# Estimates for approximate solutions to a functional differential equation model of cell division

## Authors

• Stephen William Taylor University of Auckland
• Xueshan Yang University of Auckland

## Keywords:

cell division, size-structured, functional PDE

## Abstract

The functional partial differential equation (FPDE) for cell division,

$\frac{\partial}{\partial t}n(x,t)+\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)$

$+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t),$

is not amenable to analytical solution techniques, despite being closely related to the first order partial differential equation (PDE)

$\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t),$

which, with known $$F(x,t)$$, can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $$n(\alpha x,t)$$ and $$n(\beta x,t)$$, where $$\beta \ge 2 \ge\alpha \ge 1$$, which arise because cells of size $$x$$ are created when cells of size  $$\alpha x$$ and $$\beta x$$ divide.

The nonnegative function, $$n(x,t)$$, denotes the density of cells at time $$t$$ with respect to cell size $$x$$. The functions $$g(x,t)$$, $$b(x,t)$$ and $$\mu(x,t)$$ are, respectively, the growth rate, splitting rate and death rate of cells of size $$x$$. The total number of cells, $$\int_{0}^{\infty}n(x,t)dx$$, coincides with the $$L^1$$ norm of $$n$$. The goal of this paper is to find estimates in $$L^1$$ (and, with some restrictions, $$L^p$$ for $$p>1$$) for a sequence of approximate solutions to the FPDE that are generated by solving the first order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

doi:10.1017/S1446181121000055

## Author Biographies

### Stephen William Taylor, University of Auckland

Mathematics Department, University of Auckland, New Zealand

### Xueshan Yang, University of Auckland

Mathematics Department, University of Auckland, New Zealand

## Published

2021-04-25 — Updated on 2021-04-25

## Section

Special Issue for Renown Researcher