# A Differential Equation for a Cell Growth Model

### New Zealand Journal of Mathematics

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

Christopher S. Withers

Applied Mathematics Group

Industrial Research Limited

Lower Hutt,

New Zealand

School of Mathematics

University of Manchester

Manchester M13 9PL,

United Kingdom

Abstract We give solutions to the first order differential equation $y'(x)+c(x)y(x)= g(x)^{-1} \sum_{i=1}^I p_i b \left(\alpha_i x\right) y \left(\alpha_i x\right).$ 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 αi new cells of size x / αi with probability pi, 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.