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

Yan'e Wang; Jianhua Wu

doi:10.3934/dcdsb.2011.15.849

Article Contents

2011, Volume 15, Issue 3: 849-865. Doi: 10.3934/dcdsb.2011.15.849

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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

Received: September 30, 2009

Revised: July 31, 2010

Published: September 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K57, 92D25.

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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.
[2]	A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.doi: 10.1007/BF00289234.
[3]	T. Erneux and E. Reiss, Brusselator isolas, SIAM J. Appl. Math., 43 (1983), 1240-1246.doi: 10.1137/0143082.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	J. H. Wu, Global solutions of a biological depletion model, J. Shaanxi Normal University (Nature Science Edition), 28 (2000), 26-29.
[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.
[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.
[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.
[15]	W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18.
[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.