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

Ritchie, Joshua; Krause, Andrew L.; Gorder, Robert A. Van

doi:10.1016/j.aop.2022.169033

Cited by 17 publications

(5 citation statements)

References 47 publications

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“… 2021 ; Ritchie et al. 2022 ). Regarding potential biological applications, see, for example, Cavallo et al.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry

Diez,

Krause,

Maini

et al. 2024

Bull Math Biol

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Conditions for self-organisation via Turing’s mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

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“… 2021 ; Ritchie et al. 2022 ). Regarding potential biological applications, see, for example, Cavallo et al.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry

Diez,

Krause,

Maini

et al. 2024

Bull Math Biol

Self Cite

View full text Add to dashboard Cite

show abstract

“…Another mathematical direction meriting further investigation is a consideration of Hopf/Wave bifurcations in these coupled systems, whereby linear instability leads to spatiotemporally oscillating solutions. Such instabilities are not possible in the classical two-species case, but exist in three-species models or models incorporating inertial (hyperbolic) effects [32, 57]. Regarding potential biological applications, see, for example, [10].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry

Diez

Krause

Maini

et al. 2023

Preprint

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Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

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“…Many such systems are also somewhat sensitive to initial conditions and other details of the model, so being able to tweak these interactively allows for a deeper understanding of the roles of such details in the resulting dynamics. Figure 6 presents some exemplar systems, including invasion in a spatially heterogeneous Fisher-Kolmogorov equation, spiral waves in a three-species 'rock-paper-scissors' Lotka-Volterra system (Reichenbach et al 2007), spatiotemporal chaos in the Kuramoto-Sivashinsky equation (Kalogirou et al 2015), coarsening in the Cahn-Hilliard equation (Miranville 2019), Turing wave bifurcations in hyperbolic reaction-diffusion systems (Ritchie et al 2022), and irregular vegetation patterns in the Klausmeier model (Klausmeier 1999). These examples have many connections to contemporary research questions where we feel the lightning-fast interactivity of VisualPDE can be helpful in stimulating discussion and understanding broad qualitative features of different models.…”

Section: Spatiotemporal Dynamicsmentioning

confidence: 99%

VisualPDE: Rapid Interactive Simulations of Partial Differential Equations

Walker,

Townsend,

Chudasama

et al. 2023

Bull Math Biol

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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.

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

Cited by 17 publications

References 47 publications

Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry

Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry

Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry

VisualPDE: Rapid Interactive Simulations of Partial Differential Equations

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