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

In this paper, the existence of solutions to a class of retarded functional di↵erential equations... more In this paper, the existence of solutions to a class of retarded functional di↵erential equations is established along with a result concerning the continuous dependence of solutions on a parameter. These results generalize theorems concerning existence and continuous dependence in a Lebesgue integral setting to a Henstock-Kurzweil integral setting.

Forest Science, 1995

ABSTRACT

Journal of Thermal Science, 1992

The determination of critical conditions for thermal ignition of combustible materials has been t... more The determination of critical conditions for thermal ignition of combustible materials has been traditionally studied by the use of one overall reaction with bounded parameter values for the activation energy and other chemical constants. Significant errors can occur in the values of the threshold parameters for ignition when there are two (or more) simultaneous reactions present with distinct values of the chemical constants. Recent work with simultaneous parallel reactions showed the thresholds for ignition could be lowered in this case. In this paper, motivated by experimental results for forest litter and coal, it is shown that for sequential reactions (different values of parameters in different temperature ranges) that the threshold conditions are changed (safer for lower ambient temperatures and less safe for higher ambient temperatures). The mathematical analysis is summarised and a detailed analysis is given for the forest litter and crushed coal applications. The experimental results show that variable activation energy does occur and that this extension of the classical Frank-Kamenetskii theory is needed. Here the analysis is confined to the slab geometry only but the ideas developed can euily be extended to more general systems, including tho~e involving mass transport, consumption, and phase changes.

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

The quasilinear equation, which arises in combustion theory in the investigation of the steady-st... more The quasilinear equation, which arises in combustion theory in the investigation of the steady-state energy balance has an intriguing solution set in spherical domains where it becomes n-l du-++eU=O, P dr O<r<l u'(1) + Bi. u(1) = 0, u'(0) = 0. This can be shown to have an infinite number of finite positive solutions when X = X, = 2(n-2)em21B ', when 2 < n < 10. Phase plane techniques are used. All but the minimal solutions are unstable as solutions of the timedependent version of the above equations. The extension of these methods to spherically annular domains 0 < (L < T < 1, with an inner boundary condition ~'(a) = A(5 0), shows strikingly different behaviour. First, the infinite multiplicity disappears and is approached asymptotically as (Y-) O+. Second, the uniqueness of the solution for small X also disappears. This last fact has implications for the basis of stability of the minimal steady-state.

Journal of Theoretical Biology, 2002

The effects of milking frequency on milk production is a key question for the dairy industry. Mil... more The effects of milking frequency on milk production is a key question for the dairy industry. Milk production is related to the number of active alveoli in the mammary gland and movement between active and quiescent alveolar pools is influenced by the milking frequency. In this paper, we analyse a mechanistic model based on known biological inputs that describes the effect of milking frequency on the alveolar composition of the mammary gland. It is shown that the model can qualitatively reproduce the correct alveolar dynamics. We also investigate the model robustness and parameter sensitivity. Additionally, by making the plausible assumption that the senescence rate of alveoli is proportional to the number of quiescent alveoli present, we obtain an analytical solution requiring periodic resetting.

Inverse Problems, 1991

ABSTRACT

International Journal of Intelligent Systems Technologies and Applications, 2007

This paper presents an in vivo detection method to estimate the size and the position of a breast... more This paper presents an in vivo detection method to estimate the size and the position of a breast tumour using microwave frequencies. At these frequencies there is a significant difference in dielectric properties between a malignant tumour and healthy breast tissue. By considering these properties we solve the forward problem of the signal's scattering effect for a two-dimensional breast model. We use Newton's multidimensional iterative method to solve the inverse problem and compute the unknown location and size of the tumour. Our analytical study suggests that the approach will be capable of detecting a millimetre size tumour inside the breast. Due to the complex scattering from the non-homogeneous internal structure and other complications the microwave measurements can have errors. However, in tests, our algorithm can calculate tumour distance with 0.54% error when there is a 10% error in the value of the microwave field measurements.

IET Science, Measurement & Technology, 2009

A two-dimensional inverse computing method to detect an internal object using microwave measureme... more A two-dimensional inverse computing method to detect an internal object using microwave measurements is presented. The modelling of the application system has been directed towards the eventual in vivo detection of breast tumours, in particular. This procedure will enable the non-destructive determination of the internal object size and location. The model procedure has been tested by using complex reflection coefficients measured in a microwave experimental application system. Measured data agree with the theoretical calculations.

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

A rather unusual but realistic two-dimensional dynamical system is formulated, analysed, and inte... more A rather unusual but realistic two-dimensional dynamical system is formulated, analysed, and interpreted for the interactions of earthworms and plant litter in soils. The model proposes the use of litter quality as an independent variable as well as the elapsed time. The system is searched for thresholds beyond which the system is sustainable with an asymptotically stable steady state. Only one such state is possible. The outcome is that there is a threshold on the average quality value of the litter input which has to be above a certain value for the system to be sustainable. For special descriptions of the litter input and palatability dependence on quality, explicit solutions are possible for the stability analysis.

ANZIAM Journal, 2016

We study a cell growth model with a division function that models cells which divide only after t... more We study a cell growth model with a division function that models cells which divide only after they have reached a certain minimum size. In contrast to the cases studied in the literature, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. We show that there is a steady size distribution solution to this problem.

