Optimal control of a chemotaxis system

Fister, K. Renee; McCarthy, C. Maeve

doi:10.1090/qam/1976365

Authors: K. Renee Fister and C. Maeve McCarthy
Journal: Quart. Appl. Math. 61 (2003), 193-211
MSC: Primary 49J20; Secondary 35K50, 35K57, 49K20, 49N90, 92C17
DOI: https://doi.org/10.1090/qam/1976365
MathSciNet review: MR1976365
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Chemotaxis is the process by which cells aggregate under the force of a chemical attractant. The cell and chemoattractant concentrations are governed by a coupled system of parabolic partial differential equations. We investigate the optimal control of the proportion of cells being generated in two settings. One involves harvesting the actual cells and the other depicts removing a proportion of the chemoattractant. The optimality system for each problem contains forward and backward reaction-diffusion and convection-diffusion equations. Numerical results are presented.

W. Allegretta, H. Xie, and S. Yang, Properties of solutions for a chemotaxis system, Mathematical Biology 35 (1979), 949–966.

B. Allweiss, J. Dostal, K. Carey, T. Edwards, and R. Freter, The role of chemotaxis in the ecology of bacterial pathogens of mucosal surfaces, Nature 266 (1977), 448 450.

Wolfgang Alt and Douglas A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation, J. Math. Biol. 24 (1987), no. 6, 691–722. MR 880453, DOI https://doi.org/10.1007/BF00275511

I. Chet and R. Mitchell, Ecological aspects of microbial chemotactic behavior, Annual Review of Microbiology 30 (1976), 221 239.

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981), no. 3-4, 217–237. MR 632161, DOI https://doi.org/10.1016/0025-5564%2881%2990055-9

M. Fisackerly and B. McCartin, A one-dimensional numerical model of chemotaxis, Kettering University, Flint MI, 1997.

E. S. Fisher and D. A. Lauffenburger, Analysis of the effects of immune cell motility and chemotaxis on target elimination dynamics, Math. Biosci. 98 (1990), no. 1, 73–102. MR 1045958, DOI https://doi.org/10.1016/0025-5564%2890%2990012-N
Renee Fister, Optimal control of harvesting in a predator-prey parabolic system, Houston J. Math. 23 (1997), no. 2, 341–355. MR 1682649
W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations, Computing 20 (1978), no. 3, 229–240 (English, with German summary). MR 620052, DOI https://doi.org/10.1007/BF02251947
Miguel A. Herrero and Juan J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), no. 2, 177–194. MR 1478048, DOI https://doi.org/10.1007/s002850050049
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), no. 2, 819–824. MR 1046835, DOI https://doi.org/10.1090/S0002-9947-1992-1046835-6

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology 26 (1970), 399–415.

---, Model for chemotaxis, Journal of Theoretical Biology 30 (1971), 225–234.

---, Traveling bands of chemotactic bacteria: A theoretical analysis, Journal of Theoretical Biology 30 (1971), 235–248.

I. Richard Lapidus, Microbial chemotaxis in flowing water in the vicinity of a source of attractant or repellent, J. Theoret. Biol. 85 (1980), no. 3, 543–547. MR 582731, DOI https://doi.org/10.1016/0022-5193%2880%2990326-4

D. Lauffenburger and R. Aris, Measurement of leukocyte motility and chemotaxis parameters using a quantitative analysis of the under-agarose migration assay, Mathematical Biosciences 44 (1979), 121–138.

D. Lauffenburger and K. Keller, Effects of leukocyte random motility and chemotaxis in tissue inflammatory response, Journal of Theoretical Biology 81 (1979), 475–503.

J.-L. Lions, Optimal control of systems governed by partial differential equations., Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971. Translated from the French by S. K. Mitter. MR 0271512

J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Springer-Verlag, Berlin, 1972.

B.J. McCartin and M.W. Fisackerly, Semidiscretization of the unsteady convection-diffusion equation, Proceedings of the Thirteenth Annual Conference on Applied Mathematics (CAM97).

J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Error estimates in the maximum norm for linear problems in one and two dimensions. MR 1439750
J. D. Murray, Mathematical biology, 2nd ed., Biomathematics, vol. 19, Springer-Verlag, Berlin, 1993. MR 1239892

W. Orr, J. Varani, M. Gondek, P. Ward, and G. Mundy, Chemotactic responses of tumor cells to products of resorbing bone, Science 203 (1979), 176–178.

G. Oster and J.D. Murray, Pattern formation models and developmental constraints, Journal of Experimental Zoology 25 (1989), 186–202.

James A. Powell, Tyler McMillen, and Peter White, Connecting a chemotactic model for mass attack to a rapid integro-difference emulation strategy, SIAM J. Appl. Math. 59 (1999), no. 2, 547–572. MR 1654411, DOI https://doi.org/10.1137/S0036139996313459
S. A. Levin and T. G. Hallam (eds.), Mathematical ecology, Lecture Notes in Biomathematics, vol. 54, Springer-Verlag, Berlin, 1984. MR 760282
Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
Atsushi Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon. 45 (1997), no. 2, 241–265. MR 1441498