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This study investigates transient wave dynamics in Turing pattern formation, focusing on waves emerging from localised disturbances. While the traditional focus of diffusion-driven instability has primarily centred on stationary solutions, considerable attention has also been directed towards understanding spatio-temporal behaviours, particularly the propagation of patterning from localised disturbances. We analyse these waves of patterning using both the well-established marginal stability criterion and weakly nonlinear analysis with envelope equations. Both methods provide estimates for the wave speed but the latter method, in addition, approximates the wave profile and amplitude. We then compare these two approaches analytically near a bifurcation point and reveal that the marginal stability criterion yields exactly the same estimate for the wave speed as the weakly nonlinear analysis. Furthermore, we evaluate these estimates against numerical results for Schnakenberg and CDIMA (chlorine dioxide–iodine–malonic acid) kinetics. In particular, our study emphasises the importance of the characteristic speed of pattern propagation, determined by diffusion dynamics and a complex relation with the reaction kinetics in Turing systems. This speed serves as a vital parameter for comparison with experimental observations, akin to observed pattern length scales. Furthermore, more generally, our findings provide systematic methodologies for analysing transient wave properties in Turing systems, generating insight into the dynamic evolution of pattern formation. • Near a bifurcation point, marginal stability criterion and envelope equations give exactly the same results. • There is a unique travelling wave depositing a pattern from a localised perturbation. • The travelling wave speed may also be used to discriminate models in applications. [ABSTRACT FROM AUTHOR]

Abstract: We study the change in the speed of pushed and bistable fronts of the reaction–diffusion equation in the presence of a small cut-off. We give explicit formulas for the shift in the speed for arbitrary reaction terms . The dependence of the speed shift on the cut-off parameter is a function of the front speed and profile in the absence of the cut-off. In order to determine the speed shift we solve the leading order approximation to the front profile in the neighborhood of the leading edge and use a variational principle for the speed. We apply the general formula to the Nagumo equation and recover the results which have been obtained recently by geometric analysis. The formulas given are of general validity and we also apply them to a class of reaction terms which have not been considered elsewhere. [Copyright &y& Elsevier]

Abstract: We investigate the effects of a linear cut-off on front propagation in the Nagumo equation at a so-called Maxwell point, where the corresponding front solution in the absence of a cut-off is stationary. We show that the correction to the propagation speed induced by the cut-off is positive in this case; moreover, we determine the leading-order asymptotics of that correction in terms of the cut-off parameter, and we calculate explicitly the corresponding coefficient. Our analysis is based on geometric techniques from dynamical systems theory and, in particular, on the method of geometric desingularization (‘blow-up’). [Copyright &y& Elsevier]

Abstract: Global and localized granular surface flow is analyzed in a model that takes into account the basic forces between a flowing granular layer and an underlying granular bed. Starting from a quantitatively correct description of global stick–slip avalanches on the surfaces of heaps we are able to describe localized flows of various types as seen in experiments of Daerr and Douady [A. Daerr, S. Douady, Two types of avalanche behavior in granular media, Nature (London) 399 (1999) 241]. The descriptive level takes into account the free surface and the velocity of the flow solely. We limit our model to cases where this surface is effectively one-dimensional so that our model equations are consequently one-dimensional in space. With this kind of description we are able to account for the tight coupling between local grain velocity and local height of the so-called lonely waves. Our model shows the nature of a wave of mobilization and deposition. Granular material is set into motion in a domino-like manner. In addition destabilization of material behind an avalanche is taken into account. The front velocities are found to be constant and in general head and rear front travel with the same velocity in our model. The front velocities depend on the tilt angle of the heap that those localized structures run down on. A critical angle is observed above which the rear front velocity abruptly switches sign from positive (downslope) to negative (upslope) values. Furthermore, we find global influences of the surface shape of a flowing granular layer on Bagnold friction. This characteristic collisional friction between flowing layers and beds is shown to cause S-shaping of the free surface for rapid flow velocities. Finally, our model also explains the power spectrum of avalanches as experimentally detected by Jaeger et al. [H. Jaeger, C. Liu, S. Nagel, Relaxation at the angle of repose, Phys. Rev. Lett. 62 (1989) 40]. [ABSTRACT FROM AUTHOR]

Abstract: We study the propagation of pH fronts in a narrow, pseudo-one-dimensional strip of gel containing bromate, sulfite, ferrocyanide and aluminum ions. The front propagation speed decreases with [ Al3+] as , where is a constant. We determine the diffusion coefficients of protons and ferricyanide ions and find that Al(OH)3 loaded in agarose or acrylamide gels binds protons reversibly, thereby slowing the propagating fronts without significantly affecting the diffusion of ferricyanide ions. We present a model that qualitatively reproduces the experimental behavior, and we suggest some general principles governing autocatalytic fronts in buffered systems. [Copyright &y& Elsevier]

Abstract: We study reactive front speeds in randomly perturbed cellular flows using a variational representation for the front speed. We develop this representation into a computational tool for computing the front speeds without resorting to closure approximations. We demonstrate that the front speeds depend on flow statistics and topologies in a complex and dramatic manner. [Copyright &y& Elsevier]

A simplified model of clonal plant growth is formulated, motivated by observations of spatial structures in Posidonia oceanica meadows in the Mediterranean Sea. Two levels of approximation are considered for the scale-dependent feedback terms. Both take into account mortality and clonal, or vegetative, growth as well as competition and facilitation, but the first version is nonlocal in space while the second is local. Study of the two versions of the model in the one-dimensional case reveals that both cases exhibit qualitatively similar behavior (but quantitative differences) and describe the competition between three spatially extended states, the bare soil state, the populated state, and a pattern state, and the associated spatially localized structures. The latter are of two types, holes in the populated state and vegetation patches on bare ground, and are organized within distinct snaking bifurcations diagrams. Fronts between the three extended states are studied and a transition between pushed and pulled fronts identified. Numerical simulations in one spatial dimension are used to determine front speeds and confront the predictions from the marginal stability condition for pulled fronts. • A one-dimensional nonlocal model of clonal vegetation growth is studied. • A local approximation for competitive interactions is also considered. • Homogeneous states, extended patterns and localized structures are identified. • Fronts between the different states are studied. [ABSTRACT FROM AUTHOR]

The paper considers the transition between spatially localized and propagating stick–slip motion in frictional systems. To overcome the difficulties related to the non-smoothness of the friction law we have studied two models: a friction-excited chain of weakly coupled oscillators excited by a frictional moving belt (model I) and a chain of Van der Pol oscillators (model II). The two models show very similar dynamical features. Both exhibit discrete breathers (i.e. spatially localized periodic solutions) for low coupling. For higher coupling, both exhibit propagation of switch fronts from low to high-amplitude limit cycle oscillations. Besides, in both models the transition from discrete breathers to propagating limit cycles happens through a nucleation process and, above a certain critical coupling, the velocity of propagation scales linearly with the elastic coupling coefficient. We suggest that dynamical features similar to what was demonstrated in this work are expected for models that are different quantitatively but preserve the same structure of the state space in every single unit. • Self-excited frictional systems possess various similar dynamical features. • Spatially localized solutions lose stability to propagating fronts. • Front propagation is initiated through a process of nucleation. • Velocity of propagation depends linearly on the coupling between the oscillators. [ABSTRACT FROM AUTHOR]