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May  2014, 19(3): 817-826. doi: 10.3934/dcdsb.2014.19.817

Traveling wave solutions of competitive models with free boundaries

1. 

Department of Mathematics, Tongji University, Shanghai, 200092, China

2. 

Department of Mathematics, Tongji University, Shanghai 200092

Received  July 2013 Revised  September 2013 Published  February 2014

We study two systems of reaction diffusion equations with monostable or bistable type of nonlinearities and with free boundaries. These systems are used as multi-species competitive model. For two-species models, we prove the existence of traveling wave solutions, each of which consists of two semi-waves intersecting at the free boundary. For three-species models, we also obtain some traveling wave solutions. In this case, however, every traveling wave solution consists of two semi-waves and one compactly supported wave in between, each intersecting with its neighbors at the free boundaries.
Citation: Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
References:
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D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in Partial Differential Equations and Related Topics, 446 (1975), 5. Google Scholar

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

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C. H. Chang and C. C. Chen, Traveling wave solutions of a free boundary problem for a two-species competitive model,, Commun. Pure Appl. Anal., 12 (2013), 1065. doi: 10.3934/cpaa.2013.12.1065. Google Scholar

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E. C. M. Crooks, E. N. Dancer, D. Hilhorst, M. Mimura and H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions,, Nonlinear Anal. Real World Appl., 5 (2004), 645. doi: 10.1016/j.nonrwa.2004.01.004. Google Scholar

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E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system,, European J. Appl. Math., 10 (1999), 97. doi: 10.1017/S0956792598003660. Google Scholar

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Y. Du and Z. G. 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

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Y. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., (). Google Scholar

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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. Google Scholar

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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

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

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

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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, 446 (1975), 5. Google Scholar

[2]

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

[3]

C. H. Chang and C. C. Chen, Traveling wave solutions of a free boundary problem for a two-species competitive model,, Commun. Pure Appl. Anal., 12 (2013), 1065. doi: 10.3934/cpaa.2013.12.1065. Google Scholar

[4]

E. C. M. Crooks, E. N. Dancer, D. Hilhorst, M. Mimura and H. Ninomiya, Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions,, Nonlinear Anal. Real World Appl., 5 (2004), 645. doi: 10.1016/j.nonrwa.2004.01.004. Google Scholar

[5]

E. N. Dancer, D. Hilhorst, M. Mimura and L. A. Peletier, Spatial segregation limit of a competition-diffusion system,, European J. Appl. Math., 10 (1999), 97. doi: 10.1017/S0956792598003660. Google Scholar

[6]

Y. Du and Z. G. 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

[7]

Y. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., (). Google Scholar

[8]

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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

H. Murakawa and H. Ninomiya, Fast reaction limit of a three-component reaction-diffusion system,, J. Math. Anal. Appl., 379 (2011), 150. doi: 10.1016/j.jmaa.2010.12.040. Google Scholar

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