arXiv subjects

Andrew L. Krause

Publications and source records attributed to Andrew L. Krause.

At least 19 recordsLinked to original sources

arXiv2026

Patterns in Time and Space from a Single Morphogen via Nonlinear Layering

Spatial and temporal pattern formation in reaction-diffusion systems is typically studied with two or more equations, as scalar reaction-diffusion equations confined to convex domains do not admit stable inhomogeneous states in time or space on long timescales. Here, we show that a single morphogen diffusing across layered two-dimensional media, with nonlinear coupling between layers, is able to generate stable patterns in time and space. This $N$-layer model is analysed via a thin-domain limit, which reduces to an $N$-component reaction-diffusion system on a homogeneous one-dimensional domain. This reduced model can be analysed via linear stability techniques, showing that non-diffusive, or reactive, coupling between regions is necessary for pattern-forming instabilities, at least in the reduced model. This reduced system can exhibit Turing, Hopf, and Turing-wave instabilities, with emergent structures that are numerically shown to persist even away from the thin-domain regime of the full 2D single-morphogen system. These results suggest that heterogeneous stratification and nonlinear coupling can broaden the class of systems which exhibit complex spatiotemporal behaviours, which may be relevant in scenarios where only a single morphogen is known to act.

nlin.PS↗

arXiv2025

Transient Instability and Patterns of Reactivity in Diffusive-Chemotaxis Soil Carbon Dynamics

We study pattern formation in a chemotaxis model of bacteria and soil carbon dynamics as an example system where transient dynamics can give rise to pattern formation outside of Turing unstable regimes. We use a detailed analysis of the reactivity of the non-spatial and spatial dynamics, stability analyses, and numerical continuation to uncover detailed aspects of this system's pattern-forming potential. In addition to patterning in Turing unstable parameter regimes, reactivity of the spatial system can itself lead to a range of parameters where a spatially uniform state is asymptotically stable, but exhibits transient growth that can induce pattern formation. We show that this occurs in the bistable region of a subcritical Turing bifurcation. Intriguingly, such bistable regions appear in two spatial dimensions, but not in a one-dimensional domain, suggesting important interplays between geometry, transient growth, and the emergence of multistable patterns. We discuss the implications of our analysis for the bacterial soil organic carbon system, as well as for reaction-transport modeling more generally.

math.NA↗

arXiv2025

Chase-and-Run and Chirality in Nonlocal Models of Pattern Formation

Chase-and-run dynamics, in which one population pursues another that flees from it, are found throughout nature, from predator-prey interactions in ecosystems to the collective motion of cells during development. Intriguingly, in many of these systems, the movement is not straight; instead, 'runners' veer off at an angle from their pursuers. This angled movement often exhibits a consistent left-right asymmetry, known as lateralisation or chirality. Inspired by such phenomena in zebrafish skin patterns and evasive animal motion, we explore how chirality shapes the emergence of patterns in nonlocal (integro-differential) advection-diffusion models. We extend such models to allow movement at arbitrary angles, uncovering a rich landscape of behaviours. We find that chirality can enhance pattern formation, suppress oscillations, and give rise to entirely new dynamical structures, such as rotating pulses of chasers and runners. We also uncover how chase-and-run dynamics can cause populations to mix or separate. Through linear stability analysis, we identify physical mechanisms that drive some of these effects, whilst also exposing striking limitations of this theory in capturing more complex dynamics. Our findings suggest that chirality could have roles in ecological and cellular patterning beyond simply breaking left-right symmetry.

nlin.PS↗

arXiv2025

Designing reaction-cross-diffusion systems with Turing and wave instabilities

General conditions are established under which reaction-cross-diffusion systems can undergo spatiotemporal pattern-forming instabilities. Recent work has focused on designing systems theoretically and experimentally to exhibit patterns with specific features, but the case of non-diagonal diffusion matrices has yet to be analysed. Here, a framework is presented for the design of general $n$-component reaction-cross-diffusion systems that exhibit Turing and wave instabilities of a given wavelength. For a fixed set of reaction kinetics, it is shown how to choose diffusion matrices that produce each instability; conversely, for a given diffusion tensor, how to choose linearised kinetics. The theory is applied to several examples including a hyperbolic reaction-diffusion system, two different 3-component models, and a spatio-temporal version of the Ross-Macdonald model for the spread of malaria.

nlin.PS↗

arXiv2025

Pattern Localisation in the Swift-Hohenberg Equation via Slowly Varying Spatial Heterogeneity

Theories of localised pattern formation are important to understand a broad range of natural patterns, but are less well-understood than more established mechanisms of domain-filling pattern formation. Here, we extend recent work on pattern localisation via slow spatial heterogeneity in reaction-diffusion systems to the Swift-Hohenberg equation. We use a WKB asymptotic approach to show that, in the limit of a large domain and slowly varying heterogeneity, conditions for Turing-type linear instability localise in a simple way, with the spatial variable playing the role of a parameter. For nonlinearities locally corresponding to supercritical bifurcations in the spatially homogeneous system, this analysis asymptotically predicts regions where patterned states are confined, which we confirm numerically. We resolve the inner region of this asymptotic approach, finding excellent agreement with the tails of these confined pattern regions. In the locally subcritical case, however, this theory is insufficient to fully predict such confined regions, and so we propose an approach based on numerical continuation of a local homogeneous analog system. Pattern localisation in the heterogeneous system can then be determined based on the Maxwell point of this system, with the spatial variable parameterizing this point. We compare this theory of localisation via spatial heterogeneity to localised patterns arising from homoclinic snaking, and suggest a way to distinguish between different localisation mechanisms in natural systems based on how these structures decay to the background state (i.e. how their tails decay). We also explore cases where both of these local theories of pattern formation fail to capture the interaction between spatial heterogeneity and underlying pattern-forming mechanisms, suggesting that more work needs to be done to fully disentangle exogenous and intrinsic heterogeneity.

nlin.PS↗

arXiv2025

Pattern Formation as a Resilience Mechanism in Cancer Immunotherapy

Mathematical and computational modelling in oncology has played an increasingly important role in not only understanding the impact of various approaches to treatment on tumour growth, but in optimizing dosing regimens and aiding the development of treatment strategies. However, as with all modelling, only an approximation is made in the description of the biological and physical system. Here we show that tissue-scale spatial structure can have a profound impact on the resilience of tumours to immunotherapy using a classical model incorporating IL-2 compounds and effector cells as treatment parameters. Using linear stability analysis, numerical continuation, and direct simulations, we show that diffusing cancer cell populations can undergo pattern-forming (Turing) instabilities, leading to spatially-structured states that persist far into treatment regimes where the corresponding spatially homogeneous systems would uniformly predict a cancer-free state. These spatially-patterned states persist in a wide range of parameters, as well as under time-dependent treatment regimes. Incorporating treatment via domain boundaries can increase this resistance to treatment in the interior of the domain, further highlighting the importance of spatial modelling when designing treatment protocols informed by mathematical models. Counter-intuitively, this mechanism shows that increased effector cell mobility can increase the resilience of tumours to treatment. We conclude by discussing practical and theoretical considerations for understanding this kind of spatial resilience in other models of cancer treatment, in particular those incorporating more realistic spatial transport.

q-bio.TO↗

arXiv2025

Integral Asymptotics, Coalescing Saddles, and Multiple-scales Analysis of a Generalised Swift-Hohenberg Equation

