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

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.
Citation: 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
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. doi: 10.3934/nhm.2012.7.583. Google Scholar

[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. doi: 10.3934/cpaa.2013.12.1065. Google Scholar

[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. doi: 10.1137/S0036141099351693. Google Scholar

[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. doi: 10.1016/j.jde.2011.02.011. Google Scholar

[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. doi: 10.1016/j.jfa.2013.07.016. Google Scholar

[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. doi: 10.1137/090771089. Google Scholar

[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), (). Google Scholar

[8]

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

[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. doi: 10.3934/cpaa.2007.6.1145. Google Scholar

[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. doi: 10.1007/s10884-012-9267-0. Google Scholar

[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. doi: 10.1016/S1468-1218(02)00009-3. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[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. doi: 10.3934/dcds.2013.33.2007. Google Scholar

[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. doi: 10.1007/BF00283257. Google Scholar

[19]

M. X. Wang, On some free boundary problems of the prey-predator model,, preprint, (). Google Scholar

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. doi: 10.3934/nhm.2012.7.583. Google Scholar

[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. doi: 10.3934/cpaa.2013.12.1065. Google Scholar

[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. doi: 10.1137/S0036141099351693. Google Scholar

[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. doi: 10.1016/j.jde.2011.02.011. Google Scholar

[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. doi: 10.1016/j.jfa.2013.07.016. Google Scholar

[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. doi: 10.1137/090771089. Google Scholar

[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), (). Google Scholar

[8]

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

[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. doi: 10.3934/cpaa.2007.6.1145. Google Scholar

[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. doi: 10.1007/s10884-012-9267-0. Google Scholar

[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. doi: 10.1016/S1468-1218(02)00009-3. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[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. doi: 10.3934/dcds.2013.33.2007. Google Scholar

[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. doi: 10.1007/BF00283257. Google Scholar

[19]

M. X. Wang, On some free boundary problems of the prey-predator model,, preprint, (). Google Scholar

[1]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083

[2]

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

[3]

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

[4]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171

[5]

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

[6]

Chiun-Chuan Chen, Li-Chang Hung. Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1451-1469. doi: 10.3934/cpaa.2016.15.1451

[7]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111

[8]

Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405

[9]

Yuzo Hosono. Traveling waves for a diffusive Lotka-Volterra competition model I: singular perturbations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 79-95. doi: 10.3934/dcdsb.2003.3.79

[10]

Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027

[11]

Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591

[12]

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

[13]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[14]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043

[15]

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

[16]

Peixuan Weng. Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 883-904. doi: 10.3934/dcdsb.2009.12.883

[17]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

[18]

Yubin Liu, Peixuan Weng. Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 505-518. doi: 10.3934/dcdsb.2015.20.505

[19]

Sebastian Acosta. A control approach to recover the wave speed (conformal factor) from one measurement. Inverse Problems & Imaging, 2015, 9 (2) : 301-315. doi: 10.3934/ipi.2015.9.301

[20]

Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]