Graeme C Wake - Academia.edu (original) (raw)

Papers by Graeme C Wake

Springer eBooks, 2014

Living cell populations which are simultaneously growing and dividing are usually structured by s... more Living cell populations which are simultaneously growing and dividing are usually structured by size, which can be, for example, mass, volume, or DNA content. The evolution of the number density \(n(x,t)\) of cells by size \(x\), in an unperturbed situation, is observed experimentally to exhibit the attribute of that of an asymptotic “Steady-Size-Distribution” (SSD). That is, \(n(x,t) \sim \) scaled (by time \(t\)) multiple of a constant shape \(y(x)\) as \(t \rightarrow \infty \), and \(y(x)\) is then the SSD distribution, with constant shape for large time. A model describing this is given, enabling parameters to be evaluated. The model involves a linear non-local partial differential equation. Similar to the well-known pantograph equation, the solution gives rise to an unusual first order singular eigenvalue problem. Some results and conjectures are given on the spectrum of this problem. The principal eigenfunction gives the steady-size distribution and serves to provide verification of the observation about the asymptotic growth of the size-distribution.

Differential and Integral Equations

ABSTRACT A model for the simultaneous growth and division of a cell population structured by size... more ABSTRACT A model for the simultaneous growth and division of a cell population structured by size is examined. The case considered here is that of asymmetrical cell division when cells are dividing into beta_1\beta_1beta1 and beta2\beta_2beta_2 daughter cells at a constant rate and the parameters for growth and mortality are constants. The model has a steady-size distribution solution which satisfies a nonlocal differential equation. The solution is in the form of a Dirichlet series which is shown to be the unique probability density function for the steady-size distribution.

Differential and Integral Equations

Forest Science, 1995

ABSTRACT

Journal of Thermal Science, 1992

Journal of Applied Mathematics and Decision Sciences, 1997

To examine the long-term effects of fertiliser application on pasture growth under grazing, a mat... more To examine the long-term effects of fertiliser application on pasture growth under grazing, a mathematical representation of the pasture ecosystem is created and analysed mathematically. From this the nutrient application level needed to maintain a given stocking rate can be determined, along with its profitability. Feasible stocking levels and fertiliser application rates are investigated and the optimal combination found, along with the sensitivity of this combination. It is shown that profitability is relatively insensitive to fertiliser level compared with stocking rate.

Trends in Mathematics, 2015

In this note we discuss the following topics: 1. Epigenetics: How to alter your genes? This is ev... more In this note we discuss the following topics: 1. Epigenetics: How to alter your genes? This is evolution within a lifetime. Epigenetics is a relatively new scientific field; research only began in the mid nineties, and has only found traction in the wider scientific community in the last decade or so. We have long been told our genes are our destiny. But it is now thought a genotype’s expression (that is, its phenotype), can change during its lifetime by habit, lifestyle, even finances. What does this mean for our children? So we consider phenotype change: (a) firstly in a stochastic setting, where we consider the expected value of the mean fitness; (b) then we consider a Plastic Adaptive Response (PAR) in which the response to an environmental cue is initiated after a period of waiting; (c) finally, we consider the steady-fitness states, when the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state. 2. Consider the steady-size distribution of an evolving cohort of cells and therein establish thresholds for growth or decay of the cohort.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1997

The bifurcational behaviour is investigated of a simple mathematical model of the self-heating of... more The bifurcational behaviour is investigated of a simple mathematical model of the self-heating of combustible vapour from the evaporation of combustible fluid within (fibrous) lagging material. The lagging is considered to be completely soaked in the combustible fluid so that the fibres are completely covered; hence the evaporation term in this model is not dependent on the amount of liquid present and the main ignition event (due to oxidation of vapour) is countered by the endothermic evaporation and Newtonian cooling. This leads to a simpler equation set in the temperature and amount of vapour only (the liquid equation is decoupled). It is found that depending on the dimensions of the material (proportional to the volume to surface area ratio in this well-stirred approach), there are not only saddle-node bifurcations but important Hopf bifurcations leading to stable limit cycles in the temperature-fuel vapour concentration phase plane.

Obesity Research & Clinical Practice, 2010

Mathematical and Computer Modelling, 1993

Journal of Theoretical Biology, 2002

Inverse Problems, 1991

ABSTRACT

International Journal of Intelligent Systems Technologies and Applications, 2007

IET Science, Measurement & Technology, 2009

European Journal of Applied Mathematics, 2011

In this paper we study the probability density function solutions to a second-order pantograph eq... more In this paper we study the probability density function solutions to a second-order pantograph equation with a linear dispersion term. The functional equation comes from a cell growth model based on the Fokker–Planck equation. We show that the equation has a unique solution for constant positive growth and splitting rates and construct the solution using the Mellin transform.

