Spreading speed revisited: Analysis of a free boundarymodel

Gary Bunting; Yihong Du; Krzysztof Krakowski

doi:10.3934/nhm.2012.7.583

Article Contents

2012, Volume 7, Issue 4: 583-603. Doi: 10.3934/nhm.2012.7.583

This issue Previous Article Preface Next Article On the optimal control for a semilinear equation with cost depending on the free boundary

Spreading speed revisited: Analysis of a free boundary model

1.
School of Science and Technology, University of New England, Armidale, NSW 2351

Received: January 2012

Revised: July 2012

Published online: December 2012

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 35K20, 35R35, 35J60, 92B05.

Citation:

Related Papers

Cited by

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.