Analysis of hysteretic reaction-diffusion systems

Chiu, Chichia; Walkington, Noel

doi:10.1090/qam/1604805

Authors: Chichia Chiu and Noel Walkington
Journal: Quart. Appl. Math. 56 (1998), 89-106
MSC: Primary 92D25; Secondary 35K57, 47D06, 47N20
DOI: https://doi.org/10.1090/qam/1604805
MathSciNet review: MR1604805
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Abstract: In this paper, we consider a mathematical model motivated by patterned growth of bacteria. The model is a system of differential equations that consists of two sub-systems. One is a system of ordinary differential equations and the other one is a reaction-diffusion system. Pattern formation in this model is caused by an initial instability of the ordinary differential equations. However, nonlinear coupling to the reaction-diffusion system stabilizes the ordinary differential equations resulting in stationary long-time behavior. We establish existence, uniqueness, and characterize long-time behavior of the solutions.

[1] N. F. Britton, Reaction-diffusion equations and their applications to biology, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. MR 866143
[2] E. O. Budriené, A. A. Polezhaev, and M. O. Ptitsyn, Mathematical modelling of intercellular regulation causing the formation of spatial structures in bacterial colonies, J. Theoret. Biol. 135 (1988), no. 3, 323–341. MR 971888, https://doi.org/10.1016/S0022-5193(88)80248-0
[3] Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
[4] C. Chiu, F. C. Hoppensteadt, and W. Jäger, Analysis and computer simulation of accretion patterns in bacterial cultures, to appear
[5] Chichia Chiu and Noel Walkington, An ADI method for hysteretic reaction-diffusion systems, SIAM J. Numer. Anal. 34 (1997), no. 3, 1185–1206. MR 1451120, https://doi.org/10.1137/S0036142994270181
[6] F. C. Hoppensteadt and W. Jäger, Pattern formation by bacteria, Biological growth and spread (Proc. Conf., Heidelberg, 1979) Lecture Notes in Biomath., vol. 38, Springer, Berlin-New York, 1980, pp. 68–81. MR 609347
[7] F. C. Hoppensteadt, W. Jäger, and C. Pöppe, A hysteresis model for bacterial growth patterns, Modelling of patterns in space and time (Heidelberg, 1983) Lecture Notes in Biomath., vol. 55, Springer, Berlin, 1984, pp. 123–134. MR 813709, https://doi.org/10.1007/978-3-642-45589-6_11
[8] G. Hauser, Uber Fäulnisslacterien und deren Beriehungen zur Septicämie, Leipzig: F. G. W. Vogel, 1885
[9] U. Hornung and R. E. Showalter, Pde-models with hysteresis on the boundary, in A. Visintin, editor, Models of Hysteresis, volume 286, Research Notes in Mathematics, Pitman, 1983, pp. 30-38
[10] Ulrich Hornung and R. E. Showalter, Elliptic-parabolic equations with hysteresis boundary conditions, SIAM J. Math. Anal. 26 (1995), no. 4, 775–790. MR 1338362, https://doi.org/10.1137/S0036141093228290
[11] D. L. Lewis and D. K. Gattie, The ecology of quiescent microbes, ASM News 57, 27-32 (1991)
[12] P. Gerhardt, et al., Manual of Methods for General Bacteriology, P. Gerhardt, editor-in-chief, Amer. Soc. for Microbiology, Washington, DC 20006, 1981
[13] H. Melvin Lieberstein, Theory of partial differential equations, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 93. MR 0355280
[14] T. D. Little and R. E. Showalter, Semilinear parabolic equations with Preisach hysteresis, Differential Integral Equations 7 (1994), no. 3-4, 1021–1040. MR 1270116
[15] J. W. T. Wimpenney, J. P. Coombs, R. W. Lovitt, and S. G. Whittaker, A gel-stabilized model ecosystem for investigating microbial growth in spatially ordered solute gradients, J. Gen. Microbio. 127, 277-287 (1981)
[16] Jack W. Macki, Paolo Nistri, and Pietro Zecca, Mathematical models for hysteresis, SIAM Rev. 35 (1993), no. 1, 94–123. MR 1207799, https://doi.org/10.1137/1035005
[17] J. Monod, Recherches sur La Croissance de Cultures Bacteriennes, Hermann et Cie, Paris, 1942
[18] J. D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer-Verlag, Berlin, 1989. MR 1007836
[19] T. E. Shehata and A. G. Marr, Synchronous growth of enteric bacteria, J. Bacteriology 103, 789 (1970)