On the stability of the cell-size distribution. II. Time-periodic development rates.

*(English)*Zbl 0667.92015The authors study the asymptotic behaviour of solutions for the equation:
\[
\partial_ tn(t,x)+\partial_ x(gn)(t,x)+(\mu +b)n(t,x)=4b n(t,x),
\]
where, e.g.,
\[
(\mu+b)n(t,x)= (\mu(t,x)+b(t,x))n(t,x).
\]
Here n stands for the density of the number of cells with respect to size x. The equation serves to describe the dynamics of n at time t. The functions \(\mu\),b and g (which are assumed to be known) are the rates of death, division and growth, respectively. Several biological restrictions here take the form of boundary conditions.

For the time-periodic functions \(\mu\), b and g (with the same period), which are assumed to meet several additional restrictions, the following asymptotic estimate is proved: \[ n(t,x)\sim C e^{\sigma t} \bar n(t,x)\quad as\quad t\to \infty, \] where the time-periodic function \(\bar n(t,x)\) and \(\sigma\) do not depend on the initial distribution. In addition, the authors relax a condition on g which was imposed in the homogeneous case of this assertion (with \(\mu\), b and g not depending on time). [See part I by the authors, J. Math. Biol. 19, No.2, 227-248 (1984; Zbl 0543.92021)].

The proof of the results is based on the investigation of solution operators corresponding to an appropriate integral equation. Their spectral properties are studied by means of the theory of strongly positive linear operators on Banach lattices.

For the time-periodic functions \(\mu\), b and g (with the same period), which are assumed to meet several additional restrictions, the following asymptotic estimate is proved: \[ n(t,x)\sim C e^{\sigma t} \bar n(t,x)\quad as\quad t\to \infty, \] where the time-periodic function \(\bar n(t,x)\) and \(\sigma\) do not depend on the initial distribution. In addition, the authors relax a condition on g which was imposed in the homogeneous case of this assertion (with \(\mu\), b and g not depending on time). [See part I by the authors, J. Math. Biol. 19, No.2, 227-248 (1984; Zbl 0543.92021)].

The proof of the results is based on the investigation of solution operators corresponding to an appropriate integral equation. Their spectral properties are studied by means of the theory of strongly positive linear operators on Banach lattices.

##### MSC:

92D25 | Population dynamics (general) |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

##### Keywords:

boundary conditions; integral equation; spectral properties; strongly positive linear operators on Banach lattices
PDF
BibTeX
XML
Cite

\textit{O. Diekmann} et al., Comput. Math. Appl., Part A 12, 491--512 (1986; Zbl 0667.92015)

Full Text:
DOI

##### References:

[1] | Diekmann, O.; Heijmans, H.J.A.M.; Thieme, H.R., On the stability of the cell size distribution, J. math. biol., 19, 227-248, (1984) · Zbl 0543.92021 |

[2] | Anderson, E.C.; Bell, G.I.; Petersen, D.F.; Tobey, R.A., Cell growth and division IV. determination of volume growth and rate probability, Biophys. J., 9, 246-263, (1969) |

[3] | Anderson, E.C.; Petersen, D.F., Cell growth and division II. experimental studies of cell volume distributions in Mammalian suspension cultures, Biophys. J., 7, 353-364, (1967) |

[4] | Bell, G.I., Cell growth and division III. conditions for balanced exponential growth in a mathematical model, Biophys. J., 8, 431-444, (1968) |

[5] | Bell, G.I.; Anderson, E.C., Cell growth and division I. A mathematical model with applications to cell volume distributions in Mammalian suspension cultures, Biophys. J., 7, 329-351, (1967) |

[6] | Sinko, J.W.; Streifer, W., A new model for the age-size structure of a population, Ecology, 48, 910-918, (1967) |

[7] | Sinko, J.W.; Streifer, W., A model for populations reproducing by fission, Ecology, 52, 330-335, (1971) |

[8] | Streifer, W., () |

[9] | Heijmans, H.J.A.M., On the stable size distribution of populations reproducing by fission into two unequal parts, Math. biosc., 72, 19-50, (1984) · Zbl 0568.92015 |

[10] | Lasota, A.; Mackey, M.C., Global asymptotic properties of proliferating cell populations, J. math. biol., 19, 43-62, (1984) · Zbl 0529.92011 |

[11] | Tyson, J.J., The coordination of cell growth and divisionâ€”intentional or incidental?, Bio essays, 2, 72-77, (1985) |

[12] | Tyson, J.J., The coordination of cell growth and division: a comparison of models, (), 291-295 |

[13] | Tyson, J.J.; Hanssgen, K.B., The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions, J. theor. biol., 113, 29-62, (1985) |

[14] | J. J. Tyson and O. Diekmann, Sloppy size control of the cell division cycle, J. Theor. Biol. (in press). |

[15] | Tyson, J.J.; Hannsgen, K.B., Global asymptotic stability of the size distribution in probabilistic models of the cell cycle, J. math. biol., 22, 61-68, (1985) · Zbl 0558.92012 |

[16] | Hannsgen, K.B.; Tyson, J.J.; Watson, L.T., Steady-state size distributions in probabilistic models of the cell division cycle, SIAM J. appl. math., 45, 523-540, (1985) · Zbl 0577.92019 |

[17] | Krasnoselskii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen |

[18] | Schaefer, H.H., Banach lattices and positive operators, (1974), Springer-Verlag · Zbl 0291.46008 |

[19] | Diekmann, O.; Lauwerier, H.A.; Aldenberg, T.; Metz, J.A.J., Growth, fission and the stable size distribution, J. math. biol., 18, 135-148, (1983) · Zbl 0533.92016 |

[20] | Courant, R.; Hilbert, D., () |

[21] | Garabedian, R.R., Partial differential equations, (1964), John Wiley · Zbl 0124.30501 |

[22] | Lang, S., Analysis II, (1969), Addison-Wesley |

[23] | Yosida, K., Functional analysis, () · Zbl 0152.32102 |

[24] | Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag Berlin · Zbl 0516.47023 |

[25] | Tanabe, H., Equations of evolution, (1979), Pitman |

[26] | Dowson, H.R., Spectral theory of linear operators, (1978), Academic Press · Zbl 0384.47001 |

[27] | Heijmans, H.J.A.M., An eigenvalue problem related to cell growth, J. math. anal. appl., 111, 253-280, (1985) · Zbl 0594.92010 |

[28] | Kato, T., Perturbation theory for linear operators, () · Zbl 0148.12601 |

[29] | Schaefer, H.H., Topological vector spaces, (1966), Macmillan · Zbl 0141.30503 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.