2,026 research outputs found
Accurate and efficient calculation of response times for groundwater flow
We study measures of the amount of time required for transient flow in
heterogeneous porous media to effectively reach steady state, also known as the
response time. Here, we develop a new approach that extends the concept of mean
action time. Previous applications of the theory of mean action time to
estimate the response time use the first two central moments of the probability
density function associated with the transition from the initial condition, at
, to the steady state condition that arises in the long time limit, as . This previous approach leads to a computationally convenient
estimation of the response time, but the accuracy can be poor. Here, we outline
a powerful extension using the first raw moments, showing how to produce an
extremely accurate estimate by making use of asymptotic properties of the
cumulative distribution function. Results are validated using an existing
laboratory-scale data set describing flow in a homogeneous porous medium. In
addition, we demonstrate how the results also apply to flow in heterogeneous
porous media. Overall, the new method is: (i) extremely accurate; and (ii)
computationally inexpensive. In fact, the computational cost of the new method
is orders of magnitude less than the computational effort required to study the
response time by solving the transient flow equation. Furthermore, the approach
provides a rigorous mathematical connection with the heuristic argument that
the response time for flow in a homogeneous porous medium is proportional to
, where is a relevant length scale, and is the aquifer
diffusivity. Here, we extend such heuristic arguments by providing a clear
mathematical definition of the proportionality constant.Comment: 22 pages, 3 figures, accepted version of paper published in Journal
of Hydrolog
New homogenization approaches for stochastic transport through heterogeneous media
The diffusion of molecules in complex intracellular environments can be
strongly influenced by spatial heterogeneity and stochasticity. A key challenge
when modelling such processes using stochastic random walk frameworks is that
negative jump coefficients can arise when transport operators are discretized
on heterogeneous domains. Often this is dealt with through homogenization
approximations by replacing the heterogeneous medium with an
homogeneous medium. In this work, we present a new class
of homogenization approximations by considering a stochastic diffusive
transport model on a one-dimensional domain containing an arbitrary number of
layers with different jump rates. We derive closed form solutions for the th
moment of particle lifetime, carefully explaining how to deal with the internal
interfaces between layers. These general tools allow us to derive simple
formulae for the effective transport coefficients, leading to significant
generalisations of previous homogenization approaches. Here, we find that
different jump rates in the layers gives rise to a net bias, leading to a
non-zero advection, for the entire homogenized system. Example calculations
show that our generalized approach can lead to very different outcomes than
traditional approaches, thereby having the potential to significantly affect
simulation studies that use homogenization approximations.Comment: 9 pages, 2 figures, accepted version of paper published in The
Journal of Chemical Physic
Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art
Stochasticity is a key characteristic of intracellular processes such as gene
regulation and chemical signalling. Therefore, characterising stochastic
effects in biochemical systems is essential to understand the complex dynamics
of living things. Mathematical idealisations of biochemically reacting systems
must be able to capture stochastic phenomena. While robust theory exists to
describe such stochastic models, the computational challenges in exploring
these models can be a significant burden in practice since realistic models are
analytically intractable. Determining the expected behaviour and variability of
a stochastic biochemical reaction network requires many probabilistic
simulations of its evolution. Using a biochemical reaction network model to
assist in the interpretation of time course data from a biological experiment
is an even greater challenge due to the intractability of the likelihood
function for determining observation probabilities. These computational
challenges have been subjects of active research for over four decades. In this
review, we present an accessible discussion of the major historical
developments and state-of-the-art computational techniques relevant to
simulation and inference problems for stochastic biochemical reaction network
models. Detailed algorithms for particularly important methods are described
and complemented with MATLAB implementations. As a result, this review provides
a practical and accessible introduction to computational methods for stochastic
models within the life sciences community
Critical length for the spreading-vanishing dichotomy in higher dimensions
We consider an extension of the classical Fisher-Kolmogorov equation, called
the \textit{Fisher-Stefan} model, which is a moving boundary problem on . A key property of the Fisher-Stefan model is the
\textit{spreading-vanishing dichotomy}, where solutions with will eventually spread as , whereas solutions
where will vanish as . In one
dimension is it well-known that the critical length is . In this work we re-formulate the Fisher-Stefan model in higher
dimensions and calculate as a function of spatial dimensions
in a radially symmetric coordinate system. Our results show how
depends upon the dimension of the problem and numerical
solutions of the governing partial differential equation are consistent with
our calculations
