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December  2011, 15(3): 849-865. doi: 10.3934/dcdsb.2011.15.849

Stability of positive constant steady states and their bifurcation in a biological depletion model

Yan'e Wang 1, and  Jianhua Wu 1,

1. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China, China

Received  October 2009 Revised  August 2010 Published  February 2011

This paper is concerned with a biological depletion model in a bounded domain. The stability of the positive constant steady states is discussed. In one dimensional case, we make a detailed description for the global bifurcation structure from two positive constant solutions. The result indicates that if $d$ is properly small, the system has at least one non-constant positive steady-state. The main tools used here include the stability theory, bifurcation theory and simulations. From extensive numerical simulations, the predictions from linear theory are confirmed and the influence of parameters $d,D,\sigma$ on these patterns is depicted.
Keywords: Depletion model, stability, simulation., bifurcation.
Mathematics Subject Classification: Primary: 35K57, 92D2.
Citation: Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849
References:
[1]

H. L. Smith and P. Waltman, "The theory of the Chemostat: Dynamics of Microbial Competition," Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[2]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar

[3]

T. Erneux and E. Reiss, Brusselator isolas, SIAM J. Appl. Math., 43 (1983), 1240-1246. doi: 10.1137/0143082.  Google Scholar

[4]

I. Lengyel and I. R. Epstein, Modeling of Turing structure in the Chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652. doi: 10.1126/science.251.4994.650.  Google Scholar

[5]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[6]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar

[7]

W. M. Ni and J. C. Wei, On positive solutions concentrating on spheres for the Gierer-Meinhardt system, J. Diff. Eqns., 221 (2006), 158-189. doi: 10.1016/j.jde.2005.03.004.  Google Scholar

[8]

R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166. doi: 10.1016/j.jmaa.2004.12.026.  Google Scholar

[9]

W. M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Transactions of the American Mathematical Society, 357 (2005), 3953-3969. doi: 10.1090/S0002-9947-05-04010-9.  Google Scholar

[10]

J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89. doi: 10.1007/s00285-007-0146-y.  Google Scholar

[11]

J. H. Wu, Global solutions of a biological depletion model, J. Shaanxi Normal University (Nature Science Edition), 28 (2000), 26-29.  Google Scholar

[12]

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

[13]

M. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[14]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[15]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.  Google Scholar

[16]

I. Takagi, Point-condensation for a reaction-diffusion system, J. Diff. Eqns., 61 (1986), 208-249. doi: 10.1016/0022-0396(86)90119-1.  Google Scholar

show all references

References:
[1]

H. L. Smith and P. Waltman, "The theory of the Chemostat: Dynamics of Microbial Competition," Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[2]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar

[3]

T. Erneux and E. Reiss, Brusselator isolas, SIAM J. Appl. Math., 43 (1983), 1240-1246. doi: 10.1137/0143082.  Google Scholar

[4]

I. Lengyel and I. R. Epstein, Modeling of Turing structure in the Chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991), 650-652. doi: 10.1126/science.251.4994.650.  Google Scholar

[5]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[6]

J. H. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.  Google Scholar

[7]

W. M. Ni and J. C. Wei, On positive solutions concentrating on spheres for the Gierer-Meinhardt system, J. Diff. Eqns., 221 (2006), 158-189. doi: 10.1016/j.jde.2005.03.004.  Google Scholar

[8]

R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166. doi: 10.1016/j.jmaa.2004.12.026.  Google Scholar

[9]

W. M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Transactions of the American Mathematical Society, 357 (2005), 3953-3969. doi: 10.1090/S0002-9947-05-04010-9.  Google Scholar

[10]

J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89. doi: 10.1007/s00285-007-0146-y.  Google Scholar

[11]

J. H. Wu, Global solutions of a biological depletion model, J. Shaanxi Normal University (Nature Science Edition), 28 (2000), 26-29.  Google Scholar

[12]

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

[13]

M. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[14]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[15]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.  Google Scholar

[16]

I. Takagi, Point-condensation for a reaction-diffusion system, J. Diff. Eqns., 61 (1986), 208-249. doi: 10.1016/0022-0396(86)90119-1.  Google Scholar

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