American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution
  • CPAA Home
  • This Issue
  • Next Article
    Existence of a rotating wave pattern in a disk for a wave front interaction model
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.  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.  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.  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.  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.  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.   Google Scholar

[7]

Z. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883.  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.  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, (2007).   Google Scholar

[10]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology,, Japan J. Appl. Math., 2 (1985), 151.  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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Translations of Mathematical Monographs, 140 (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.  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.  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.  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.  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.  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.   Google Scholar

[7]

Z. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883.  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.  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, (2007).   Google Scholar

[10]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology,, Japan J. Appl. Math., 2 (1985), 151.  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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Translations of Mathematical Monographs, 140 (1994).   Google Scholar

[1]

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
[2]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087
[3]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213
[4]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
[5]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441
[6]

Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199
[7]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[8]

Harunori Monobe, Hirokazu Ninomiya. Traveling wave solutions with convex domains for a free boundary problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 905-914. doi: 10.3934/dcds.2017037
[9]

Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317
[10]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[11]

Mudassar Imran, Youssef Raffoul, Muhammad Usman, Chi Zhang. A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de Vries-Kuramoto Sivashinsky type equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 691-705. doi: 10.3934/dcdss.2018043
[12]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667
[13]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122
[14]

Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations & Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355
[15]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[16]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033
[17]

Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067
[18]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
[19]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10
[20]

Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi. Stability of the travelling wave in a 2D weakly nonlinear Stefan problem. Kinetic & Related Models, 2009, 2 (1) : 109-134. doi: 10.3934/krm.2009.2.109

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences