May  2012, 11(3): 911-934. doi: 10.3934/cpaa.2012.11.911

Existence of traveling waves with transition layers for some degenerate cross-diffusion systems

1. 

Department of Mathematics, Capital Normal University, Beijing 100048, China

Received  June 2010 Revised  April 2011 Published  December 2011

This paper is concerned with the existence of traveling waves for a class of degenerate reaction-diffusion systems with cross-diffusion. By applying the analytic singular perturbation method and the theorem of center manifold approximation, we prove the existence of traveling waves with transition layers for the more general degenerate systems with cross-diffusion. Especially for the degenerate S-K-T competition system with cross-diffusion we prove that some new wave patterns exhibiting competition exclusion are induced by the cross-diffusion. We also extend some of the existence results in [5] for the non-cross diffusion systems to the more general degenerate biological systems with cross-diffusion, however the detailed fast-slow structure of the waves for the systems with cross-diffusion is a little different from those for the systems without cross-diffusion.
Citation: Yanxia Wu, Yaping Wu. Existence of traveling waves with transition layers for some degenerate cross-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 911-934. doi: 10.3934/cpaa.2012.11.911
References:
[1]

J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences, (1981). Google Scholar

[2]

C. Conley and R. A. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model,, Indiana Univ. Math. J., 33 (1984), 319. Google Scholar

[3]

R. A. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretical approach,, J. Differential Equations, 44 (1982), 343. Google Scholar

[4]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations,, Discrete Contin. Dyn. Syst. Ser. B., 3 (2003), 79. Google Scholar

[5]

Y. Hosono, Traveling waves for some biological systems with density dependent diffusion,, Japan J. Appl. Math., 4 (1987), 297. Google Scholar

[6]

Y. Hosono, Traveling wave solutions for some density-dependent diffusion equations,, Japan J. Appl. Math., 3 (1986), 163. Google Scholar

[7]

Y. Hosono and M. Mimura, Singular perturbation approach to traveling waves in competing and diffusion species models,, J. Math. Kyoto Univ., 22 (1982), 435. Google Scholar

[8]

M. Ito, A remark on singular perturbation methods,, Hiroshima Math. J., 14 (1985), 619. Google Scholar

[9]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonlinear Anal., 28 (1997), 145. Google Scholar

[10]

Y. Kan-on and Q. Fang, Stability of monotone traveling waves for competition-diffusion equation,, Japan, 13 (1996), 343. Google Scholar

[11]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theret. Biol., 79 (1979), 83. Google Scholar

[12]

M. M. Tang and P. C. Fife, Propagation fronts for competiting species equations with diffusion,, Arch. Rational. Mech. Anal., 73 (1980), 69. Google Scholar

[13]

Y. Wu, Traveling waves for a class of cross-diffusion systems with small parameters,, J. Differential Equations, 123 (1995), 1. Google Scholar

[14]

Y. Wu, The existence of traveling waves for a cross-diffusion system with small parameter,, Beijing Math., 3 (1997), 74. Google Scholar

[15]

Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross diffusion systems,, Phys. D, 200 (2005), 325. Google Scholar

[16]

Y. P. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion,, Sci. China Math., 53 (2010), 1161. Google Scholar

show all references

References:
[1]

J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences, (1981). Google Scholar

[2]

C. Conley and R. A. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model,, Indiana Univ. Math. J., 33 (1984), 319. Google Scholar

[3]

R. A. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretical approach,, J. Differential Equations, 44 (1982), 343. Google Scholar

[4]

Y. Hosono, Traveling waves for a diffusive Lotka-Volterra competition model. I. Singular perturbations,, Discrete Contin. Dyn. Syst. Ser. B., 3 (2003), 79. Google Scholar

[5]

Y. Hosono, Traveling waves for some biological systems with density dependent diffusion,, Japan J. Appl. Math., 4 (1987), 297. Google Scholar

[6]

Y. Hosono, Traveling wave solutions for some density-dependent diffusion equations,, Japan J. Appl. Math., 3 (1986), 163. Google Scholar

[7]

Y. Hosono and M. Mimura, Singular perturbation approach to traveling waves in competing and diffusion species models,, J. Math. Kyoto Univ., 22 (1982), 435. Google Scholar

[8]

M. Ito, A remark on singular perturbation methods,, Hiroshima Math. J., 14 (1985), 619. Google Scholar

[9]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonlinear Anal., 28 (1997), 145. Google Scholar

[10]

Y. Kan-on and Q. Fang, Stability of monotone traveling waves for competition-diffusion equation,, Japan, 13 (1996), 343. Google Scholar

[11]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theret. Biol., 79 (1979), 83. Google Scholar

[12]

M. M. Tang and P. C. Fife, Propagation fronts for competiting species equations with diffusion,, Arch. Rational. Mech. Anal., 73 (1980), 69. Google Scholar

[13]

Y. Wu, Traveling waves for a class of cross-diffusion systems with small parameters,, J. Differential Equations, 123 (1995), 1. Google Scholar

[14]

Y. Wu, The existence of traveling waves for a cross-diffusion system with small parameter,, Beijing Math., 3 (1997), 74. Google Scholar

[15]

Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross diffusion systems,, Phys. D, 200 (2005), 325. Google Scholar

[16]

Y. P. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion,, Sci. China Math., 53 (2010), 1161. Google Scholar

[1]

Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152

[2]

Peng Feng, Zhengfang Zhou. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1145-1165. doi: 10.3934/cpaa.2007.6.1145

[3]

F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239

[4]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185

[5]

Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147

[6]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193

[7]

Michio Urano, Kimie Nakashima, Yoshio Yamada. Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity. Conference Publications, 2005, 2005 (Special) : 868-877. doi: 10.3934/proc.2005.2005.868

[8]

Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157

[9]

Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265

[10]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[11]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

[12]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

[13]

Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133

[14]

Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163

[15]

Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921

[16]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

[17]

Hiroshi Matsuzawa. On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments. Conference Publications, 2009, 2009 (Special) : 516-525. doi: 10.3934/proc.2009.2009.516

[18]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

[19]

Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275

[20]

Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]