Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems

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.

Localized pattern formation: semi-strong interaction asymptotic analysis for three components model

“…The procedure that we apply to find the spike equilibrium solution using asymptotic approximation is similar to that used in [21,22]. The first step is considering the steady-state system of (1.1)…”

Stationary localised patterns without Turing instability

“…To investigate pattern solutions, we must extend this analysis in $$$ A $$$ to the third order. After the required calculations (see Al Saadi et al 27 for more details), the following equation is obtained: $$$$ {\partial}_tA=\rho {C}_1A+{C}_3A{\left|A\right|}^2+O\left(A{\left|A\right|}^4\right). $$$$ …”

“…Our focus is on obtaining analytical expression for a particular codimension two points at which the spatial instability curve changes from being a subcritical to a supercritical bifurcation. This point is identified by undertaking a normal form analysis, as detailed in Haragus and Iooss 25 and applied in Al Saadi et al 27 and Verschueren and Champneys. 28 The first step is to determine the amplitude equation arising from any spatial instability.…”

“…To investigate pattern solutions, we must extend this analysis in A to the third order. After the required calculations (see Al Saadi et al 27 for more details), the following equation is obtained:…”

Analysing transitions from a Turing instability to large periodic patterns in a reaction-diffusion system

“…Singular perturbation theory [19,27,28] is one of the few techniques that has yielded new insights into the complex phenomenology of patterns far from onset; see, for instance, [14,16,23] and references therein. Recently, there has been an increasing interest in linking the small amplitude homoclinic patterns, that emerge near a Turing instability, to localized patterns found near the singular limit away from onset through a mixture of numerical investigations, return-map analysis and singular perturbation theory [1,2,3,10,51]. The aim of this paper is to investigate analytically spatially periodic patterns emerging from a Turing instability using geometric singular perturbation theory.…”