Spreading speed revisited: Analysis of a free boundary model

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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

Abstract
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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 & 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, 446 (1975), 5.
[2]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33. 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. 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. 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. 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. doi: 10.1137/S0036141099351693.
[7]	Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations,", 1, 1 (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. 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. 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. 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. 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.
[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.
[16]	R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335.
[17]	K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations,, J. Math. Biol., 2 (1975), 251.
[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. 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), A1 (1937), 1.
[20]	A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects,, Popul. Ecol., 51 (2009), 341.
[21]	M. A. Lewis and P. Kareiva, Allee dynamics and the spreading of invasive organisms,, Theor. Population Bio., 43 (1993), 141.
[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. doi: 10.1002/cpa.20154.
[23]	Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883. 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. 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., ().
[27]	L. I. Rubinstein, "The Stefan Problem,", Amer. Math. Soc., (1971).
[28]	N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford Series in Ecology and Evolution, (1997).
[29]	J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.
[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. 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. doi: 10.1007/s00285-007-0078-6.
[32]	J. X. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161. 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, 446 (1975), 5.
[2]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33. 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. 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. 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. 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. doi: 10.1137/S0036141099351693.
[7]	Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations,", 1, 1 (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. 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. 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. 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. 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.
[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.
[16]	R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335.
[17]	K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations,, J. Math. Biol., 2 (1975), 251.
[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. 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), A1 (1937), 1.
[20]	A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects,, Popul. Ecol., 51 (2009), 341.
[21]	M. A. Lewis and P. Kareiva, Allee dynamics and the spreading of invasive organisms,, Theor. Population Bio., 43 (1993), 141.
[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. doi: 10.1002/cpa.20154.
[23]	Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883. 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. 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., ().
[27]	L. I. Rubinstein, "The Stefan Problem,", Amer. Math. Soc., (1971).
[28]	N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford Series in Ecology and Evolution, (1997).
[29]	J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.
[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. 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. doi: 10.1007/s00285-007-0078-6.
[32]	J. X. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161. doi: 10.1137/S0036144599364296.

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