Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-24T02:46:44.687Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A CELL DIVISION EQUATION WITH A LINEAR GROWTH RATE

Part of: General first-order equations and systems

Published online by Cambridge University Press:  26 February 2018

B. VAN BRUNT ,
A. ALMALKI ,
T. LYNCH  and
A. ZAIDI Open the ORCID record for A. ZAIDI [Opens in a new window]
B. VAN BRUNT
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand email b.vanbrunt@massey.ac.nz, alone-h1@hotmail.com, t.a.lynch@massey.ac.nz
A. ALMALKI
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand email b.vanbrunt@massey.ac.nz, alone-h1@hotmail.com, t.a.lynch@massey.ac.nz
T. LYNCH
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand email b.vanbrunt@massey.ac.nz, alone-h1@hotmail.com, t.a.lynch@massey.ac.nz
A. ZAIDI*
Affiliation:
Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan email ali.zaidi@lums.edu.pk
*
✉ali.zaidi@lums.edu.pk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.

Keywords

functional partial differential equationsnonlocal partial differential equations

MSC classification

Primary: 35F16: Initial-boundary value problems for linear first-order equations
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 3 , January 2018 , pp. 293 - 312
Copyright
© 2018 Australian Mathematical Society 

References

Abner, K., Aaviksaar, T., Adamberg a, K. and Vilu, nd R, “Single-cell model of prokaryotic cell cycle”, J. Theoret. Biol. 341 (2014) 78–87; doi:10.1016/j.jtbi.2013.09.035.Google Scholar
Andrews, G. E., “The theory of partitions”, in: Encyclopaedia of mathematics and its applications, Volume 2 (Addison-Wesley, Reading, MA, 1976) 17–19; doi:10.1017/CBO9780511608650.Google Scholar
Bernard, E., Doumic, M. and Gabriel, P., “Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts”, Cornell University Library, Ithaca, NY, September 2016; arXiv:1609.03846v2.Google Scholar
Borok, V. and Zitomirskii, J., “On the Cauchy problem for linear partial differential equations with linearly transformed argument”, Sov. Math. Dokl. 12 (1971) 1412–1416.Google Scholar
Cooper, S., “Distinguishing between linear and exponential growth during the division cycle: Single-cell studies, cell culture studies, and the object of cell-cycle research”, Theor. Biol. Med. Model 3(10) (2006); doi:10.1186/1742-4682-3-10.Google Scholar
da Costa, F. P., Grinfeld, M. and McLeod, J. B., “Unimodality of steady size distributions of growing cell populations”, J. Evol. Equ. 1 (2001) 405–409; doi:10.1007/PL00001379.Google Scholar
Derfel, G. and Zitomirskii, J., “Behaviour at zero of the solutions of functional differential equations”, Theory Funct. Funct. Anal. Appl. N31 (1979) 44–49; Teor. Funsktii, Funktsional Anal. i Prilozhen (in Russian).Google Scholar
Diekmann, O., Heijmans, H. and Thieme, H., “On the stability of cell size distribution”, J. Math. Biol. 19 (1984) 227–248; doi:10.1007/BF00277748.Google Scholar
Doumic, M. and Escobedo, M., “Time asymptotics for a critical case in fragmentation and growth-fragmentation equations”, Kinet. Relat. Models 9 (2016) 251–297; doi:10.3934/krm.2016.9.251.Google Scholar
Doumic, M. and Gabriel, P., “Eigenelements of a general aggregation-fragmentation model”, Math. Models Methods Appl. Sci. 20 (2010) 757–783; doi:10.1142/S021820251000443X.Google Scholar
Flajolet, P., Gourdon, X. and Dumas, P., “Mellin transforms and asymptotics: Harmonic sums”, Theoret. Comput. Sci. 144 (1995) 3–58; doi:10.1016/0304-3975(95)00002-E.Google Scholar
Hall, A. J. and Wake, G. C., “A functional differential equation arising in the modelling of cell-growth”, J. Aust. Math. Soc. Ser. B 30 (1989) 424–435; doi:10.1017/S0334270000006366.Google Scholar
Hall, A. J. and Wake, G. C., “A functional differential equation determining steady size distributions for populations of cells growing exponentially”, J. Aust. Math. Soc. Ser. B 31 (1990) 434–453; doi:10.1017/S0334270000006779.Google Scholar
Hall, A. J., Wake, G. C. and Gandar, P. W., “Steady size distributions for cells in one dimensional plant tissues”, J. Math. Biol. 30 (1991) 101–123; doi:10.1007/BF00160330.Google Scholar
Iserles, A., “On the generalized pantograph functional differential equation”, Euro. J. Appl. Math. 4 (1993) 1–38; doi:10.1017/S0956792500000966.Google Scholar
Kato, T. and McLeod, J. B., “The functional-differential equation $y^{\prime }(x)=ay(\unicode[STIX]{x1D706}x)+by(x)$ ”, Bull. Amer. Math. Soc. 77 (1971) 891–937; doi:S0002-9904-1971-12805-7.Google Scholar
Koch, A., “Biomass growth rate during the prokaryote cell cycle”, Crit. Rev. Microbiol. 19 (1993) 17–42; doi:10.3109/10408419309113521.Google Scholar
Kubitschek, H., “Linear cell growth in Escherichia coli ”, Biophys. J. 8(7) (1968) 792–804;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1367559/?page=1.Google Scholar
Michel, P., “Existence of a solution to the cell division eigenproblem”, Math. Models Methods Appl. Sci. 16(7, supp.) (2006) 1125–1153; doi:S0218202506001480.Google Scholar
Michel, P., Mischler, S. and Perthame, B., “General relative entropy inequality: An illustration on growth models”, J. Math. Pures Appl. 84 (2005) 1235–1260; doi:10.1016/j.matpur.2005.04.001.Google Scholar
Morgan, D., “A remarkable sequence derived from Euler products”, J. Math. Phys. 41 (2000) 7109–7121; doi:10.1063/1.1290380.Google Scholar
Perthame, B., Transport equations in biology (Birkhäuser, Basel, 2007).Google Scholar
Perthame, B. and Ryzhik, I., “Exponential decay for the fragmentation or cell-division equation”, J. Differential Equations 210 (2005) 155–177; doi:10.1016/j.jde.2004.10.018.Google Scholar
Sinko, J. and Streifer, W., “A new model for age-size structure of a population”, Ecology 48 (1967) 910–918; doi:10.2307/1934533.Google Scholar
Titchmarsh, E. C., Introduction to the theory of Fourier integrals (Clarendon Press, Oxford, 1937) 1–14.Google Scholar
van Brunt, B. and Wake, G. C., “A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model”, European J. Appl. Math. 22 (2011) 151–168; doi:10.1017/S0956792510000367.Google Scholar
Zaidi, A. A., van Brunt, B. and Wake, G. C., “Solutions to an advanced functional partial differential equation of the pantograph type”, Proc. R. Soc. A 471(2179) (2015); doi:10.1098/rspa.2014.0947.Google Scholar