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We study the existence and stability of propagating fronts in Meinhardt’s multivariable reaction-diffusion model of branching in one spatial dimension. We identify a saddle-node-infinite-period bifurcation of fronts that leads to episodic front propagation in the parameter region below propagation failure and show that this state is stable. Stable constant speed fronts exist only above this parameter value. We use numerical continuation to show that propagation failure is a consequence of the presence of a T-point corresponding to the formation of a heteroclinic cycle in a spatial dynamics description. Additional T-points are identified that are responsible for a large multiplicity of different unstable traveling front-peak states. The results indicate that multivariable models may support new types of behavior that are absent from typical two-variable models but may nevertheless be important in developmental processes such as branching and somitogenesis. 

We study the linear stability properties of spatially localized single- and multi-peak states generated in a subcritical Turing bifurcation in the Meinhardt model of branching. In one spatial dimension, these states are organized in a foliated snaking structure owing to peak–peak repulsion but are shown to be all linearly unstable, with the number of unstable modes increasing with the number of peaks present. Despite this, in two spatial dimensions, direct numerical simulations reveal the presence of stable single- and multi-spot states whose properties depend on the repulsion from nearby spots as well as the shape of the domain and the boundary conditions imposed thereon. Front propagation is shown to trigger the growth of new spots while destabilizing others. The results indicate that multi-variable models may support new types of behavior that are absent from typical two-variable models. 

Abstract A simple equation modelling an inextensible elastic lining of an inner-lined tube subject to an imposed pressure difference is derived from a consideration of the idealised elastic properties of the lining and the pressure and soft-substrate forces. Two cases are considered in detail, one with prominent wrinkling and a second one in which wrinkling is absent and only buckling remains. Bifurcation diagrams are computed via numerical continuation for both cases. Wrinkling, buckling, folding, and mixed-mode solutions are found and organised according to system-response measures including tension, in-plane compression, maximum curvature and energy. Approximate wrinkle solutions are constructed using weakly nonlinear theory, in excellent agreement with numerics. Our approach explains how the wavelength of the wrinkles is selected as a function of the parameters in compressed wrinkling systems and shows how localised folds and mixed-mode states form in secondary bifurcations from wrinkled states. Our model aims to capture the wrinkling response of arterial endothelium to blood pressure changes but applies much more broadly. 

Abstract Systems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical and interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.