American Institute of Mathematical Sciences

Advanced
November  2014, 19(9): 2941-2961. doi: 10.3934/dcdsb.2014.19.2941

On the steady state of a shadow system to the SKT competition model

Qi Wang 1,

1. 

Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China

Received  December 2013 Revised  February 2014 Published  September 2014

We study a boundary value problem with an integral constraint that arises from the modelings of species competition proposed by Lou and Ni in [10]. Through local and global bifurcation theories, we obtain the existence of non-constant positive solutions to this problem, which are small perturbations from its positive constant solution, over a one-dimensional domain. Moreover, we investigate the stability of these bifurcating solutions. Finally, for the diffusion rate being sufficiently small, we construct infinitely many positive solutions with single transition layer, which is represented as an approximation of a step function. The transition-layer solution can be used to model the segregation phenomenon through interspecific competition.
Keywords: transition layer., shadow system, Competition model, nonlinear boundary value problem.
Mathematics Subject Classification: Primary: 35B40, 35J57; Secondary: 92D25, 35B32, 35B3.
Citation: Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941
References:
[1]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, Journal of Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[2]

________, Bifurcation, perturbation of simple eigenvalues, and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.   Google Scholar

[3]

S. Ei, Two-timing methods with applications to heterogeneous reaction-diffusion systems,, Hiroshima Math. J., 18 (1988), 127.   Google Scholar

[4]

P. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations,, J. Math. Anal. Appl., 54 (1976), 497.  doi: 10.1016/0022-247X(76)90218-3.  Google Scholar

[5]

J. K. Hale and K. Sakamoto, Existence and stability of transition layers,, Japan J. Appl. Math., 5 (1988), 367.  doi: 10.1007/BF03167908.  Google Scholar

[6]

M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system,, Japan J. Indust. Appl. Math., 15 (1998), 233.  doi: 10.1007/BF03167402.  Google Scholar

[7]

Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations,, Hiroshima Math. J., 23 (1993), 193.   Google Scholar

[8]

T. Kato, Functional Analysis,, Springer Classics in Mathematics, (1996).   Google Scholar

[9]

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

[10]

________, Diffusion vs cross-diffusion: An elliptic approach,, J. Differential Equations, 154 (1999), 157.  doi: 10.1006/jdeq.1998.3559.  Google Scholar

[11]

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

[12]

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.  doi: 10.3934/dcds.2004.10.435.  Google Scholar

[13]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains,, Publ. Res. Inst. Math. Sci., 19 (1983), 1049.  doi: 10.2977/prims/1195182020.  Google Scholar

[14]

M. Mimura, S.-I Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system,, J. Math. Biol., 29 (1991), 219.  doi: 10.1007/BF00160536.  Google Scholar

[15]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49.  doi: 10.1007/BF00276035.  Google Scholar

[16]

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.   Google Scholar

[17]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[18]

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

[19]

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

show all references

References:
[1]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, Journal of Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[2]

________, Bifurcation, perturbation of simple eigenvalues, and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.   Google Scholar

[3]

S. Ei, Two-timing methods with applications to heterogeneous reaction-diffusion systems,, Hiroshima Math. J., 18 (1988), 127.   Google Scholar

[4]

P. Fife, Boundary and interior transition layer phenomena for pairs of second-order differential equations,, J. Math. Anal. Appl., 54 (1976), 497.  doi: 10.1016/0022-247X(76)90218-3.  Google Scholar

[5]

J. K. Hale and K. Sakamoto, Existence and stability of transition layers,, Japan J. Appl. Math., 5 (1988), 367.  doi: 10.1007/BF03167908.  Google Scholar

[6]

M. Iida, T. Muramatsu, H. Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system,, Japan J. Indust. Appl. Math., 15 (1998), 233.  doi: 10.1007/BF03167402.  Google Scholar

[7]

Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations,, Hiroshima Math. J., 23 (1993), 193.   Google Scholar

[8]

T. Kato, Functional Analysis,, Springer Classics in Mathematics, (1996).   Google Scholar

[9]

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

[10]

________, Diffusion vs cross-diffusion: An elliptic approach,, J. Differential Equations, 154 (1999), 157.  doi: 10.1006/jdeq.1998.3559.  Google Scholar

[11]

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

[12]

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.  doi: 10.3934/dcds.2004.10.435.  Google Scholar

[13]

H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains,, Publ. Res. Inst. Math. Sci., 19 (1983), 1049.  doi: 10.2977/prims/1195182020.  Google Scholar

[14]

M. Mimura, S.-I Ei and Q. Fang, Effect of domain-shape on coexistence problems in a competition-diffusion system,, J. Math. Biol., 29 (1991), 219.  doi: 10.1007/BF00160536.  Google Scholar

[15]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49.  doi: 10.1007/BF00276035.  Google Scholar

[16]

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.   Google Scholar

[17]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[18]

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

[19]

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

[1]

Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89
[2]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011
[3]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108
[4]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084
[5]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[6]

X. Liang, Roderick S. C. Wong. On a Nested Boundary-Layer Problem. Communications on Pure & Applied Analysis, 2009, 8 (1) : 419-433. doi: 10.3934/cpaa.2009.8.419
[7]

Shin-Ichiro Ei, Kota Ikeda, Yasuhito Miyamoto. Dynamics of a boundary spike for the shadow Gierer-Meinhardt system. Communications on Pure & Applied Analysis, 2012, 11 (1) : 115-145. doi: 10.3934/cpaa.2012.11.115
[8]

John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276
[9]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416
[10]

Kateryna Marynets. Study of a nonlinear boundary-value problem of geophysical relevance. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4771-4781. doi: 10.3934/dcds.2019194
[11]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045
[12]

Hideo Ikeda, Masayasu Mimura, Tommaso Scotti. Shadow system approach to a plankton model generating harmful algal bloom. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 829-858. doi: 10.3934/dcds.2017034
[13]

Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212
[14]

Byung-Hoon Hwang, Seok-Bae Yun. Stationary solutions to the boundary value problem for the relativistic BGK model in a slab. Kinetic & Related Models, 2019, 12 (4) : 749-764. doi: 10.3934/krm.2019029
[15]

Yuanshi Wang, Hong Wu. Transition of interaction outcomes in a facilitation-competition system of two species. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1463-1475. doi: 10.3934/mbe.2017076
[16]

Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331
[17]

Liping Wang, Chunyi Zhao. Solutions with clustered bubbles and a boundary layer of an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2333-2357. doi: 10.3934/dcds.2014.34.2333
[18]

Liping Wang, Juncheng Wei. Solutions with interior bubble and boundary layer for an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 333-351. doi: 10.3934/dcds.2008.21.333
[19]

Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234
[20]

Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences