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The role of memory and cognition in the movement of individuals (e.g. animals) within a population, is thought to play an important role in population dispersal. In response, there has been increasing interest in incorporating spatial…
Generative diffusion models have achieved remarkable success in producing high-quality images. However, these models typically operate in continuous intensity spaces, diffusing independently across pixels and color channels. As a result,…
Stochastic models of diffusion with excluded-volume effects are used to model many biological and physical systems at a discrete level. The average properties of the population may be described by a continuum model based on partial…
Mathematically modelling diffusive and advective transport of particles in heterogeneous layered media is important to many applications in computational, biological and medical physics. While deterministic continuum models of such…
Self-organisation of individuals within large collectives occurs throughout biology. Mathematical models can help elucidate the individual-level mechanisms behind these dynamics, but analytical tractability often comes at the cost of…
Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs),…
Continuum models for the spatial dynamics of growing cell populations have been widely used to investigate the mechanisms underpinning tissue development and tumour invasion. These models consist of nonlinear partial differential equations…
Mathematical models of biological populations commonly use discrete structure classes to capture trait variation among individuals (e.g. age, size, phenotype, intracellular state). Upscaling these discrete models into continuum descriptions…
We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model involves a partial differential equation describing the…
Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either…
Mathematical models are vital interpretive and predictive tools used to assist in the understanding of cell migration. There are typically two approaches to modelling cell migration: either micro-scale, discrete or macro-scale, continuum.…
Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or…
Numerous processes across both the physical and biological sciences are driven by diffusion. Partial differential equations (PDEs) are a popular tool for modelling such phenomena deterministically, but it is often necessary to use…
We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution…
We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have numerical results in one and two dimensions. In…
Reaction-diffusion equations describe various spatially extended processes that unfold as traveling fronts moving at constant velocity. We introduce and solve analytically a model that, besides such fronts, supports solutions advancing as…
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
Understanding whether a population will survive and flourish or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction-diffusion equations, where…
We show that propagation speeds in invasion processes modeled by reaction-diffusion systems are determined by marginal spectral stability conditions, as predicted by the marginal stability conjecture. This conjecture was recently settled in…
Abridged abstract: Inert interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system. These interactions affect only the movement of the entities, and examples…