Janet Dyson

Asynchronous exponential growth in an age structured population of proliferating and quiescent cells

A model of a proliferating cell population is analyzed. The model distinguishes individual cells by cell age, which corresponds to phase of the cell cycle. The model also distinguishes individual cells by proliferating or quiescent status. The model allows cells to transit between these two states at any age, that is, any phase of the cell cycle. The model also allows newly divided cells to enter quiescence at cell birth, that is, cell age 0. Sufficient conditions are established to assure that the cell population has asynchronous exponential growth. As a consequence of this asynchronous exponential growth the population stabilizes in the sense that the proportion of the population in any age range, or the fraction in proliferating or quiescent state, converges to a limiting value as time evolves, independently of the age distribution and proliferating or quiescent fractions of the initial cell population. The asynchronous exponential growth is proved by demonstrating that the strongly continuous linear semi-group associated with the partial differential equations of the model is positive, irreducible, and eventually compact.

20.—Functional Differential Equations and Non-linear Evolution Operators *

The abstract non-linear non-autonomous functional differential equation is considered. An evolution operator is associated with the solutions of this equation and existence and stability results are obtained.

Semigroups of translations associated with functional and functional differential equations

Functional and functional differential equations in a Banach space X are related to systems of operators A ( t ) in C = C (− r , 0; X ), given by Conditions are sought on F such that A ( t ) generates an evolution system U ( t, s )φThis system gives the segments of solution for φ in a certain domain which is determined.

An Age and Spatially Structured Model of Tumor Invasion with Haptotaxis II

An analysis of a model of tumor growth into surrounding tissue is continued from an earlier treatment, in which the global existence of unique solutions to the model was established. The model consists of a system of nonlinear partial differential equations for the population densities of tumor cells, extracellular matrix macromolecules, oxygen concentration, and extracellular matrix degradative enzyme concentration. The spatial growth of the tumor involves the directed movement of tumor cells toward the extracellular matrix through haptotaxis. Cell age is used to track progression of cells through the cell cycle. Regularity, positivity, and global bounds of the solutions of the model are proved.

Existence and Asymptotic Properties of Solutions of a Nonlocal Evolution Equation Modeling Cell-Cell Adhesion

In this paper we consider some fundamental properties of a new type of nonlocal reaction-diffusion equation originally proposed a few years ago in [N. J. Armstrong, K. J. Painter, and J. A. Sherratt, J. Theoret. Biol., 243 (2006), pp. 98–113] as a possible continuum mathematical model for cell-cell adhesion. The basic model is on an infinite domain and contains a nonlocal flux term which models the component of cell motion attributable to the cell having formed bonds with nearby cells within its sensing radius, and the nonlocal term is both nonlinear and involves spatial derivatives, making the analysis challenging. We establish the local existence of a classical solution working in spaces of uniformly continuous functions. We then establish that the model has a positivity preserving property and we find bounds on the solution, and we then establish the existence of a unique global solution in each of the biologically realistic cases when the cell density

Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions

n(x,t)

Some remarks on the asymptotic behavior of a nonautonomous, nonlinear functional differential equation

satisfies

A SEMILINEAR TRANSPORT EQUATION WITH DELAYS

n(x,0)\rightarrow0

A semigroup approach to nonlinear nonautonomous neutral functional differential equations

and

A non-local evolution equation model of cell–cell adhesion in higher dimensional space

n(x,0...