Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Optimal control of a chemotaxis system


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.


References [Enhancements On Off] (What's this?)

  • [1] W. Allegretta, H. Xie, and S. Yang, Properties of solutions for a chemotaxis system, Mathematical Biology 35 (1979), 949-966.
  • [2] 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.
  • [3] W. Alt and D. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation, Journal of Mathematical Biology 24 (1985), 691-722. MR 880453
  • [4] I. Chet and R. Mitchell, Ecological aspects of microbial chemotactic behavior, Annual Review of Microbiology 30 (1976), 221 239.
  • [5] S. Childress and J.K. Percus, Nonlinear aspects of chemotaxis, Mathematical Biosciences 56 (1981). 217-237. MR 632161
  • [6] M. Fisackerly and B. McCartin, A one-dimensional numerical model of chemotaxis, Kettering University, Flint MI, 1997.
  • [7] E.S. Fisher and D. Lauffenburger, Analysis of the effects of immune cell motility and chemotaxis on target elimination dynamics, Mathematical Biosciences 98 (1990), 73 102. MR 1045958
  • [8] K.R. Fister, Optimal control of harvesting in a predator-prey parabolic system, Houston Journal of Mathematics 23:2 (1997), 341-355. MR 1682649
  • [9] W.K. Hackbusch, A numerical method for solving parabolic equations with opposite orientations, Computing 20 (1978), 229-240. MR 620052
  • [10] M. Herrero and J.J.L. Velázquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology 35 (1996), 177-194. MR 1478048
  • [11] W. Jager and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of American Mathematical Society 329 (1992), 819-824. MR 1046835
  • [12] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology 26 (1970), 399-415.
  • [13] -, Model for chemotaxis, Journal of Theoretical Biology 30 (1971), 225-234.
  • [14] -, Traveling bands of chemotactic bacteria: A theoretical analysis, Journal of Theoretical Biology 30 (1971), 235-248.
  • [15] I. R. Lapidus, Microbial chemotaxis in flowing water in the vicinity of a source of attractant or repellent, Journal of Theoretical Biology 85 (1980), 543-547. MR 582731
  • [16] 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.
  • [17] D. Lauffenburger and K. Keller, Effects of leukocyte random motility and chemotaxis in tissue inflammatory response, Journal of Theoretical Biology 81 (1979), 475-503.
  • [18] J.L. Lions, Optimal control of systems governed by partial differential equation, Springer-Verlag, New York, 1971. MR 0271512
  • [19] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Springer-Verlag, Berlin, 1972.
  • [20] B.J. McCartin and M.W. Fisackerly, Semidiscretization of the unsteady convection-diffusion equation, Proceedings of the Thirteenth Annual Conference on Applied Mathematics (CAM97).
  • [21] J.J.H. Miller, E. O'Riordan, and G.I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, 1996. MR 1439750
  • [22] J.D. Murray, Mathematical biology, Springer-Verlag, Berlin, 1980. MR 1239892
  • [23] 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.
  • [24] G. Oster and J.D. Murray, Pattern formation models and developmental constraints, Journal of Experimental Zoology 25 (1989), 186-202.
  • [25] J. Powell, T. McMillen, and P. White, Connecting a chemotactic model for mass attack to a rapid integro-difference emulation strategy, SIAM Journal on Applied Mathematics 59:2 (1998), 547-572. MR 1654411
  • [26] L. Segel, Taxis in cellular ecology, Mathematical Ecology (S.A. Levin and T.G. Hallam, eds.), Springer-Verlag, 1984, pp. 407-424. MR 760282
  • [27] J. Simon, Compact sets in the space $ {l^p}\left( 0, t; b \right)$, Annali di Matematica Pura and Applicata (1987), 65-96. MR 916688
  • [28] A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Mathematics Japonica 45:2 (1997), 241-265. MR 1441498

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 49J20, 35K50, 35K57, 49K20, 49N90, 92C17

Retrieve articles in all journals with MSC: 49J20, 35K50, 35K57, 49K20, 49N90, 92C17


Additional Information

DOI: https://doi.org/10.1090/qam/1976365
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society