• Previous Article
    Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers
  • DCDS Home
  • This Issue
  • Next Article
    Spatial dynamics of a nonlocal and delayed population model in a periodic habitat
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

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.
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.

[1]

Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941

[2]

Thomas Lepoutre, Salomé Martínez. Steady state analysis for a relaxed cross diffusion model. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613

[3]

Jing Liu, Xiaodong Liu, Sining Zheng, Yanping Lin. Positive steady state of a food chain system with diffusion. Conference Publications, 2007, 2007 (Special) : 667-676. doi: 10.3934/proc.2007.2007.667

[4]

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

[5]

Qing Li, Yaping Wu. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3657-3682. doi: 10.3934/dcds.2020051

[6]

Wei-Ming Ni, Yaping Wu, Qian Xu. The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5271-5298. doi: 10.3934/dcds.2014.34.5271

[7]

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

[8]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875

[9]

Philippe Souplet, Juan-Luis Vázquez. Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 221-234. doi: 10.3934/dcds.2006.14.221

[10]

Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597

[11]

Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587

[12]

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

[13]

Lorena Bociu, Jean-Paul Zolésio. Existence for the linearization of a steady state fluid/nonlinear elasticity interaction. Conference Publications, 2011, 2011 (Special) : 184-197. doi: 10.3934/proc.2011.2011.184

[14]

Lijuan Wang, Hongling Jiang, Ying Li. Positive steady state solutions of a plant-pollinator model with diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1805-1819. doi: 10.3934/dcdsb.2015.20.1805

[15]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations and Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39

[16]

Yangyang Shi, Hongjun Gao. Homogenization for stochastic reaction-diffusion equations with singular perturbation term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2401-2426. doi: 10.3934/dcdsb.2021137

[17]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301

[18]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198

[19]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228

[20]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]