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

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.
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

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