A Fokker-Planck equation for growing cell populations. (English) Zbl 0644.92019
The purpose of the paper is to give closed form solutions to a Fokker- Planck equation for the number of cells with a certain degree of maturity as a function of maturation velocity v and time \(t: f_ t+vf_{\mu}=Df_{vv}\) (\(\mu\) is the degree of maturation and D a diffusion coefficient). The equation is supplemented with the boundary conditions
\[
f_ v(\mu,v,t)|_{v=0}=0,\quad \lim_{v\to \infty}f(\mu,v,t)=0
\]
and a reproduction equation (of several types). The method is based on the separation of variables and on eigenfunction expansions.
Reviewer: G.Di Blasio
MSC:
| 92D25 | Population dynamics (general) |
| 35Q99 | Partial differential equations of mathematical physics and other areas of application |
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
Keywords:
model for cell growth; Airy functions; time dependent linear transport theory; closed form solutions; Fokker-Planck equation; degree of maturity; maturation velocity; reproduction equation; separation of variables; eigenfunction expansionsReferences:
| [1] | Rotenberg, M.: Transport theory for growing cell populations. J. Theor. Biol. 103, 181 (1983) · doi:10.1016/0022-5193(83)90024-3 |
| [2] | Coddington, E. A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955 · Zbl 0064.33002 |
| [3] | Hille, E.: Lectures on ordinary differential equations. London: Addison-Wesley 1969 · Zbl 0179.40301 |
| [4] | Lebowitz, J. L., Rubinow, S. I.: A theory for the age and generation time distribution of a microbial population. J. Math. Biol. 1, 17 (1974) · Zbl 0402.92023 · doi:10.1007/BF02339486 |
| [5] | Larsen, E. W., Zweifel, P. F.: On the spectrum of the linear transport operator. J. Math. Phys. 15, 1987 (1974) · Zbl 0367.47002 · doi:10.1063/1.1666570 |
| [6] | Palczewski, A.: Spectral properties of the space nonhomogeneous linearized Boltzmann operator. Transp. Theor. Stat. Phys. 13, 409 (1984) · Zbl 0585.47023 · doi:10.1080/00411458408214486 |
| [7] | Beals, R., Protopopescu, V.: Abstract time-dependent transport equation. J. Math. Anal. Appl., to appear · Zbl 0657.45007 |
| [8] | Greenberg, W., van der Mee, C. V. M., Protopopescu, V.: Boundary value problems in abstract kinetic theory. Basel: Birkhäuser, in preparation, cf. Chaps. 12-14 |
| [9] | Protopopescu, V.: On the spectral decomposition of the transport operator with anisotropic scattering and periodic boundary conditions. Transp. Theor. Stat. Phys. 14, 103 (1985) · Zbl 0613.45013 · doi:10.1080/00411458508211672 |
| [10] | Case, K. M., Zweifel, P. F.: Linear transport theory. Reading (Mass.): Addison-Wesley 1967 · Zbl 0162.58903 |
| [11] | Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. New York: Dover 1964 · Zbl 0171.38503 |
| [12] | Weinberger, H. F.: Partial differential equations. Waltham (Mass.): Blaisdell 1965 · Zbl 0127.04805 |
| [13] | Kato, T.: Perturbation theory for linear operators. Heidelberg: Springer 1966 · Zbl 0148.12601 |
| [14] | Beals, R., Protopopescu, V.: Half-range completeness for the Fokker-Planck equation. J. Stat. Phys. 32, 565 (1983) · Zbl 0571.35088 · doi:10.1007/BF01008957 |
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