Analysis of hysteretic reaction-diffusion systems

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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. E. Britton,*Reaction-Diffusion Equations and Their Applications to Biology*, Academic Press, New York, 1986 MR**866143****[2]**E. O. Budriené, A. A. Polezhaev, and M. O. Ptitsyn,*Mathematical modeling of intercellular regulation causing the formation of spatial structures in bacterial colonies*, J. Theor. Biol.**135**, 323-341 (1988) MR**971888****[3]**J. 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]**C. Chiu and N. J. Walkington,*An ADI method for hysteretic reaction-diffusion systems*, SIAM J. Numer. Anal.**34**, 1185-1206 (1997) MR**1451120****[6]**F. C. Hoppensteadt and W. Jäger,*Pattern formation by bacteria*, Lecture Notes in Biomath.**38**, 68-81 (1980) MR**609347****[7]**F. C. Hoppensteadt, W. Jäger, and C. Pöppe,*A hysteresis model for bacterial growth patterns*, in Modelling of Patterns in Space and Time, W. Jäger and J. D. Murray, eds., Lecture Notes in Biomath. vol. 55, Springer-Verlag, Berlin, New York, 1984, pp. 123-134 MR**813709****[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]**U. Hornung and R. E. Showalter,*Elliptic-parabolic equations with hysteresis boundary conditions*, SIAM J. Mathematical Analysis**26**, 775-790 (1995) MR**1338362****[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. M. Lieberstein,*Theory of Partial Differential Equations*, Mathematics in Science and Engineering, vol. 93, Academic Press, New York and London, 1972 MR**0355280****[14]**T. D. Little and R. E. Showalter,*Semilinear parabolic equations with Preisach hysteresis*, Differential and Integral Equations**7**, 1021-1040 (1994) 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]**J. W. Macki, P. Nistri, and P. Zecca,*Mathematical Models for Hysteresis*, SIAM Review, Vol. 35, No. 1, 1993, pp. 94-123 MR**1207799****[17]**J. Monod, Recherches sur La Croissance de Cultures Bacteriennes, Hermann et Cie, Paris, 1942**[18]**J. D. Murray,*Mathematical Biology*, Biomathematics Texts, Springer-Verlag, 1989 MR**1007836****[19]**T. E. Shehata and A. G. Marr,*Synchronous growth of enteric bacteria*, J. Bacteriology**103**, 789 (1970)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
92D25,
35K57,
47D06,
47N20

Retrieve articles in all journals with MSC: 92D25, 35K57, 47D06, 47N20

Additional Information

DOI:
https://doi.org/10.1090/qam/1604805

Article copyright:
© Copyright 1998
American Mathematical Society