# A Differential Equation for a Cell Growth Model

### From NZJM

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**New Zealand Journal of Mathematics**

**Vol.** 43, (2013), **Pages** 43-63

**Christopher S. Withers**

Applied Mathematics Group

Industrial Research Limited

Lower Hutt,

New Zealand

**Saralees Nadarajah**

School of Mathematics

University of Manchester

Manchester M13 9PL,

United Kingdom

mailto:Saralees.Nadarajah@manchester.ac.uk

**Abstract** We give solutions to the first order differential equation
In the cell growth model, *y*(*x*) is the probability density function
for the size of a cell, *b*(*x*) is the rate at which a cell
of size *x* divides and creates α_{i} new cells of size *x* / α_{i}
with probability *p*_{i}, *c*(*x*) is a function determined by *b*(*x*) and the
growth rate of a cell, *g*(*x*).

**Keywords** Cell growth model; Differential equations; Probability density function.

**Classification** (MSC2000) 34A99.

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