biosciences, 236 (2012), 64-76.
    [12] Theoretical Ecology, 4 (2011), 37-53.
    [13] Journal of Theoretical Biology, 245 (2007), 220-229.
    [14] Bulletin of Mathematical Biology, 55 (1993), 365-384.
    [15] Ecology, 73 (1992), 1530-1535.
    [16] Mathematical Methods in the Applied Sciences, 36 (2013), 1768-1775.
    [17] Journal of Theoretical Biology, 218 (2002), 375-394.
    [18] Theoretical Population Biology, 72 (2007), 136-147.
    [19] Springer, 2003.
    [20] Journal of Differential Equations, 40 (1981), 232-252.
    [21] International Journal of Biomathematics, 5 (2012), 1250023, 11 pp.
    [22] Wiley, London, 2003.
    [23] SIAM Journal of Appllied Mathematics, 35 (1978), 1-16.
    [24] Ecology, 63 (1982), 1802-1813.
    [25] Natural Resource Modeling, 3 (1989), 481-538.
    [26] Journal of Mathematical Analysis and Applications, 339 (2008), 1220-1230.
    [27] Journal of Theoretical Biology, 69 (1977), 613-623.
    [28] Bulletin of Mathematical Biology, 69 (2007), 931-956.
    [29] Springer-Verlag, Berlin and New York, 1983.
    [30] Nonlinear Analysis: Real World Applications, 12 (2011), 2931-2942.
    [31] Journal of Mathematical Biology, 60 (2010), 59-74.
    [32] Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
    [33] Journal of Mathematical Biology, 42 (2001), 489-506.
    [34] Journal of Mathematical Analysis and Applications, 238 (1999), 179-195.
    [35] Journal of Mathematical Biology, 67 (2013), 1227-1259.
    [36] Journal of Mathematical Analysis and Applications, 344 (2008), 217-230.
    [37] Journal of Mathematical Biology, 28 (1990), 463-474.
    [38] Fields Institute Communication, 21 (1999), 325-337.
    [39] Journal of Mathematical Biology, 36 (1998), 389-406.
    [40] Mathematical Biosciences, 88 (1988), 67-84.
    [41] Ecology, 73 (1992), 1943-1967.
    [42] Theoretical Population Biology, 43 (1993), 141-158.
    [43] Journal of Differential Equations, 72 (1988), 1-27.
    [44] Proceedings of the American Mathematical Society, 133 (2005), 3619-3626.
    [45] Journal of Differential Equations, 131 (1996), 79-131.
    [46] SIAM Review, 44 (2002), 311-370.
    [47] Springer, 2000.
    [48] Chaos, Solitons & Fractals, 23 (2005), 55-65.
    [49] Physica D: Nonlinear Phenomena, 188 (2004), 134-151.
    [50] Chaos, Solitons & Fractals, 37 (2008), 1343-1355.
    [51] Bulletin of Mathematical Biology, 71 (2009), 863-887.
    [52] Proceedings of the Royal Society of London-B: Biological Sciences, 271 (2004), 1407-1414.
    [53] Bulletin of Mathematical Biology, 52 (1990), 119-152.
    [54] Springer, New York, USA, 2002.
    [55] Notices of the AMS, 48 (2001), 1304-1314.
    [56] Proceedings of the Royal Society of Edinburgh-A-Mathematics, 133 (2003), 919-942.
    [57] Journal of Differential Equations, 200 (2004), 245-273.
    [58] Journal of Differential Equations, 247 (2009), 866-886.
    [59] SIAM Journal on Applied Mathematics, 67 (2007), 1479-1503.
    [60] Nonlinearity, 21 (2008), 1471.
    [61] Theoretical Population Biology, 53 (1998), 108-130.
    [62] Transactions of the American Fisheries Society, 111 (1982), 255-266.
    [63] Mathematical Biosciences, 122 (1994), 1-23.
    [64] Springer-Verlag, New York, 1994.
    [65] Ecology, 58 (1977), 1237-1253.
    [66] Trends in Ecology & Evolution, 14 (1999), 401-405.
    [67] Oikos, (1999), 185-190.
    [68] Theoretical Ecology, 5 (2012), 297-309.
    [69] Princeton University Press, 2003.
    [70] Philosophical Transactions of the Royal Society of London-B, 237 (1952), 37-72.
    [71] Journal of Biological Dynamics, 6 (2012), 524-538.
    [72] Journal of Differential Equations, 251 (2011), 1276-1304.
    [73] Science Press, Beijing, 1993.
    [74] Journal of Mathematical Analysis and Applications, 292 (2004), 484-505.
    [75] Physica D, 196 (2004), 172-192.
    [76] Computers and Mathematics with Applications, 52 (2006), 707-720.
    [77] Physical Review E, 75 (2007), 051913.
    [78] Ecological Modelling, 221 (2010), 131-140.
    [79] World Scientific, 2006.
    [80] SIAM Journal on Applied Mathematics, 61 (2001), 1445-1472.
    [81] Applied Mathematics and Computation, 159 (2004), 863-880.
    [82] Theoretical Population Biology, 67 (2005), 23-31.
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