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

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