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December  2012, 7(4): 583-603. doi: 10.3934/nhm.2012.7.583

Spreading speed revisited: Analysis of a free boundary model

Gary Bunting 1, , Yihong Du 1, and  Krzysztof Krakowski 1,

1. 

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia, Australia, Australia

Received  January 2012 Revised  July 2012 Published  December 2012

We investigate, from a more ecological point of view, a free boundary model considered in [11] and [8] that describes the spreading of a new or invasive species, with the free boundary representing the spreading front. We derive the free boundary condition by considering a "population loss" at the spreading front, and correct some mistakes regarding the range of spreading speed in [11]. Then we use numerical simulation to gain further insights to the model, which may help to determine its usefulness in concrete ecological situations.
Keywords: free boundary, invasive population, Diffusive logistic equation, spreading speed., spreading-vanishing dichotomy.
Mathematics Subject Classification: 35K20, 35R35, 35J60, 92B0.
Citation: Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks and Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" Lecture Notes in Math., 446, Springer, Berlin, (1975), 5-49.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[3]

H. Berestycki, F. Hamel and H. Matano, Bistable traveling waves around an obstacle, Comm. Pure Appl. Math., 62 (2009), 729-788. doi: 10.1002/cpa.20275.

[4]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189. doi: 10.1016/j.jfa.2008.06.030.

[5]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26.

[6]

X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693.

[7]

Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations," 1, Maximum Principles and Applications, World Scientific, Singapore, 2006. doi: 10.1142/9789812774446.

[8]

Y. Du and Z. M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, II, J. Diff. Eqns., 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011.

[9]

Y. Du and Z. M. Guo, The Stefan problem for the Fisher-KPP equation, J. Diff. Eqns., 253 (2012), 996-1035. doi: 10.1016/j.jde.2012.04.014.

[10]

Y. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, preprint, 2011.

[11]

Y. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 1305-1333. doi: 10.1137/090771089.

[12]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, preprint, 2011.

[13]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. European Math. Soc., 12 (2010), 279-312. doi: 10.4171/JEMS/198.

[14]

X. Fauvergue, J-C. Malausa, L. Giuge and F. Courchamp, Invading parasitoids suffer no Allee effect: A manipulative field experiment, Ecology, 88 (2008), 2392-2403.

[15]

I. Filin, R. D. Holt and M. Barfield, The relation of density regulation to habitat specialization, evolution of a speciesrange, and the dynamics of biological invasions, Am. Nat., 172 (2008), 233-247.

[16]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.

[17]

K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.

[18]

D. Hilhorst, M. Iida, M. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180. doi: 10.1007/BF03168569.

[19]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantitéde matière et son application à un problème biologique, Bull. Univ. Moscou Sér. Internat. A1 (1937), 1-26; English transl. in: "Dynamics of Curved Fronts" (ed. P. Pelcé), Academic Press, (1988), 105-130.

[20]

A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects, Popul. Ecol., 51 (2009), 341-354.

[21]

M. A. Lewis and P. Kareiva, Allee dynamics and the spreading of invasive organisms, Theor. Population Bio., 43 (1993), 141-158.

[22]

X. Liang and X-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.

[23]

Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.

[24]

J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, "Invasion Ecology," Blackwell Publishing, 2007.

[25]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042.

[26]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Cont. Dyn. Syst. A., to appear.

[27]

L. I. Rubinstein, "The Stefan Problem," Amer. Math. Soc., Providence, RI, 1971.

[28]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford Series in Ecology and Evolution, Oxford Univ. Press., Oxford, 1997.

[29]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.

[30]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.

[31]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.

[32]

J. X. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" Lecture Notes in Math., 446, Springer, Berlin, (1975), 5-49.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[3]

H. Berestycki, F. Hamel and H. Matano, Bistable traveling waves around an obstacle, Comm. Pure Appl. Math., 62 (2009), 729-788. doi: 10.1002/cpa.20275.

[4]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189. doi: 10.1016/j.jfa.2008.06.030.

[5]

H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. I. Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26.

[6]

X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693.

[7]

Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations," 1, Maximum Principles and Applications, World Scientific, Singapore, 2006. doi: 10.1142/9789812774446.

[8]

Y. Du and Z. M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, II, J. Diff. Eqns., 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011.

[9]

Y. Du and Z. M. Guo, The Stefan problem for the Fisher-KPP equation, J. Diff. Eqns., 253 (2012), 996-1035. doi: 10.1016/j.jde.2012.04.014.

[10]

Y. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, preprint, 2011.

[11]

Y. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 1305-1333. doi: 10.1137/090771089.

[12]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, preprint, 2011.

[13]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. European Math. Soc., 12 (2010), 279-312. doi: 10.4171/JEMS/198.

[14]

X. Fauvergue, J-C. Malausa, L. Giuge and F. Courchamp, Invading parasitoids suffer no Allee effect: A manipulative field experiment, Ecology, 88 (2008), 2392-2403.

[15]

I. Filin, R. D. Holt and M. Barfield, The relation of density regulation to habitat specialization, evolution of a speciesrange, and the dynamics of biological invasions, Am. Nat., 172 (2008), 233-247.

[16]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.

[17]

K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.

[18]

D. Hilhorst, M. Iida, M. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem, Japan J. Indust. Appl. Math., 18 (2001), 161-180. doi: 10.1007/BF03168569.

[19]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantitéde matière et son application à un problème biologique, Bull. Univ. Moscou Sér. Internat. A1 (1937), 1-26; English transl. in: "Dynamics of Curved Fronts" (ed. P. Pelcé), Academic Press, (1988), 105-130.

[20]

A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects, Popul. Ecol., 51 (2009), 341-354.

[21]

M. A. Lewis and P. Kareiva, Allee dynamics and the spreading of invasive organisms, Theor. Population Bio., 43 (1993), 141-158.

[22]

X. Liang and X-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.

[23]

Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.

[24]

J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, "Invasion Ecology," Blackwell Publishing, 2007.

[25]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042.

[26]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Cont. Dyn. Syst. A., to appear.

[27]

L. I. Rubinstein, "The Stefan Problem," Amer. Math. Soc., Providence, RI, 1971.

[28]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford Series in Ecology and Evolution, Oxford Univ. Press., Oxford, 1997.

[29]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.

[30]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3.

[31]

H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.

[32]

J. X. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.

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