Weakly nonlinear analysis of minimal models for Turing patterns

The simplest particle-based mass-action models for Turing instability – i.e. those with only two component species undergoing instantaneous interactions of at most two particles, with the smallest number of distinct interactions – fall into a surprisingly small number of classes of reaction schemes. In previous work we have computed this classification, with different schemes distinguished by the structure of the interactions. Within a given class the reaction stoichiometry and rates remain as parameters that determine the linear and nonlinear evolution of the system.

Adopting the usual weakly nonlinear scalings and analysis reveals that, under suitable choices of reaction stoichiometry, and in nine of the 11 classes of minimal scheme exhibiting a spatially in-phase (“true activator-inhibitor”) Turing instability, stable patterns are indeed generated in open regions of parameter space via a generically supercritical bifurcation from the spatially uniform state. In three of these classes the instability is always supercritical while in six there is an open region in which it is subcritical. Intriguingly, however, in the remaining two classes of minimal scheme we require different weakly nonlinear scalings, since the coefficient in the usual cubic normal form unexpectedly vanishes identically. In these cases, a different set of asymptotic scalings is required.

We present a complete analysis through deriving the normal form for these two cases also, which involves quintic terms. This fifth-order normal form also captures the behaviour along the boundaries between the supercritical and subcritical cases of the cubic normal form. The details of these calculations reveal the distinct roles played by reaction rate parameters as compared to stoichiometric parameters.

We quantitatively validate our analysis via numerical simulations and confirm the two different scalings for the amplitude of predicted stable patterned states.