Turing pattern outside of the Turing domain

Coloured Noise from Stochastic Inflows in Reaction-Diffusion Systems

In this paper, we present a framework for investigating coloured noise in reaction-diffusion systems. We start by considering a deterministic reaction-diffusion equation and show how external forcing can cause temporally correlated or coloured noise. Here, the main source of external noise is considered to be fluctuations in the parameter values representing the inflow of particles to the system. First, we determine which reaction systems, driven by extrinsic noise, can admit only one steady state, so that effects, such as stochastic switching, are precluded from our analysis. To analyse the steady-state behaviour of reaction systems, even if the parameter values are changing, necessitates a parameter-free approach, which has been central to algebraic analysis in chemical reaction network theory. To identify suitable models, we use tools from real algebraic geometry that link the network structure to its dynamical properties. We then make a connection to internal noise models and show how power spectral methods can be used to predict stochastically driven patterns in systems with coloured noise. In simple cases, we show that the power spectrum of the coloured noise process and the power spectrum of the reaction-diffusion system modelled with white noise multiply to give the power spectrum of the coloured noise reaction-diffusion system.

More than Skew: Asymmetric Wave Propagation in a Reaction-Diffusion-Convection System

Convection-induced instability in reaction-diffusion systems produces complicated patterns of oscillations behind propagating wavefronts. We transform the system twice: into lambda-omega form, then into polar variables. We find analytical estimates for the wavefront speed which we confirm numerically. Our previous work examined a simpler system [E. H. Flach, S. Schnell, and J. Norbury, Phys. Rev. E 76, 036216 (2007)]; the onset of instability is qualitatively different in numerical solutions of this system. We modify our estimates and connect the two different behaviours. Our estimate explains how the Turing instability fits with pattern found in reaction-diffusion-convection systems. Our results can have important applications to the pattern formation analysis of biological systems.

Dynamical patterning modules: physico-genetic determinants of morphological development and evolution

The shapes and forms of multicellular organisms arise by the generation of new cell states and types and changes in the numbers and rearrangements of the various kinds of cells. While morphogenesis and pattern formation in all animal species are widely recognized to be mediated by the gene products of an evolutionarily conserved 'developmental-genetic toolkit', the link between these molecular players and the physics underlying these processes has been generally ignored. This paper introduces the concept of 'dynamical patterning modules' (DPMs), units consisting of one or more products of the 'toolkit' genes that mobilize physical processes characteristic of chemically and mechanically excitable meso- to macroscopic systems such as cell aggregates: cohesion, viscoelasticity, diffusion, spatiotemporal heterogeneity based on lateral inhibition and multistable and oscillatory dynamics. We suggest that ancient toolkit gene products, most predating the emergence of multicellularity, assumed novel morphogenetic functions due to change in the scale and context inherent to multicellularity. We show that DPMs, acting individually and in concert with each other, constitute a 'pattern language' capable of generating all metazoan body plans and organ forms. The physical dimension of developmental causation implies that multicellular forms during the explosive radiation of animal body plans in the middle Cambrian, approximately 530 million years ago, could have explored an extensive morphospace without concomitant genotypic change or selection for adaptation. The morphologically plastic body plans and organ forms generated by DPMs, and their ontogenetic trajectories, would subsequently have been stabilized and consolidated by natural selection and genetic drift. This perspective also solves the apparent 'molecular homology-analogy paradox', whereby widely divergent modern animal types utilize the same molecular toolkit during development by proposing, in contrast to the Neo-Darwinian principle, that phenotypic disparity early in evolution occurred in advance of, rather than closely tracked, genotypic change.

Limit cycles in the presence of convection: a traveling wave analysis

We consider a diffusion model with limit cycle reaction functions. In an unbounded domain, diffusion spreads the pattern outwards from the source. Convection adds instability to the reaction-diffusion system. The result of this instability is a readiness to create a pattern. We choose the Lambda-Omega reaction functions for their simple limit cycle. We carry out a transformation of the dependent variables into polar form. From this we consider the initiation of the pattern to approximate a traveling wave. We carry out numerical experiments to test our analysis. These confirm the premise of the analysis, that the initiation can be modeled by a traveling wave. Furthermore, the analysis produces a good estimate of the numerical results. Most significantly, we confirm that the pattern consists of two different types.