Integral asymptotics play an important role in the analysis of differential equations and in a variety of other settings. In this work, we apply an integral asymptotics approach to study spatially localized solutions of a heterogeneous generalised Swift-Hohenberg equation. The outer solution is obtained via WKBJ asymptotics, while the inner solution requires the method of coalescing saddles. We modify the classic method of Chester et al. to account for additional technicalities, such as complex branch selection and local transformation to a cubic polynomial. By integrating our results, we construct an approximate global solution to the generalised Swift-Hohenberg problem and validate it against numerical contour integral solutions. We also demonstrate an alternative approach that circumvents the complexity of integral asymptotics by analyzing the original differential equation directly through a multiple-scales analysis and show that this generates the same leading-order inner solution obtained using the coalescing saddles method at least for one of the cases considered via integral asymptotics. Our findings reinforce the significance of integral asymptotics in approximating the fourth order differential equations found in the linear stability analysis for generalisations of the Swift Hohenberg equations. This study has also highlighted a conjecture that, in certain cases, the method of coalescing saddles can be systematically replaced by multiple-scales analysis using an intermediary differential equation, a hypothesis for future investigation.

nlin.PS↗

arXiv2023

Interdependent Total Factor Productivity in an Input-Output model

Industries learn productivity improvements from their suppliers. The observed empirical importance of these interactions, often omitted by input-output models, mandates larger attention. This article embeds interdependent total factor productivity (TFP) growth into a general non-parametric input-output model. TFP growth is assumed to be Cobb-Douglas in TFP-stocks of adjacent sectors, where elasticities are the input-output coefficients. Studying how the steady state of the system reacts to changes in research effort bears insight for policy and the input-output literature. First, industries higher in the supply chain see a greater multiplication of their productivity gains. Second, the presence of `laggard' industries can bottleneck the the rest of the economy. By deriving these insights formally, we review a canonical method for aggregating TFP -- Hulten's Theorem -- and show the potential importance of backward linkages.

econ.TH↗

arXiv2023

Turing instabilities are not enough to ensure pattern formation

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction--diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. \ak{While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria.} Given that biological systems \ak{such as} gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

nlin.PS↗

arXiv2023

VisualPDE: rapid interactive simulations of partial differential equations

Computing has revolutionised the study of complex nonlinear systems, both by allowing us to solve previously intractable models and through the ability to visualise solutions in different ways. Using ubiquitous computing infrastructure, we provide a means to go one step further in using computers to understand complex models through instantaneous and interactive exploration. This ubiquitous infrastructure has enormous potential in education, outreach and research. Here, we present VisualPDE, an online, interactive solver for a broad class of 1D and 2D partial differential equation (PDE) systems. Abstract dynamical systems concepts such as symmetry-breaking instabilities, subcritical bifurcations and the role of initial data in multistable nonlinear models become much more intuitive when you can play with these models yourself, and immediately answer questions about how the system responds to changes in parameters, initial conditions, boundary conditions or even spatiotemporal forcing. Importantly, VisualPDE is freely available, open source and highly customisable. We give several examples in teaching, research and knowledge exchange, providing high-level discussions of how it may be employed in different settings. This includes designing web-based course materials structured around interactive simulations, or easily crafting specific simulations that can be shared with students or collaborators via a simple URL. We envisage VisualPDE becoming an invaluable resource for teaching and research in mathematical biology and beyond. We also hope that it inspires other efforts to make mathematics more interactive and accessible.

physics.ed-ph↗

arXiv2023

Patterning of nonlocal transport models in biology: the impact of spatial dimension

Throughout developmental biology and ecology, transport can be driven by nonlocal interactions. Examples include cells that migrate based on contact with pseudopodia extended from other cells, and animals that move based on their vision of other animals. Nonlocal integro-PDE models have been used to investigate contact attraction and repulsion in cell populations in 1D. In this paper, we generalise the analysis of pattern formation in such a model from 1D to higher spatial dimensions. Numerical simulations in 2D demonstrate complex behaviour in the model, including spatio-temporal patterns, multi-stability, and the selection of spots or stripes heavily depending on interactions being attractive or repulsive. Through linear stability analysis in $N$ dimensions, we demonstrate how, unlike in local Turing reaction-diffusion models, the capacity for pattern formation fundamentally changes with dimensionality for this nonlocal model. Most notably, pattern formation is possible only in higher than one spatial dimension for both the single species system with repulsive interactions, and the two species system with `run-and-chase' interactions. The latter case may be relevant to zebrafish stripe formation, which has been shown to be driven by run-and-chase dynamics between melanophore and xanthophore pigment cells.

nlin.PS↗

arXiv2023

Spatial Heterogeneity Localizes Turing Patterns in Reaction-Cross-Diffusion Systems

Motivated by bacterial chemotaxis and multi-species ecological interactions in heterogeneous environments, we study a general one-dimensional reaction-cross-diffusion system in the presence of spatial heterogeneity in both transport and reaction terms. Under a suitable asymptotic assumption that the transport is slow over the domain, while gradients in the reaction heterogeneity are not too sharp, we study the stability of a heterogeneous steady state approximated by the system in the absence of transport. Using a WKB ansatz, we find that this steady state can undergo a Turing-type instability in subsets of the domain, leading to the formation of localized patterns. The boundaries of the pattern-forming regions are given asymptotically by `local' Turing conditions corresponding to a spatially homogeneous analysis parameterized by the spatial variable. We developed a general open-source code which is freely available, and show numerical examples of this localized pattern formation in a Schnakenberg cross-diffusion system, a Keller-Segel chemotaxis model, and the Shigesada-Kawasaki-Teramoto model with heterogeneous parameters. We numerically show that the patterns may undergo secondary instabilities leading to spatiotemporal movement of spikes, though these remain approximately within the asymptotically predicted localized regions. This theory can elegantly differentiate between spatial structure due to background heterogeneity, from spatial patterns emergent from Turing-type instabilities.

nlin.PS↗

arXiv2023

Concentration-Dependent Domain Evolution in Reaction-Diffusion Systems

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction-diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution, and extend this to $N$-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

nlin.PS↗

arXiv2022

Fixed and Distributed Gene Expression Time Delays in Reaction-Diffusion Systems

Time delays, modelling the process of intracellular gene expression, have been shown to have important impacts on the dynamics of pattern formation in reaction-diffusion systems. In particular, past work has shown that such time delays can shrink the Turing space, thereby inhibiting patterns from forming across large ranges of parameters. Such delays can also increase the time taken for pattern formation even when Turing instabilities occur. Here we consider reaction-diffusion models incorporating fixed or distributed time delays, modelling the underlying stochastic nature of gene expression dynamics, and analyze these through a systematic linear instability analysis and numerical simulations for several sets of different reaction kinetics. We find that even complicated distribution kernels (skewed Gaussian probability density functions) have little impact on the reaction-diffusion dynamics compared to fixed delays with the same mean delay. We show that the location of the delay terms in the model can lead to changes in the size of the Turing space (increasing or decreasing) as the mean time delay, $τ$, is increased. We show that the time to pattern formation from a perturbation of the homogeneous steady state scales linearly with $τ$, and conjecture that this is a general impact of time delay on reaction-diffusion dynamics, independent of the form of the kinetics or location of the delayed terms. Finally we show that while initial and boundary conditions can influence these dynamics, particularly the time-to-pattern, the effects of delay appear robust under variations of initial and boundary data. Overall our results help clarify the role of gene expression time delays in reaction-diffusion patterning, and suggest clear directions for further work in studying more realistic models of pattern formation.

nlin.PS↗

arXiv2022

Turing and wave instabilities in hyperbolic reaction-diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation

Hyperbolic reaction-diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction-diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction-diffusion systems which are distinct from the well studied Turing case.

nlin.PS↗

arXiv2021

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations, and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of `trivial' base states. We emphasise important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality.

nlin.PS↗

arXiv2021

Isolating Patterns in Open Reaction-Diffusion Systems

Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of `open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization, and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions, and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain, and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions, and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

nlin.PS↗

arXiv2020

Turing Patterning in Stratified Domains

Reaction-diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal-mesenchymal coupling in development, and symmetry-breaking in cell polarisation. We develop a modelling framework for bi-layer reaction-diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction-diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally-relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

nlin.PS↗