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December  2014, 34(12): 5271-5298. doi: 10.3934/dcds.2014.34.5271

The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion

Wei-Ming Ni 1, , Yaping Wu 2, and  Qian Xu 3,

1. 

Center for PDE, East China Normal University, Shanghai, 200241, China
2. 

College of Mathematical Sciences, Capital Normal University, Beijing 100048, China
3. 

Department of Basic Courses, Beijing Union University, Beijing 100101, China

Received  July 2013 Revised  April 2014 Published  June 2014

This paper concerns with the existence and stability properties of non-constant positive steady states in one dimensional space for the following competition system with cross diffusion $$\left\{ \begin{array}{ll} u_t=[(d_{1}+\rho_{12}v)u]_{xx}+u(a_{1}-b_{1}u-c_{1}v),&x\in(0,1), t>0, \\ v_t= d_{2}v_{xx}+v(a_{2}-b_{2}u-c_{2}v),& x\in(0,1),t>0,                    (1) \\ u_{x}=v_{x}=0, &x=0,1, t>0. \end{array}\right. $$ First, by Lyapunov-Schmidt method, we obtain the existence and the detailed structure of a type of small nontrivial positive steady states to the shadow system of (1) as $\rho_{12}\to \infty$ and when $d_2$ is near $a_2/\pi^2$, which also verifies some related existence results obtained earlier in [11] by a different method. Then, based on the detailed structure of the steady states, we further establish the stability of the small nontrivial positive steady states for the shadow system by spectral analysis. Finally, we prove the existence and stability of the corresponding nontrivial positive steady states for the original cross diffusion system (1) when $\rho_{12}$ is large enough and $d_2$ is near $a_2/\pi^2$.
Keywords: steady states, cross diffusion, spectral analysis, shadow system., Existence, stability.
Mathematics Subject Classification: Primary: 35B25, 35B35, 35B36; Secondary: 35K59, 35K57, 35J5.
Citation: 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 & Continuous Dynamical Systems, 2014, 34 (12) : 5271-5298. doi: 10.3934/dcds.2014.34.5271
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: 10.3934/dcds.2004.10.719.  Google Scholar

[2]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[3]

M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641. doi: 10.1007/s00285-006-0013-2.  Google Scholar

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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: 10.1216/rmjm/1194275939.  Google Scholar

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Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.  Google Scholar

[6]

H. Kielhofer, Bifurcation Theory: An Introduction with Applications to PDEs, Applied Math. Sci. Vol. 156, Springer Verlag, New York Inc. 2004. doi: 10.1007/b97365.  Google Scholar

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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: 10.1016/0022-0396(85)90020-8.  Google Scholar

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Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157.  Google Scholar

[9]

Y. Lou and W. M. Ni, Diffusion vs cross diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559.  Google Scholar

[10]

Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross diffusion system, Discrete and Continuous Dynamical Systems, 4 (1998), 193-203. Google Scholar

[11]

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. Google Scholar

[12]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079. Google Scholar

[13]

M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  Google Scholar

[14]

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.  Google Scholar

[15]

W. M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, Hand. Differ. Equ., North-Holland, Amesterdam, 1 (2004), 157-233. doi: 10.1016/S1874-5733(04)80005-6.  Google Scholar

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N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[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.  Google Scholar

[18]

Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340. doi: 10.1016/j.jde.2004.08.015.  Google Scholar

[19]

Y. Wu and Q. Xu, The Existence and structure of large spiky steady states for S-K-T competition system with cross diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385. doi: 10.3934/dcds.2011.29.367.  Google Scholar

[20]

Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross diffusion, Science China, 53 (2010), 1161-1184. doi: 10.1007/s11425-010-0141-4.  Google Scholar

[21]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross diffusion, Stationary Partial Differential Equations, Hand. Differ. Equ., Elsevier, Amsterdam, VI (2008), 411-501. doi: 10.1016/S1874-5733(08)80023-X.  Google Scholar

[22]

Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, 282-299, World Sci. Publ. Hackensack, NJ, (2009). doi: 10.1142/9789812834744_0013.  Google Scholar

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: 10.3934/dcds.2004.10.719.  Google Scholar

[2]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[3]

M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641. doi: 10.1007/s00285-006-0013-2.  Google Scholar

[4]

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: 10.1216/rmjm/1194275939.  Google Scholar

[5]

Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.  Google Scholar

[6]

H. Kielhofer, Bifurcation Theory: An Introduction with Applications to PDEs, Applied Math. Sci. Vol. 156, Springer Verlag, New York Inc. 2004. doi: 10.1007/b97365.  Google Scholar

[7]

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: 10.1016/0022-0396(85)90020-8.  Google Scholar

[8]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157.  Google Scholar

[9]

Y. Lou and W. M. Ni, Diffusion vs cross diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190. doi: 10.1006/jdeq.1998.3559.  Google Scholar

[10]

Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross diffusion system, Discrete and Continuous Dynamical Systems, 4 (1998), 193-203. Google Scholar

[11]

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. Google Scholar

[12]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079. Google Scholar

[13]

M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  Google Scholar

[14]

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.  Google Scholar

[15]

W. M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, Hand. Differ. Equ., North-Holland, Amesterdam, 1 (2004), 157-233. doi: 10.1016/S1874-5733(04)80005-6.  Google Scholar

[16]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[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.  Google Scholar

[18]

Y. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340. doi: 10.1016/j.jde.2004.08.015.  Google Scholar

[19]

Y. Wu and Q. Xu, The Existence and structure of large spiky steady states for S-K-T competition system with cross diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385. doi: 10.3934/dcds.2011.29.367.  Google Scholar

[20]

Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross diffusion, Science China, 53 (2010), 1161-1184. doi: 10.1007/s11425-010-0141-4.  Google Scholar

[21]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross diffusion, Stationary Partial Differential Equations, Hand. Differ. Equ., Elsevier, Amsterdam, VI (2008), 411-501. doi: 10.1016/S1874-5733(08)80023-X.  Google Scholar

[22]

Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, 282-299, World Sci. Publ. Hackensack, NJ, (2009). doi: 10.1142/9789812834744_0013.  Google Scholar

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