# American Institute of Mathematical Sciences

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

 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.
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:
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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.  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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