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June  2020, 40(6): 3657-3682. doi: 10.3934/dcds.2020051

Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion

1. 

College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China

2. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

* Corresponding author: Yaping Wu

Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday

Received  April 2019 Revised  July 2019 Published  October 2019

This paper is concerned with the existence and stability of nontrivial positive steady states of Shigesada-Kawasaki-Teramoto competition model with cross diffusion under zero Neumann boundary condition. By applying the special perturbation argument based on the Lyapunov-Schmidt reduction method, we obtain the existence and the detailed asymptotic behavior of two branches of nontrivial large positive steady states for the specific shadow system when the random diffusion rate of one species is near some critical value. Further by applying the detailed spectral analysis with the special perturbation argument, we prove the spectral instability of the two local branches of nontrivial positive steady states for the limiting system. Finally, we prove the existence and instability of the two branches of nontrivial positive steady states for the original SKT cross-diffusion system when both the cross diffusion rate and random diffusion rate of one species are large enough, while the random diffusion rate of another species is near some critical value.

Citation: 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
References:
[1]

Y. S. ChoiR. 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.

[2]

A.-K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Anal., 13 (1989), 1091-1113.  doi: 10.1016/0362-546X(89)90097-7.

[3]

Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.  doi: 10.32917/hmj/1206392779.

[4]

K. Kishimoto and H. F. Weiberger, 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.

[5]

K. Kuto, Limiting structure of shrinking solutions to the stationary Shigesada-Kawasaki-Teramoto model with large cross-diffusion, SIAM J. Math. Anal., 47 (2015), 3993-4024.  doi: 10.1137/140991455.

[6]

D. Le, Cross diffusion systems on n spatial dimensional domains, Indiana Univ. Math. J., 51 (2002), 625-644.  doi: 10.1512/iumj.2002.51.2198.

[7]

Q. Li and Y. P. Wu, Stability analysis on a type of steady state for the SKT competition model with large cross diffusion, J. Math. Anal. Appl., 462 (2018), 1048-1072.  doi: 10.1016/j.jmaa.2018.01.023.

[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.

[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.

[10]

Y. LouW.-M. Ni and Y. P. Wu, On the global existence of a cross diffusion system, Discrete and Contin. Dyn. Syst., 4 (1998), 193-203.  doi: 10.3934/dcds.1998.4.193.

[11]

Y. LouW.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross diffusion, Discrete and Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.

[12]

Y. LouW. M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete and Contin. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589.

[13]

M. MimuraY. NishiuraA. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449.  doi: 10.32917/hmj/1206133048.

[14]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.

[15]

W.-M. NiY. P. Wu and Q. Xu, The existence and stability of nontrivial steady states for S-K-T competition model with cross-diffusion, Discrete and Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271.

[16]

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

[17]

L. WangY. P. Wu and Q. Xu, Instability of spiky steady states for S-K-T biological competing model with cross-diffusion, Nonlinear Anal., 159 (2017), 424-457.  doi: 10.1016/j.na.2017.02.026.

[18]

Y. P. 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.

[19]

Y. P. 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.

[20]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, Handbook of Differential Equations: Stationary Partial Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 6 (2008), 411-501.  doi: 10.1016/S1874-5733(08)80023-X.

[21]

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

show all references

Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday

References:
[1]

Y. S. ChoiR. 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.

[2]

A.-K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Anal., 13 (1989), 1091-1113.  doi: 10.1016/0362-546X(89)90097-7.

[3]

Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.  doi: 10.32917/hmj/1206392779.

[4]

K. Kishimoto and H. F. Weiberger, 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.

[5]

K. Kuto, Limiting structure of shrinking solutions to the stationary Shigesada-Kawasaki-Teramoto model with large cross-diffusion, SIAM J. Math. Anal., 47 (2015), 3993-4024.  doi: 10.1137/140991455.

[6]

D. Le, Cross diffusion systems on n spatial dimensional domains, Indiana Univ. Math. J., 51 (2002), 625-644.  doi: 10.1512/iumj.2002.51.2198.

[7]

Q. Li and Y. P. Wu, Stability analysis on a type of steady state for the SKT competition model with large cross diffusion, J. Math. Anal. Appl., 462 (2018), 1048-1072.  doi: 10.1016/j.jmaa.2018.01.023.

[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.

[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.

[10]

Y. LouW.-M. Ni and Y. P. Wu, On the global existence of a cross diffusion system, Discrete and Contin. Dyn. Syst., 4 (1998), 193-203.  doi: 10.3934/dcds.1998.4.193.

[11]

Y. LouW.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross diffusion, Discrete and Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.

[12]

Y. LouW. M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete and Contin. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589.

[13]

M. MimuraY. NishiuraA. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449.  doi: 10.32917/hmj/1206133048.

[14]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.

[15]

W.-M. NiY. P. Wu and Q. Xu, The existence and stability of nontrivial steady states for S-K-T competition model with cross-diffusion, Discrete and Contin. Dyn. Syst., 34 (2014), 5271-5298.  doi: 10.3934/dcds.2014.34.5271.

[16]

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

[17]

L. WangY. P. Wu and Q. Xu, Instability of spiky steady states for S-K-T biological competing model with cross-diffusion, Nonlinear Anal., 159 (2017), 424-457.  doi: 10.1016/j.na.2017.02.026.

[18]

Y. P. 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.

[19]

Y. P. 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.

[20]

Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, Handbook of Differential Equations: Stationary Partial Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 6 (2008), 411-501.  doi: 10.1016/S1874-5733(08)80023-X.

[21]

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

Figure 1.  (a): $ B<C $ i.e. strong competition; (b): $ B>C $ i.e. weak competition
Figure 2.  (a): spiky steady state near positive constant steady states $ (u^*, v^*) $ for small $ d_2 $, large enough $ \rho_{12} $ and $ \rho_{12}/d_1 $, (b): large spiky steady state for small $ d_2 $, large enough $ \rho_{12} $ and $ \rho_{12}/d_1 $, (c): positive steady state with singular bifurcation structure when $ d_2 $ is near $ a_2/\pi^2 $, $ \rho_{12} $ and $ \rho_{12}/d_1 $ are large enough
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