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March  2013, 12(2): 1065-1074. doi: 10.3934/cpaa.2013.12.1065

Travelling wave solutions of a free boundary problem for a two-species competitive model

Chueh-Hsin Chang 1, and  Chiun-Chuan Chen 2,

1. 

Department of Mathematics, National Taiwan University, National Taiwan University, Taipei, 10617, Taiwan
2. 

Department of Mathematics, National Taiwan University, and National Center for Theoretical Sciences (Taipei Office), No. 1, Sec. 4, Roosevelt Road, Taipei, 10617

Received  August 2011 Revised  April 2012 Published  September 2012

We study a di usive logistic system with a free boundary in ecology proposed by Mimura, Yamada and Yotsutani [10]. Motivated by the spreading-vanishing dichotomy obtained by Du and Lin [1], we suppose the spreading speed of the free boundary tends to a constant as time tends to in nity and consider the corresponding travelling wave problem. We establish the existence and uniqueness of a travelling wave solution for this free boundary problem.
Keywords: KPP type equation, free boundary problem, spreading speed., travelling wave.
Mathematics Subject Classification: 35K57, 35C07, 35R3.
Citation: Chueh-Hsin Chang, Chiun-Chuan Chen. Travelling wave solutions of a free boundary problem for a two-species competitive model. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1065-1074. doi: 10.3934/cpaa.2013.12.1065
References:
[1]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. Correction in: http://turing.une.edu.au/ ydu/papers/DuLin-siam-10-correction.pdf. doi: 10.1137/090771089.  Google Scholar

[2]

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

[3]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432.  Google Scholar

[4]

D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285. doi: 10.1016/S1468-1218(02)00009-3.  Google Scholar

[5]

K. I. Kim and Z. Lin, A free boundary problem for a parabolic system describing an ecological model, Nonlinear Anal. Real World Appl., 10 (2009), 428-436. doi: 10.1016/j.nonrwa.2007.10.003.  Google Scholar

[6]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equations de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. Moscou S'er. Internat., A1 (1937), 1-26. English transl. in Dynamics of Curved Fronts, P. Pelc'e (ed.), Academic Press, 1988, 105-130. Google Scholar

[7]

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

[8]

Z. Ling, Q. Tang and Z. Lin, A free boundary problem for two-species competitive model in ecology, Nonlinear Anal. Real World Appl., 11 (2010), 1775-1781. doi: 10.1016/j.nonrwa.2009.04.001.  Google Scholar

[9]

J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, "Invasion Ecology," Blackwell Pub-lishing, Oxford, 2007. Google Scholar

[10]

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.  Google Scholar

[11]

M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  Google Scholar

[12]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.  Google Scholar

[13]

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

[14]

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

[15]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, 1994.  Google Scholar

show all references

References:
[1]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. Correction in: http://turing.une.edu.au/ ydu/papers/DuLin-siam-10-correction.pdf. doi: 10.1137/090771089.  Google Scholar

[2]

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

[3]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432.  Google Scholar

[4]

D. Hilhorst, M. Mimura and R. Schätzle, Vanishing latent heat limit in a Stefan-like problem arising in biology, Nonlinear Anal. Real World Appl., 4 (2003), 261-285. doi: 10.1016/S1468-1218(02)00009-3.  Google Scholar

[5]

K. I. Kim and Z. Lin, A free boundary problem for a parabolic system describing an ecological model, Nonlinear Anal. Real World Appl., 10 (2009), 428-436. doi: 10.1016/j.nonrwa.2007.10.003.  Google Scholar

[6]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equations de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. Moscou S'er. Internat., A1 (1937), 1-26. English transl. in Dynamics of Curved Fronts, P. Pelc'e (ed.), Academic Press, 1988, 105-130. Google Scholar

[7]

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

[8]

Z. Ling, Q. Tang and Z. Lin, A free boundary problem for two-species competitive model in ecology, Nonlinear Anal. Real World Appl., 11 (2010), 1775-1781. doi: 10.1016/j.nonrwa.2009.04.001.  Google Scholar

[9]

J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, "Invasion Ecology," Blackwell Pub-lishing, Oxford, 2007. Google Scholar

[10]

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.  Google Scholar

[11]

M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.  Google Scholar

[12]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.  Google Scholar

[13]

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

[14]

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

[15]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, 1994.  Google Scholar

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