Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems

Woolley, Thomas E.

doi:10.1007/s11538-022-01055-x

Cited by 5 publications

(3 citation statements)

References 98 publications

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“…From these critical points, arise two inhomogeneous solution branches, which are symmetrical to the axis of the homogeneous solution. This is a known symmetry-breaking phenomenon, due to the Turing bifurcation, which with the zero flux boundary conditions has the characteristic of a pitchfork bifurcation [37,19,38].…”

Section: Discussionmentioning

confidence: 99%

Numerical Bifurcation Analysis of Turing and Symmetry Broken Patterns of a Vegetation PDE Model

Spiliotis¹,

Russo²,

Giannino³

et al. 2023

Preprint

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We study the mechanisms of pattern formation for vegetation dynamics in water-limited regions. Our analysis is based on a set of two partial differential equations (PDEs) of reaction-diffusion type for the biomass and water and one ordinary differential equation (ODE) describing the dependence of the toxicity on the biomass. We perform a linear stability analysis in the one-dimensional finite space, we derive analytically the conditions for the appearance of Turing instability that gives rise to spatio-temporal patterns emanating from the homogeneous solution, and provide its dependence with respect to the size of the domain. Furthermore, we perform a numerical bifurcation analysis in order to study the pattern formation of the inhomogeneous solution, with respect to the precipitation rate, thus analyzing the stability and symmetry properties of the emanating patterns. Based on the numerical bifurcation analysis, we have found new patterns, which form due to the onset of secondary bifurcations from the primary Turing instability, thus giving rise to a multistability of asymmetric solutions.

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Section: Discussionmentioning

confidence: 99%

Numerical Bifurcation Analysis of Turing and Symmetry Broken Patterns of a Vegetation PDE Model

Spiliotis¹,

Russo²,

Giannino³

et al. 2023

Preprint

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“…In reaction-diffusion equations, gene-regulatory networks are represented using the reaction terms, and pre-pattern morphogens can be taken into account through the associated reaction parameters [18]. Such equations display self-organization in space and time via bifurcations [30,31], which often cause switch-like transitions in the solution that may correspond to biological decisions [13] or cell-fate choices [32]. Mathematically, dynamics are typically observed to slow down near bifurcations [33].…”

Section: Introductionmentioning

confidence: 99%

Universal dynamics of biological pattern formation in spatio-temporal morphogen variations

Dalwadi

Pearce

2023

Proc. R. Soc. A.

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In biological systems, chemical signals termed morphogens self-organize into patterns that are vital for many physiological processes. As observed by Turing in 1952, these patterns are in a state of continual development, and are usually transitioning from one pattern into another. How do cells robustly decode these spatio-temporal patterns into signals in the presence of confounding effects caused by unpredictable or heterogeneous environments? Here, we answer this question by developing a general theory of pattern formation in spatio-temporal variations of ‘pre-pattern’ morphogens, which determine gene-regulatory network parameters. Through mathematical analysis, we identify universal dynamical regimes that apply to wide classes of biological systems. We apply our theory to two paradigmatic pattern-forming systems, and predict that they are robust with respect to non-physiological morphogen variations. More broadly, our theoretical framework provides a general approach to classify the emergent dynamics of pattern-forming systems based on how the bifurcations in their governing equations are traversed.

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“…In reaction-diffusion equations, gene-regulatory networks are represented using the reaction terms, and pre-pattern morphogens can be taken into account through the associated reaction parameters [15]. Such equations display self-organisation in space and time via bifurcations [27,28], which often cause switch-like transitions in the solution that may correspond to biological decisions [12] or cell-fate choices [29]. Mathematically, dynamics are typically observed to slow down near bifurcations [30,31], and these 'delayed bifurcation' effects have previously been considered in reaction-diffusion contexts [32][33][34], but the implications to biological pattern formation and decision-making are not understood.…”

Section: Introductionmentioning

confidence: 99%

Universal dynamics of biological pattern formation in spatio-temporal morphogen variations

Dalwadi

Pearce

2022

Preprint

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Generic features of biological pattern-forming systems have been uncovered that promote robustness with respect to variations between different cells or populations in system components, such as protein production rates and gene-regulatory network architecture. By contrast, less is understood about how such systems respond to spatio-temporal changes in morphogen concentrations within individual cells or populations over timescales faster than or similar to growth. Here, we formulate and apply a general theory of biological pattern formation in spatio-temporally varying concentrations of "pre-patterning" morphogens, which determine system parameters. Our analytical results reveal universal dynamical regimes of biological pattern-forming systems, and quantify how the dynamics in each regime depend on the mathematical properties of the relevant gene-regulatory network. By applying our theory, we predict that two paradigmatic pattern-forming systems are robust with respect to spatio-temporal morphogen variations that happen faster than growth timescales. More broadly, we predict that the dynamics of pattern-forming systems with spatio-temporally varying parameters can be classified based on the bifurcations in their governing equations.

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Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems

Cited by 5 publications

References 98 publications

Numerical Bifurcation Analysis of Turing and Symmetry Broken Patterns of a Vegetation PDE Model

Numerical Bifurcation Analysis of Turing and Symmetry Broken Patterns of a Vegetation PDE Model

Universal dynamics of biological pattern formation in spatio-temporal morphogen variations

Universal dynamics of biological pattern formation in spatio-temporal morphogen variations

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