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January  2011, 29(1): 367-385. doi: 10.3934/dcds.2011.29.367

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, China, China

Received  November 2009 Revised  June 2010 Published  September 2010

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: shadow system, singular perturbation., existence, spiky steady state, cross diffusion.
Mathematics Subject Classification: Primary: 35J55, 35J60, Secondary: 35B25, 35K5.
Citation: Yaping Wu, Qian Xu. The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 367-385. doi: 10.3934/dcds.2011.29.367
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.

show all references

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.

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