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 $t=0$, to the steady state condition that arises in the long time limit, as $t \to \infty$. 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 $k$ 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 $L^2/D$, where $L$ is a relevant length scale, and $D$ 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 $\textit{effective}$ 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 $k$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 $0 < x < L(t)$. A key property of the Fisher-Stefan model is the \textit{spreading-vanishing dichotomy}, where solutions with $L(t) > L_{\textrm{c}}$ will eventually spread as $t \to \infty$, whereas solutions where $L(t) \ngtr L_{\textrm{c}}$ will vanish as $t \to \infty$. In one dimension is it well-known that the critical length is $L_{\textrm{c}} = \pi/2$. In this work we re-formulate the Fisher-Stefan model in higher dimensions and calculate $L_{\textrm{c}}$ as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how $L_{\textrm{c}}$ 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 $c$, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative $x$-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed $c \ge c^* > 0$, the moving boundary model leads to travelling wave solutions with $|c| < c^{**}$. 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