Article Contents

2011, Volume 29, Issue 1: 367-385. Doi: 10.3934/dcds.2011.29.367

This issue Previous Article Limit theorems for optimal mass transportation and applications to networks Next Article Erratum

The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion

Yaping Wu^1, and
Qian Xu^1,

1.
Department of Mathematics, Capital Normal University, Beijing 100048

Received: November 2009

Revised: June 2010

Published: January 2011

Abstract / Introduction Related Papers Cited by

Abstract

This paper is concerned with the existence of large positive spiky steady states for S-K-T competition systems with cross-diffusion. Firstly by detailed integral and perturbation estimates, the existence and detailed fast-slow structure of a class of spiky steady states are obtained for the corresponding shadow system, which also verify and extend some existence results on spiky steady states obtained in [10] by different method of proof. Further by applying special perturbation method, we prove the existence of large positive spiky steady states for the original competition systems with large cross-diffusion rate.

Keywords:

Mathematics Subject Classification: Primary: 35J55, 35J60, Secondary: 35B25, 35K57.

Citation:

Related Papers

Cited by

References

[1]	Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.doi: doi:10.3934/dcds.2004.10.719.
[2]	H. Kuiper and L. Dung, Global attractors for cross diffusion systems on domains of arbitrary dimension, Rocky Mountain J. Math., 37 (2007), 1645-1668.doi: doi:10.1216/rmjm/1194275939.
[3]	Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.
[4]	K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.doi: doi:10.1016/0022-0396(85)90020-8.
[5]	T. Kolokolnikov, M. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model, the pulse-splitting regime, Phys. D, 202 (2005), 258-293.doi: doi:10.1016/j.physd.2005.02.009.
[6]	C.-S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.doi: doi:10.1016/0022-0396(88)90147-7.
[7]	Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.doi: doi:10.1006/jdeq.1996.0157.
[8]	Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.doi: doi:10.1006/jdeq.1998.3559.
[9]	Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dyn. Syst., 4 (1998), 193-203.doi: doi:10.3934/dcds.1998.4.193.
[10]	Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst, 10 (2004), 435-458.doi: doi:10.3934/dcds.2004.10.435.
[11]	M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449.
[12]	W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
[13]	W. M. Ni, I. Takagi and E. Yanagida, Stability analysis of point condensation solutions to a reaction-diffusion system proposed by Gierer and Meinhardt, Tohoku Math. J., to appear.
[14]	N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.doi: doi:10.1016/0022-5193(79)90258-3.
[15]	B. D. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.doi: doi:10.1137/S0036139902415117.
[16]	J.Wei, Existence and stability of spikes for the Gierer-Meinhardt systems, "Handbook of Differential Equations: Stationary Partial Differential Equations," Elsevier/ North-Holland, Amsterdam, V (2008), 487-585.
[17]	Y. Wu, Existence of stationary solutions with transition layers for a class of cross-diffusion systems, Proc. of Royal Soc. Edinburg, Sect. A, 132 (2002), 1493-1511.
[18]	Y. Wu and X. Zhao, The existence and stability of traveling waves with transition layers for some singular cross-diffuion systems, Phys. D, 200 (2005), 325-358.doi: doi:10.1016/j.physd.2004.11.010.
[19]	Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for S-K-T competition model with cross-diffusion, Sci. China Math., 53 (2010), 1161-1184.doi: doi:10.1007/s11425-010-0141-4.