American Institute of Mathematical Sciences

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November  2013, 18(9): 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

Spreading speed and traveling waves for a two-species weak competition system with free boundary

Chang-Hong Wu 1,

1. 

Department of Applied Mathematics, National University of Tainan, Tainan 700, Taiwan

Received  May 2013 Revised  July 2013 Published  September 2013

In this paper, we will focus on the spreading speed for a Lotka-Volterra type weak competition model with free boundary in one-dimensional habitat. Based on the comparison principle for free boundary problems, we provide some estimates of the spreading speed. Also, we deal with traveling wave solutions for the same model and show that there exists a traveling wave solution with monotone profile using a shooting method and the Schauder's fixed point theorem.
Keywords: free boundary, wave speed., spreading speed, traveling wave, Lotka-Volterra model.
Mathematics Subject Classification: Primary: 35K51, 35R35; Secondary: 92B0.
Citation: Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441
References:
[1]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks and Heterogeneous Media (NHM), 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583.

[2]

C.-H. Chang and C.-C. Chen, Travelling wave solutions of a free boundary problem for a two-species competitive model, Communications on Pure and Applied Analysis (CPAA), 12 (2012), 1065-1074. doi: 10.3934/cpaa.2013.12.1065.

[3]

X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM Journal on Mathematical Analysis, 32 (2000), 778-800. doi: 10.1137/S0036141099351693.

[4]

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

[5]

Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, Journal of Functional Analysis, 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016.

[6]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffsive logistic model with a free boundary, SIAM Journal on Mathematical Analysis, 42 (2010), 377-405. Correction in: http://turing.une.edu.au/~ydu/papers/DuLin-siam-10-correction.pdf. doi: 10.1137/090771089.

[7]

Y. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Cont. Dyn. Syst. (Ser. B), to appear.

[8]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., to appear.

[9]

P. Feng and Z. Zhou, Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony, Communications on Pure and Applied Analysis (CPAA), 6 (2007), 1145-1165. doi: 10.3934/cpaa.2007.6.1145.

[10]

J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, Journal of Dynamics and Differential Equations, 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0.

[11]

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

[12]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011) 467-492.

[13]

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.

[14]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan Journal of Applied Mathematics, 2 (1985), 151-186. doi: 10.1007/BF03167042.

[15]

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

[16]

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

[17]

R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007.

[18]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Archive for Rational Mechanics and Analysis, 73 (1980), 69-77. doi: 10.1007/BF00283257.

[19]

M. X. Wang, On some free boundary problems of the prey-predator model, preprint, arXiv:1301.2063.

show all references

References:
[1]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks and Heterogeneous Media (NHM), 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583.

[2]

C.-H. Chang and C.-C. Chen, Travelling wave solutions of a free boundary problem for a two-species competitive model, Communications on Pure and Applied Analysis (CPAA), 12 (2012), 1065-1074. doi: 10.3934/cpaa.2013.12.1065.

[3]

X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM Journal on Mathematical Analysis, 32 (2000), 778-800. doi: 10.1137/S0036141099351693.

[4]

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

[5]

Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, Journal of Functional Analysis, 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016.

[6]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffsive logistic model with a free boundary, SIAM Journal on Mathematical Analysis, 42 (2010), 377-405. Correction in: http://turing.une.edu.au/~ydu/papers/DuLin-siam-10-correction.pdf. doi: 10.1137/090771089.

[7]

Y. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Cont. Dyn. Syst. (Ser. B), to appear.

[8]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., to appear.

[9]

P. Feng and Z. Zhou, Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony, Communications on Pure and Applied Analysis (CPAA), 6 (2007), 1145-1165. doi: 10.3934/cpaa.2007.6.1145.

[10]

J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, Journal of Dynamics and Differential Equations, 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0.

[11]

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

[12]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011) 467-492.

[13]

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.

[14]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan Journal of Applied Mathematics, 2 (1985), 151-186. doi: 10.1007/BF03167042.

[15]

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

[16]

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

[17]

R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007.

[18]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Archive for Rational Mechanics and Analysis, 73 (1980), 69-77. doi: 10.1007/BF00283257.

[19]

M. X. Wang, On some free boundary problems of the prey-predator model, preprint, arXiv:1301.2063.

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