The analysis of fragmentation type equation for special division kernels : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, School of Fundamental Sciences, New Zealand

Abstract

The growth fragmentation equation is a linear integro-differential equation describing the evolution of cohorts that grow, divide and die or disappear in the course of time. The general formula is of first or second order, depending whether the growth process is deterministic or stochastic, respectively. We focus on a particular choice of division kernel that models size-structured cell cohorts which divide into daughter cells of equal size. This problem reduces to an initial-boundary value type that involves a modified Fokker-Planck equation with an advanced functional term. There are no general techniques for solving these problems. The constant growth rate case has been studied by a number of researchers. In particular, it was shown that the limiting solutions converge to a special solution, the separable solution.

We consider the case when the growth rate is linear and deterministic. This problem can be solved analytically for monomial splitting rates. We show that the long time dynamics for this case differ markedly from the constant growth rate case. Specifically, the solutions approach a time dependent attracting solution that is periodic in time.

The qualitative features of solutions differ when the splitting rate is constant. There are two cases. The first is when the growth rate is deterministic; the second is when the growth rate is stochastic. This case involves a constant dispersion term. In both cases, the problem can be solved directly, and the classic properties of solutions can be adapted from the previous case (with non constant splitting rate). The main distinct trait is that there is no long time attracting solution in $L^1$ for probability distribution initial data. (This result in the dispersive case follows, provided the parameters $g$, $b$ and $\alpha$ satisfy a certain inequality.) The long time asymptotic behaviour of solutions proves to be formidable to evaluate analytically for these cases. We use numerical methods to elucidate possible behaviour and examine the influence of the dispersion term. We find numerical evidence that the dispersion term plays a prominent r\^{o}le as a smoothing effect on the oscillatory behaviour, spatially and time wise, encountered in the dispersion free examples with exponential growth.