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

Previous Article
Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions
DCDS Home
This Issue
Next Article
Stochastic adding machine and $2$-dimensional Julia sets

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

Abstract
Related Papers

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 and 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.
[2]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[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.
[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.
[5]	Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.
[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.
[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.
[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. 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.
[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.
[12]	H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079.
[13]	M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.
[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.
[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.
[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.
[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, 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. 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. 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.
[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.
[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.

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.
[2]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[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.
[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.
[5]	Y. Kan-on, Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics, Hiroshima Math. J., 23 (1993), 509-536.
[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.
[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.
[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. 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.
[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.
[12]	H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079.
[13]	M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.
[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.
[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.
[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.
[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, 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. 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. 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.
[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.
[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.

[1]	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
[2]	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
[3]	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
[4]	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
[5]	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
[6]	Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189
[7]	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
[8]	Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053
[9]	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
[10]	Willian Cintra, Carlos Alberto dos Santos, Jiazheng Zhou. Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3913-3931. doi: 10.3934/dcdsb.2021211
[11]	Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133
[12]	Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536
[13]	Kimun Ryu, Inkyung Ahn. Positive steady--states for two interacting species models with linear self-cross diffusions. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1049-1061. doi: 10.3934/dcds.2003.9.1049
[14]	Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : i-i. doi: 10.3934/dcdss.20202i
[15]	Yunfeng Jia, Yi Li, Jianhua Wu. Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4785-4813. doi: 10.3934/dcds.2017206
[16]	Anton Arnold, Laurent Desvillettes, Céline Prévost. Existence of nontrivial steady states for populations structured with respect to space and a continuous trait. Communications on Pure and Applied Analysis, 2012, 11 (1) : 83-96. doi: 10.3934/cpaa.2012.11.83
[17]	Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
[18]	Dagny Butler, Eunkyung Ko, R. Shivaji. Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1181-1191. doi: 10.3934/dcdss.2014.7.1181
[19]	Qian Xu. The stability of bifurcating steady states of several classes of chemotaxis systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 231-248. doi: 10.3934/dcdsb.2015.20.231
[20]	Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Wei-Ming Ni Yaping Wu Qian Xu
on Google Scholar
Wei-Ming Ni Yaping Wu Qian Xu

[Back to Top]