13,954 research outputs found
Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry
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
Turing Instability and Pattern Formation in an Activator-Inhibitor System with Nonlinear Diffusion
In this work we study the effect of density dependent nonlinear diffusion on
pattern formation in the Lengyel--Epstein system. Via the linear stability
analysis we determine both the Turing and the Hopf instability boundaries and
we show how nonlinear diffusion intensifies the tendency to pattern formation;
%favors the mechanism of pattern formation with respect to the classical linear
diffusion case; in particular, unlike the case of classical linear diffusion,
the Turing instability can occur even when diffusion of the inhibitor is
significantly slower than activator's one. In the Turing pattern region we
perform the WNL multiple scales analysis to derive the equations for the
amplitude of the stationary pattern, both in the supercritical and in the
subcritical case. Moreover, we compute the complex Ginzburg-Landau equation in
the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal
modulation of the phase and amplitude of the homogeneous oscillatory solution.Comment: Accepted for publication in Acta Applicandae Mathematica
Turing pattern formation in the Brusselator system with nonlinear diffusion
In this work we investigate the effect of density dependent nonlinear
diffusion on pattern formation in the Brusselator system. Through linear
stability analysis of the basic solution we determine the Turing and the
oscillatory instability boundaries. A comparison with the classical linear
diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern
formation. We study the process of pattern formation both in 1D and 2D spatial
domains. Through a weakly nonlinear multiple scales analysis we derive the
equations for the amplitude of the stationary patterns. The analysis of the
amplitude equations shows the occurrence of a number of different phenomena,
including stable supercritical and subcritical Turing patterns with multiple
branches of stable solutions leading to hysteresis. Moreover we consider
traveling patterning waves: when the domain size is large, the pattern forms
sequentially and traveling wavefronts are the precursors to patterning. We
derive the Ginzburg-Landau equation and describe the traveling front enveloping
a pattern which invades the domain. We show the emergence of radially symmetric
target patterns, and through a matching procedure we construct the outer
amplitude equation and the inner core solution.Comment: Physical Review E, 201
Pattern formation driven by cross--diffusion in a 2D domain
In this work we investigate the process of pattern formation in a two
dimensional domain for a reaction-diffusion system with nonlinear diffusion
terms and the competitive Lotka-Volterra kinetics. The linear stability
analysis shows that cross-diffusion, through Turing bifurcation, is the key
mechanism for the formation of spatial patterns. We show that the bifurcation
can be regular, degenerate non-resonant and resonant. We use multiple scales
expansions to derive the amplitude equations appropriate for each case and show
that the system supports patterns like rolls, squares, mixed-mode patterns,
supersquares, hexagonal patterns
Cross-diffusion driven instability in a predator-prey system with cross-diffusion
In this work we investigate the process of pattern formation induced by
nonlinear diffusion in a reaction-diffusion system with Lotka-Volterra
predator-prey kinetics. We show that the cross-diffusion term is responsible of
the destabilizing mechanism that leads to the emergence of spatial patterns.
Near marginal stability we perform a weakly nonlinear analysis to predict the
amplitude and the form of the pattern, deriving the Stuart-Landau amplitude
equations. Moreover, in a large portion of the subcritical zone, numerical
simulations show the emergence of oscillating patterns, which cannot be
predicted by the weakly nonlinear analysis. Finally when the pattern invades
the domain as a travelling wavefront, we derive the Ginzburg-Landau amplitude
equation which is able to describe the shape and the speed of the wave.Comment: 15 pages, 5 figure
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