Now showing items 1-15 of 15

• An analytic approach to the normalized Ricci flow-like equation: Revisited

In this paper we revisit Hamilton’s normalized Ricci flow, which was thoroughly studied via a PDE approach in Kavallaris and Suzuki (2010). Here we provide an improved convergence result compared to the one presented Kavallaris and Suzuki (2010) for the critical case λ=8πλ=8π. We actually prove that the convergence towards the stationary normalized Ricci flow is realized through any time sequence.
• Data-driven selection and parameter estimation for DNA methylation mathematical models

Epigenetics is coming to the fore as a key process which underpins health. In particular emerging experimental evidence has associated alterations to DNA methylation status with healthspan and aging. Mammalian DNA methylation status is maintained by an intricate array of biochemical and molecular processes. It can be argued changes to these fundamental cellular processes ultimately drive the formation of aberrant DNA methylation patterns, which are a hallmark of diseases, such as cancer, Alzheimer's disease and cardiovascular disease. In recent years mathematical models have been used as e ective tools to help advance our understanding of the dynamics which underpin DNA methylation. In this paper we present linear and nonlinear models which encapsulate the dynamics of the molecular mechanisms which de ne DNA methylation. Applying a recently developed Bayesian algorithm for parameter estimation and model selection, we are able to estimate distributions of parameters which include nominal parameter values. Using limited noisy observations, the method also identifed which methylation model the observations originated from, signaling that our method has practical applications in identifying what models best match the biological data for DNA methylation.
• A discrete mutualism model: analysis and exploration of a financial application

We perform a stability analysis on a discrete analogue of a known, continuous model of mutualism. We illustrate how the introduction of delays affects the asymptotic stability of the system’s positive nontrivial equilibrium point. In the second part of the paper we explore the insights that the model can provide when it is used in relation to interacting financial markets. We also note the limitations of such an approach.
• Dynamics of shadow system of a singular Gierer-Meinhardt system on an evolving domain

The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer-Meinhardt model on an isotropically evolving domain. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.
• Explosive solutions of a stochastic non-local reaction–diffusion equation arising in shear band formation

In this paper, we consider a non-local stochastic parabolic equation which actually serves as a mathematical model describing the adiabatic shear-banding formation phenomena in strained metals. We first present the derivation of the mathematical model. Then we investigate under which circumstances a finite-time explosion for this non-local SPDE, corresponding to shear-banding formation, occurs. For that purpose some results related to the maximum principle for this non-local SPDE are derived and afterwards the Kaplan's eigenfunction method is employed.
• Finite-time blow-up of a non-local stochastic parabolic problem

The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up.
• Mathematical models of DNA methylation dynamics: Implications for health and ageing

DNA methylation status is a key epigenetic process which has been intimately associated with gene regulation. In recent years growing evidence has associated DNA methylation status with a variety of diseases including cancer, Alzheimers disease and cardiovascular disease. Moreover, changes to DNA methylation have also recently been implicated in the ageing process. The factors which underpin DNA methylation are complex, and remain to be fully elucidated. Over the years mathematical modelling has helped to shed light on the dynamics of this important molecular system. Although the existing models have contributed significantly to our overall understanding of DNA methylation, they fall-short of fully capturing the dynamics of this process. In this paper we develop a linear and nonlinear model which captures more fully the dynamics of the key intracellular events which characterise DNA methylation. In particular the outcomes of our linear model result in gene promoter specific methylation levels which are more biologically plausible than those revealed by previous mathematical models. In addition, our non-linear model predicts DNA methylation promoter bistability which is commonly observed experimentally. The findings from our models have implications for our current understanding of how changes to the dynamics which underpin DNA methylation affect ageing and health.
• A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted vs.\ repelled by a single chemical substance. The production vs.\ destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model we investigate the variational structures, in particular we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy-Littlewood-Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.
• Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis

This book presents new developments in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grades students in mathematics, engineering, physics, economics, and biology.
• On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics

We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for $u_t = u \Delta u + u \int_\Omega |\nabla u|^2$ in bounded domains $\Om\sub\R^n$ which arises in game theory. We prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with $\overline{\Omega}$, i.e. the finite-time blow-up is global.
• On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system

The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.

• On the quenching of a nonlocal parabolic problem arising in electrostatic MEMS control

We consider a nonlocal parabolic model for a micro-electro-mechanical system. Specifically, for a radially symmetric problem with monotonic initial data, it is shown that the solution quenches, so that touchdown occurs in the device, in a situation where there is no steady state. It is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained. Numerical simulations illustrating the results are given.
• Quenching solutions of a stochastic parabolic problem arising in electrostatic MEMS control

In the current paper, we consider a stochastic parabolic equation which actually serves as a mathematical model describing the operation of an electrostatic actuated micro-electro-mechanical system (MEMS). We first present the derivation of the mathematical model. Then after establishing the local well-posedeness of the problem we investigate under which circumstances a {\it finite-time quenching} for this SPDE, corresponding to the mechanical phenomenon of {\it touching down}, occurs. For that purpose the Kaplan's eigenfunction method adapted in the context of SPDES is employed.
• A time discretization scheme for a nonlocal degenerate problem modelling resistance spot welding

In the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm.