Article
Mathematics, Applied
Benjamin Aymard
Summary: This article introduces a unified framework to study reaction-diffusion systems with self- and cross-diffusion using a free energy approach. The framework leads to the formulation of an energy law and a numerical method. It provides an alternative method for studying nonlinear patterns and monitoring energy evolution in complex geometries.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Interdisciplinary Applications
Natham Aguirre, Michal Kowalczyk
Summary: This study investigates the problem of pattern formation using a one dimensional stochastic reaction-diffusion equation with time periodic coefficients. Large Deviations methods are applied to obtain lower bounds on the probability of developing certain evenly spaced patterns. The results suggest a correlation between the optimized number of interfaces and the length-scale parameter. Numerical simulations support the idea that the number of interfaces follows a certain law, even among unevenly spaced patterns.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
Edgar Knobloch, Arik Yochelis
Summary: This study investigates the linear stability of spatially localized single- and multi-peak states generated in a subcritical Turing bifurcation in the Meinhardt model of branching. The results show that these states exhibit a foliated snaking structure in one spatial dimension and are all linearly unstable. However, in two spatial dimensions, stable single- and multi-spot states are observed, whose properties are influenced by the repulsion between nearby spots, the shape of the domain, and the imposed boundary conditions. The findings suggest that multi-variable models may exhibit new types of behavior not present in typical two-variable models.
Article
Engineering, Mechanical
Linhe Zhu, Le He
Summary: This paper analyzes the diffusion behavior of the suspicious and infected cabins in cyberspace using a rumor propagation reaction-diffusion model. The effects of time delay and changing diffusion coefficient are considered to study the stability and instability of the system. The existence of Hopf bifurcation induced by time delay is proven, and the necessary conditions for Turing instability are studied. Numerical simulations show that variations in diffusion coefficient and time delay can change the pattern type and affect the arrangement of crowd gathering areas.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Applied
Bartosz J. Bartmanski, Ruth E. Baker
Summary: The study explores the impact of various discretisations and methods for derivation of the diffusive jump rates on the outputs of stochastic simulations of reaction-diffusion models. It shows that while minor differences are observed for simple systems, significant variations can occur in model predictions for complex systems like Turing's diffusion-driven instability model of pattern formation. Care must be taken when using the reaction-diffusion master equation framework to make predictions for stochastic reaction-diffusion systems.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Review
Engineering, Mechanical
Gui-Quan Sun, Hong-Tao Zhang, Jin-Shan Wang, Jing Li, Yi Wang, Li Li, Yong-Ping Wu, Guo-Lin Feng, Zhen Jin
Summary: This research delves into the key issue of population distribution in ecological systems and how it characterizes the relationship between populations, space-time structure, and evolution laws. It systematically summarizes the related results in pattern formation of ecological systems, showcasing the mechanisms of different patterns. This work offers valuable insights into understanding the complexity of ecosystems and can be applied in various related fields such as epidemiology, medical science, and atmospheric science.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Nicolas Verschueren, Alan R. Champneys
Summary: The investigation focuses on coupled reaction-diffusion systems in one spatial dimension that can support either isolated spike solutions or stable localized patterns in different regions of their parameter space. The distinction between the two cases is characterized by the behavior of the far field, where there is either oscillatory or monotonic decay. Two examples illustrate this transition, showing how localized patterns connected via a homoclinic snaking curve transition into a single spike solution as a second parameter is varied. This transition is caused by a Belyakov-Devaney transition between complex and real spatial eigenvalues of the far field of the primary pulse.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Materials Science, Multidisciplinary
C. McNamara, J. M. Rickman, H. M. Chan
Summary: The study investigates the microstructural evolution that initiates from a compositionally inhomogeneous patterned structure in solid-state reactions and uses reaction-diffusion formalism to model the process. By connecting the spatio-temporal evolution of the product phase with the geometry and chemistry of the starting duplex structure, a better understanding of reaction kinetics and product microstructures is obtained. The results can be used to select templates judiciously to promote the formation of desirable microstructures and quantify important experimental kinetic parameters.
Article
Mathematics, Applied
Chengxia Lei, Guanghui Zhang, Jialin Zhou
Summary: In this paper, a biomass-water reaction-diffusion model with homogeneous Neumann boundary condition is studied to determine the conditions for the existence or non-existence of non-constant stationary solutions, providing criteria for the possibility of Turing patterns in the system. The results confirm previous numerical findings and complement theoretical results for the corresponding ODE model.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Alexander Kolinichenko, Lev Ryashko
Summary: This paper investigates a spatially extended stochastic reaction-diffusion model, demonstrating a theoretical approach and comparing statistically obtained data on stochastic sensitivity functions for stable nonhomogeneous stationary patterns. It discusses variations in pattern sensitivity to noise and the phenomenon of stochastic preference in different patterns in the Brusselator.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Alan R. Champneys, Fahad Al Saadi, Victor F. Brena-Medina, Veronica A. Grieneisen, Athanasius F. M. Maree, Nicolas Verschueren, Bert Wuyts
Summary: This paper presents a synthesis of recent research on the formation of localized patterns, isolated spots, or sharp fronts in models of natural processes governed by reaction-diffusion equations. It contrasts with the well-known Turing mechanism of periodic pattern formation and provides a general picture in one spatial dimension for models on long domains exhibiting sub-critical Turing instabilities.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Mathematics, Interdisciplinary Applications
Priya Chakraborty, Mohit Kumar Jolly, Ushasi Roy, Sayantari Ghosh
Summary: Biological systems rely on bistability to exhibit non-genetic heterogeneity in cellular morphology and physiology. The spatial distribution of phenotypically heterogeneous cells, resulting from bistability, plays a significant role in phenomena such as biofilm development, adaptation, and cell motility. This paper investigates the pattern formation of a motif with non-cooperative positive feedback, which imposes a metabolic burden on its host. In-silico spatio-temporal diffusion is studied in cellular arrays in one and two dimensions with various initial conditions, and the stability of related states and the evolution of patterns are analyzed based on the variation of diffusion coefficients.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Materials Science, Multidisciplinary
Aditya Kumar, Leon M. Dean, Mostafa Yourdkhani, Allen Guo, Cole BenVau, Nancy R. Sottos, Philippe H. Geubelle
Summary: Frontal polymerization (FP) is a rapid and energy-efficient manufacturing technique for polymeric materials and composites. This paper introduces a coupled thermo-chemo-mechanical theory to describe the deformation and temperature fields during frontal polymerization of gels. Additionally, a novel bio-inspired oscillatory loading-induced patterning technique is proposed. Experimental results confirm the theoretical predictions.
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS
(2022)
Article
Multidisciplinary Sciences
Fahad Al Saadi, Alan Champneys
Summary: This study extends the research on canonical activator-inhibitor Schnakenberg-like models to include models with bistability of homogeneous equilibria, such as Gray-Scott. A homotopy from a Schnakenberg-like glycolysis model to the Gray-Scott model is studied, with numerical continuation used to understand the sequence of transitions in parameter regimes. Several distinct bifurcations are discovered, including cusp and quadruple zero points, under homotopy between bifurcation diagrams for different field feed scenarios.
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2021)
Article
Multidisciplinary Sciences
Robert A. Van Gorder
Summary: The article discusses pattern formation and suppression on temporal networks, deriving conditions for diffusive spatial and spatio-temporal pattern formation through the Turing and Benjamin-Feir mechanisms, illustrating the theory through numerical simulations. The study highlights the transient nature of Turing and Benjamin-Feir instabilities on temporal networks, allowing for transitions between different patterns or spatio-temporal states over time intervals.
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2021)
Article
Mathematics, Applied
Alan R. Champneys, Fahad Al Saadi, Victor F. Brena-Medina, Veronica A. Grieneisen, Athanasius F. M. Maree, Nicolas Verschueren, Bert Wuyts
Summary: This paper presents a synthesis of recent research on the formation of localized patterns, isolated spots, or sharp fronts in models of natural processes governed by reaction-diffusion equations. It contrasts with the well-known Turing mechanism of periodic pattern formation and provides a general picture in one spatial dimension for models on long domains exhibiting sub-critical Turing instabilities.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Mathematics, Applied
Fahad Al Saadi, Alan Champneys, Annette Worthy, Ahmed Msmali
Summary: Investigations were conducted on simple predator-prey models with rational interaction terms in one and two spatial dimensions, revealing a subcritical Turing bifurcation leading to localized patterns and isolated spots within parameter regions, as well as the occurrence of a temporal Hopf bifurcation within the localized-pattern region. Detailed spectral computations and numerical simulations showed the inheritance of the Hopf bifurcation by localized structures, resulting in temporally periodic and chaotic patterns. Simulation results in 2D confirmed the onset of complex spatio-temporal patterns within corresponding parameter regions.
IMA JOURNAL OF APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Fahad Al Saadi, Alan Champneys, Nicolas Verschueren
Summary: This study investigates activator-inhibitor reaction-diffusion equations on an infinite line, exploring the formation of Turing instability and isolated spike-like patterns. It examines the bifurcation structures connecting these two dynamical regimes and reveals a universal two-parameter state diagram. The temporal dynamics strongly depend on the diffusion ratio, with complex spatio-temporal dynamics arising from interactions between bifurcations and asymptotic analyses.
IMA JOURNAL OF APPLIED MATHEMATICS
(2021)
Article
Acoustics
R. E. White, N. A. Alexander, J. H. G. Macdonald, A. R. Champneys
Summary: The study investigates the interactions between human and civil structures during rhythmic jumping, revealing the relationship between structural vibrations and human jumping behaviors. It explores stable periodic motions and chaotic behaviors of human jumping, providing insights into the difficulty of jumping around the natural frequency of a supporting structure.
JOURNAL OF SOUND AND VIBRATION
(2021)
Article
Mathematics, Applied
Fahad Al Saadi, Annette Worthy, Ahmed Msmali, Mark Nelson
Summary: This paper investigates the existence of localized structures in two predator-prey models. The combination of linear and weakly nonlinear analysis with numerical methods identifies conditions for a wide range of complex behaviors, including the Belyckov-Devaney transition, spatial instability point, and formation of localized patterns. The role of Hopf bifurcation in mediating the stability of localized spatial solutions is revealed. Numerical solutions in two spatial dimensions confirm the onset of intricate spatio-temporal patterns within the identified parameter regions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Computer Science, Interdisciplinary Applications
Fahad Al Saadi, Annette Worthy, Haifaa Alrihieli, Mark Nelson
Summary: The paper systematically investigates the formation of localized structures in the Thomas model using modern ideas from the theory of dynamical systems. It reveals the significant changes in Turing instability at the degenerate points and the temporal stability of localized solutions depending on the presence of Hopf bifurcations and diffusion ratio.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Biophysics
Robert Hughes, Aaron Fishman, Kathryn Lamb-Riddell, Valentina Sleigh Mun, Alan Champneys, Janice Kiely, Richard Luxton
Summary: The study describes the development of an end-to-end model of a magneto-immunoassay, simulating the agglutination effect caused by the specific binding of bacteria to paramagnetic particles. Through direct imaging and microfluidic assays, the dose-specific agglutination properties and magnetophoretic transport dynamics of agglutinated clusters were observed. Mathematical modeling was used to predict the physical processes underlying the assay and show agreement with experimental results.
BIOSENSORS & BIOELECTRONICS
(2023)
Article
Biology
Edgardo Villar-Sepulveda, Alan R. R. Champneys
Summary: This study provides necessary and sufficient conditions for a diffusion-driven instability of a stable equilibrium in a reaction-diffusion system with n components and diagonal diffusion matrix. These instabilities can be either Turing or wave instabilities. The known conditions for a diffusion rate that causes a Turing bifurcation of a stable homogeneous state in the absence of diffusion are reproducible. The method of proof used here gives a constructive approach for selecting diffusion constants, based on studying dispersion relations in different limits.
JOURNAL OF MATHEMATICAL BIOLOGY
(2023)
Article
Mathematics, Applied
Fahad Al Saadi, Pedro Parra-Rivas
Summary: Spatially extended patterns and multistability are common in ecosystems. Tristability between patterned and uniform states can have important impacts on dynamical behaviors. Using a simplified model, we study the organization of localized structures in tristable regimes. We find that the coexistence of localized spots and gaps leads to the emergence of hybrid states.
Article
Multidisciplinary Sciences
Jake Ahern, Lukasz Chrobok, Alan R. Champneys, Hugh D. Piggins
Summary: Analysis of ex vivo Per2 bioluminescent rhythm recorded in the mouse dorsal vagal complex reveals a characteristic phase relationship between the area postrema (AP), the nucleus of the solitary tract (NTS), and the ependymal cells surrounding the 4th ventricle (4Vep). A phase model of the three oscillators suggests that realistic phase dynamics occur with coupling close to a synchronisation transition. The coupling topology suggests bidirectional communication of phase information between the AP, NTS, and 4Vep to synchronise the three structures. Experimental manipulations and simulations validate the model's ability to explain DVC circadian phasing.
SCIENTIFIC REPORTS
(2023)
Article
Mathematics, Applied
Edgardo Villar-Sepulveda, Alan Champneys
Summary: This paper provides explicit calculations to justify the existence and criticality of Turing bifurcations in a general class of activator-inhibitor reaction-diffusion equations on a one-dimensional infinite domain. The calculations consider two distinct scalings of parameters and reveal that the Turing bifurcation can be either subcritical or supercritical, leading to the existence of a codimension-two degenerate bifurcation. The sign of a fifth-order normal form coefficient is computed and shown to be correct for the birth of homoclinic snaking.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Medicine, General & Internal
Salvador Catsis, Alan R. Champneys, Rebecca Hoyle, Christine Currie, Jessica Enright, Katherine Cheema, Mike Woodall, Gianni Angelini, Ramesh Nadarajah, Chris Gale, Ben Gibbison
Summary: This study models the referral, diagnostic and treatment pathway for cardiovascular disease (CVD) in the English National Health Service (NHS) using a systems dynamics approach. The study uses routinely collected, publicly available data streams of primary and secondary care. The results show an increase in NHS CVD waiting lists in England, primarily due to restrictions in referrals from primary care.
Article
Mathematics, Applied
Paul Carter, Alan R. Champneys
Summary: This paper investigates a class of two-fast, one-slow multiple timescale dynamical systems and analyzes the mechanism of rapid amplitude growth in small-amplitude periodic orbits. The presence of a saddle-focus structure around the slow manifold leads to a sequence of folds as the amplitude increases. The analysis provides insights into the behavior of this type of dynamical system and is supported by numerical results.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2022)