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On the stability of the cell-size distribution. II. Time-periodic development rates. (English) Zbl 0667.92015
The 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.

92D25 Population dynamics (general)
35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: DOI
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