A functional differential equation arising in modelling of cell growth

Source abstract

Abstract A functional differential equation for the steady size distribution of a population is derived from the usual partial differential equation governing the size distribution, in the particular case where birth occurs by one individual of size x dividing into α new individuals of size x /α. This leads, in the case of constant growth and birth rate functions, to the functional differential equation y ′( x ) = − ay ( x ) + a α y (α x ) together with the integral condition We first look at a number of properties that any solution of this equation and boundary condition must have, and then proceed to find the unique solution by the method of Laplace transforms. Results from number theory on the infinite product found in the solution are presented, and it is shown that y ( x ) tends to a normal distribution as α → 1 + .