Unimodality of steady size distributions of growing cell populations (Q1602288)

The authors investigate a hyperbolic functional partial differential equation, \(n_t(x,t)=A(n)\), where \[ A(n)\equiv -(g(x)n(x,t))_x-b(x)n(x,t)+\alpha^2b(\alpha x)n(\alpha x),\quad x\in \mathbb{R}_+. \] This describes the evolution of a population of growing and dividing cells. Here a mother cell of size \(x\) divides into \(\alpha >1\) (usually \(\alpha =2\)) daughter cells of the same size \(x\). The function \(g(x)\) is the growth rate and \(b(x)\) is the division rate of a cell of size \(x\). There is no mortality of cells, so the reasonable boundary conditions for this model are \(g(0)n(0,t)=0\) and \(\lim_{x\to\infty }g(x)n(x,t)=0\) \(t>0\). The main result is that the eigenfunction \(y(x)\) of the operator \(A(n)\) is unimodal. The authors use a known result of \textit{T. Kato} and \textit{J. B. McLeod} [Bull. Am. Math. Soc. 77, 891-937 (1971; Zbl 0236.34064)] to discuss that the probability density function for the stationary size distribution is necessarily unimodal.