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

In this paper, the existence of solutions to a class of retarded functional di↵erential equations... more In this paper, the existence of solutions to a class of retarded functional di↵erential equations is established along with a result concerning the continuous dependence of solutions on a parameter. These results generalize theorems concerning existence and continuous dependence in a Lebesgue integral setting to a Henstock-Kurzweil integral setting.

Forest Science, 1995

ABSTRACT

Journal of Thermal Science, 1992

The determination of critical conditions for thermal ignition of combustible materials has been t... more The determination of critical conditions for thermal ignition of combustible materials has been traditionally studied by the use of one overall reaction with bounded parameter values for the activation energy and other chemical constants. Significant errors can occur in the values of the threshold parameters for ignition when there are two (or more) simultaneous reactions present with distinct values of the chemical constants. Recent work with simultaneous parallel reactions showed the thresholds for ignition could be lowered in this case. In this paper, motivated by experimental results for forest litter and coal, it is shown that for sequential reactions (different values of parameters in different temperature ranges) that the threshold conditions are changed (safer for lower ambient temperatures and less safe for higher ambient temperatures). The mathematical analysis is summarised and a detailed analysis is given for the forest litter and crushed coal applications. The experimental results show that variable activation energy does occur and that this extension of the classical Frank-Kamenetskii theory is needed. Here the analysis is confined to the slab geometry only but the ideas developed can euily be extended to more general systems, including tho~e involving mass transport, consumption, and phase changes.

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

The quasilinear equation, which arises in combustion theory in the investigation of the steady-st... more The quasilinear equation, which arises in combustion theory in the investigation of the steady-state energy balance has an intriguing solution set in spherical domains where it becomes n-l du-++eU=O, P dr O<r<l u'(1) + Bi. u(1) = 0, u'(0) = 0. This can be shown to have an infinite number of finite positive solutions when X = X, = 2(n-2)em21B ', when 2 < n < 10. Phase plane techniques are used. All but the minimal solutions are unstable as solutions of the timedependent version of the above equations. The extension of these methods to spherically annular domains 0 < (L < T < 1, with an inner boundary condition ~'(a) = A(5 0), shows strikingly different behaviour. First, the infinite multiplicity disappears and is approached asymptotically as (Y-) O+. Second, the uniqueness of the solution for small X also disappears. This last fact has implications for the basis of stability of the minimal steady-state.

Journal of Theoretical Biology, 2002

The effects of milking frequency on milk production is a key question for the dairy industry. Mil... more The effects of milking frequency on milk production is a key question for the dairy industry. Milk production is related to the number of active alveoli in the mammary gland and movement between active and quiescent alveolar pools is influenced by the milking frequency. In this paper, we analyse a mechanistic model based on known biological inputs that describes the effect of milking frequency on the alveolar composition of the mammary gland. It is shown that the model can qualitatively reproduce the correct alveolar dynamics. We also investigate the model robustness and parameter sensitivity. Additionally, by making the plausible assumption that the senescence rate of alveoli is proportional to the number of quiescent alveoli present, we obtain an analytical solution requiring periodic resetting.

Inverse Problems, 1991

ABSTRACT

International Journal of Intelligent Systems Technologies and Applications, 2007

This paper presents an in vivo detection method to estimate the size and the position of a breast... more This paper presents an in vivo detection method to estimate the size and the position of a breast tumour using microwave frequencies. At these frequencies there is a significant difference in dielectric properties between a malignant tumour and healthy breast tissue. By considering these properties we solve the forward problem of the signal's scattering effect for a two-dimensional breast model. We use Newton's multidimensional iterative method to solve the inverse problem and compute the unknown location and size of the tumour. Our analytical study suggests that the approach will be capable of detecting a millimetre size tumour inside the breast. Due to the complex scattering from the non-homogeneous internal structure and other complications the microwave measurements can have errors. However, in tests, our algorithm can calculate tumour distance with 0.54% error when there is a 10% error in the value of the microwave field measurements.

IET Science, Measurement & Technology, 2009

A two-dimensional inverse computing method to detect an internal object using microwave measureme... more A two-dimensional inverse computing method to detect an internal object using microwave measurements is presented. The modelling of the application system has been directed towards the eventual in vivo detection of breast tumours, in particular. This procedure will enable the non-destructive determination of the internal object size and location. The model procedure has been tested by using complex reflection coefficients measured in a microwave experimental application system. Measured data agree with the theoretical calculations.

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

A rather unusual but realistic two-dimensional dynamical system is formulated, analysed, and inte... more A rather unusual but realistic two-dimensional dynamical system is formulated, analysed, and interpreted for the interactions of earthworms and plant litter in soils. The model proposes the use of litter quality as an independent variable as well as the elapsed time. The system is searched for thresholds beyond which the system is sustainable with an asymptotically stable steady state. Only one such state is possible. The outcome is that there is a threshold on the average quality value of the litter input which has to be above a certain value for the system to be sustainable. For special descriptions of the litter input and palatability dependence on quality, explicit solutions are possible for the stability analysis.

ANZIAM Journal, 2016

We study a cell growth model with a division function that models cells which divide only after t... more We study a cell growth model with a division function that models cells which divide only after they have reached a certain minimum size. In contrast to the cases studied in the literature, the determination of the steady size distribution entails an eigenvalue that is not known explicitly, but is defined through a continuity condition. We show that there is a steady size distribution solution to this problem.