Reversible signal transmission in an active mechanical metamaterial
Mechanical metamaterials are designed to enable unique functionalities, but
are typically limited by an initial energy state and require an independent
energy input to function repeatedly. Our study introduces a theoretical active
mechanical metamaterial that incorporates a biological reaction mechanism to
overcome this key limitation of passive metamaterials. Our material allows for
reversible mechanical signal transmission, where energy is reintroduced by the
biologically motivated reaction mechanism. By analysing a coarse grained
continuous analogue of the discrete model, we find that signals can be
propagated through the material by a travelling wave. Analysis of the continuum
model provides the region of the parameter space that allows signal
transmission, and reveals similarities with the well-known FitzHugh-Nagumo
system. We also find explicit formulae that approximate the effect of the
timescale of the reaction mechanism on the signal transmission speed, which is
essential for controlling the material.Comment: 20 pages, 7 figure
A sharp-front moving boundary model for malignant invasion
We analyse a novel mathematical model of malignant invasion which takes the
form of a two-phase moving boundary problem describing the invasion of a
population of malignant cells into a population of background tissue, such as
skin. Cells in both populations undergo diffusive migration and logistic
proliferation. The interface between the two populations moves according to a
two-phase Stefan condition. Unlike many reaction-diffusion models of malignant
invasion, the moving boundary model explicitly describes the motion of the
sharp front between the cancer and surrounding tissues without needing to
introduce degenerate nonlinear diffusion. Numerical simulations suggest the
model gives rise to very interesting travelling wave solutions that move with
speed , and the model supports both malignant invasion and malignant
retreat, where the travelling wave can move in either the positive or negative
-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher
models where travelling waves move with a minimum wave speed ,
the moving boundary model leads to travelling wave solutions with . We interpret these travelling wave solutions in the phase plane and
show that they are associated with several features of the classical
Fisher-Kolmogorov phase plane that are often disregarded as being nonphysical.
We show, numerically, that the phase plane analysis compares well with long
time solutions from the full partial differential equation model as well as
providing accurate perturbation approximations for the shape of the travelling
waves.Comment: 48 pages, 21 figure
Beaver Lake Numeric Chlorophyll-a and Secchi Transparency Standards, Phases II and III: Uncertainty and Trend Analysis
The objective of Phases II and III of this study were to 1) assess the variation in chl‐a and ST across multiple spatial and temporal scales in Beaver Lake in order to validate the assessment method, and 2) quantify trends in chl‐a, ST, and nutrient (total phosphorus and total nitrogen) concentrations in Beaver Lake and the major inflowing rivers to verify any potential water quality impairment
Travelling wave solutions in a negative nonlinear diffusion-reaction model
We use a geometric approach to prove the existence of smooth travelling wave
solutions of a nonlinear diffusion-reaction equation with logistic kinetics and
a convex nonlinear diffusivity function which changes sign twice in our domain
of interest. We determine the minimum wave speed, c*, and investigate its
relation to the spectral stability of the travelling wave solutions.Comment: 23 pages, 10 figure
Rapid calculation of maximum particle lifetime for diffusion in complex geometries
Diffusion of molecules within biological cells and tissues is strongly
influenced by crowding. A key quantity to characterize diffusion is the
particle lifetime, which is the time taken for a diffusing particle to exit by
hitting an absorbing boundary. Calculating the particle lifetime provides
valuable information, for example, by allowing us to compare the timescale of
diffusion and the timescale of reaction, thereby helping us to develop
appropriate mathematical models. Previous methods to quantify particle
lifetimes focus on the mean particle lifetime. Here, we take a different
approach and present a simple method for calculating the maximum particle
lifetime. This is the time after which only a small specified proportion of
particles in an ensemble remain in the system. Our approach produces accurate
estimates of the maximum particle lifetime, whereas the mean particle lifetime
always underestimates this value compared with data from stochastic
simulations. Furthermore, we find that differences between the mean and maximum
particle lifetimes become increasingly important when considering diffusion
hindered by obstacles.Comment: 10 pages, 1 figur
Reconciling transport models across scales: the role of volume exclusion
Diffusive transport is a universal phenomenon, throughout both biological and
physical sciences, and models of diffusion are routinely used to interrogate
diffusion-driven processes. However, most models neglect to take into account
the role of volume exclusion, which can significantly alter diffusive
transport, particularly within biological systems where the diffusing particles
might occupy a significant fraction of the available space. In this work we use
a random walk approach to provide a means to reconcile models that incorporate
crowding effects on different spatial scales. Our work demonstrates that
coarse-grained models incorporating simplified descriptions of excluded volume
can be used in many circumstances, but that care must be taken in pushing the
coarse-graining process too far
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