Combustion Theory and Modelling, 1999

A model of self-heating of wet coal is presented. This involves coupled heat and mass transport w... more A model of self-heating of wet coal is presented. This involves coupled heat and mass transport within a coal pile, together with an exothermic reaction and phase changes of water. There are four state variables: temperature, oxygen, water vapour and liquid water concentrations. Heat and mass are conducted or diffused through the pile, while simultaneously undergoing chemical reaction. As demonstrated

Agricultural Systems, 1999

Springer eBooks, 2014

Living cell populations which are simultaneously growing and dividing are usually structured by s... more Living cell populations which are simultaneously growing and dividing are usually structured by size, which can be, for example, mass, volume, or DNA content. The evolution of the number density \(n(x,t)\) of cells by size \(x\), in an unperturbed situation, is observed experimentally to exhibit the attribute of that of an asymptotic “Steady-Size-Distribution” (SSD). That is, \(n(x,t) \sim \) scaled (by time \(t\)) multiple of a constant shape \(y(x)\) as \(t \rightarrow \infty \), and \(y(x)\) is then the SSD distribution, with constant shape for large time. A model describing this is given, enabling parameters to be evaluated. The model involves a linear non-local partial differential equation. Similar to the well-known pantograph equation, the solution gives rise to an unusual first order singular eigenvalue problem. Some results and conjectures are given on the spectrum of this problem. The principal eigenfunction gives the steady-size distribution and serves to provide verification of the observation about the asymptotic growth of the size-distribution.

Differential and Integral Equations

ABSTRACT A model for the simultaneous growth and division of a cell population structured by size... more ABSTRACT A model for the simultaneous growth and division of a cell population structured by size is examined. The case considered here is that of asymmetrical cell division when cells are dividing into beta_1\beta_1beta1 and beta2\beta_2beta_2 daughter cells at a constant rate and the parameters for growth and mortality are constants. The model has a steady-size distribution solution which satisfies a nonlocal differential equation. The solution is in the form of a Dirichlet series which is shown to be the unique probability density function for the steady-size distribution.

Differential and Integral Equations

Forest Science, 1995

ABSTRACT

Journal of Thermal Science, 1992

Journal of Applied Mathematics and Decision Sciences, 1997

To examine the long-term effects of fertiliser application on pasture growth under grazing, a mat... more To examine the long-term effects of fertiliser application on pasture growth under grazing, a mathematical representation of the pasture ecosystem is created and analysed mathematically. From this the nutrient application level needed to maintain a given stocking rate can be determined, along with its profitability. Feasible stocking levels and fertiliser application rates are investigated and the optimal combination found, along with the sensitivity of this combination. It is shown that profitability is relatively insensitive to fertiliser level compared with stocking rate.

Trends in Mathematics, 2015

In this note we discuss the following topics: 1. Epigenetics: How to alter your genes? This is ev... more In this note we discuss the following topics: 1. Epigenetics: How to alter your genes? This is evolution within a lifetime. Epigenetics is a relatively new scientific field; research only began in the mid nineties, and has only found traction in the wider scientific community in the last decade or so. We have long been told our genes are our destiny. But it is now thought a genotype’s expression (that is, its phenotype), can change during its lifetime by habit, lifestyle, even finances. What does this mean for our children? So we consider phenotype change: (a) firstly in a stochastic setting, where we consider the expected value of the mean fitness; (b) then we consider a Plastic Adaptive Response (PAR) in which the response to an environmental cue is initiated after a period of waiting; (c) finally, we consider the steady-fitness states, when the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state. 2. Consider the steady-size distribution of an evolving cohort of cells and therein establish thresholds for growth or decay of the cohort.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1997

The bifurcational behaviour is investigated of a simple mathematical model of the self-heating of... more The bifurcational behaviour is investigated of a simple mathematical model of the self-heating of combustible vapour from the evaporation of combustible fluid within (fibrous) lagging material. The lagging is considered to be completely soaked in the combustible fluid so that the fibres are completely covered; hence the evaporation term in this model is not dependent on the amount of liquid present and the main ignition event (due to oxidation of vapour) is countered by the endothermic evaporation and Newtonian cooling. This leads to a simpler equation set in the temperature and amount of vapour only (the liquid equation is decoupled). It is found that depending on the dimensions of the material (proportional to the volume to surface area ratio in this well-stirred approach), there are not only saddle-node bifurcations but important Hopf bifurcations leading to stable limit cycles in the temperature-fuel vapour concentration phase plane.

Obesity Research & Clinical Practice, 2010

Mathematical and Computer Modelling, 1993

Journal of Theoretical Biology, 2002

Inverse Problems, 1991

ABSTRACT

International Journal of Intelligent Systems Technologies and Applications, 2007

IET Science, Measurement & Technology, 2009

European Journal of Applied Mathematics, 2011

In this paper we study the probability density function solutions to a second-order pantograph eq... more In this paper we study the probability density function solutions to a second-order pantograph equation with a linear dispersion term. The functional equation comes from a cell growth model based on the Fokker–Planck equation. We show that the equation has a unique solution for constant positive growth and splitting rates and construct the solution using the Mellin transform.

Combustion Theory and Modelling, 1999

A model of self-heating of wet coal is presented. This involves coupled heat and mass transport w... more A model of self-heating of wet coal is presented. This involves coupled heat and mass transport within a coal pile, together with an exothermic reaction and phase changes of water. There are four state variables: temperature, oxygen, water vapour and liquid water concentrations. Heat and mass are conducted or diffused through the pile, while simultaneously undergoing chemical reaction. As demonstrated

Agricultural Systems, 1999