The extended Gray-Scott model

The original Gray-Scott model for self-replicating spots and the corresponding experiment performed by Lee et al (1993) is a two-dimensional system driven by an inflow perpendicular to the surface. In a flow reactor, the free energy that drives the system enters with the flow at one end. This may be problematic since chemical reactions may consume the free energy too quickly and there will not be sufficient free energy available along the path of the flow to initiate and sustain pattern dynamics. One mechanism that can be used to retain the original dynamics is to introduce a delay and a temporal storage for the consumption of free energy. We propose to use an additional reaction between a pre-cursor to the fuel U* and the fuel U. The resulting extended Gray-Scott (GS) model is then

 

                      U*  <—>  U  

               U + 2V   <—>  3V

                        V   —>   F  

 

Here V is a self-catalytic component and F is assumed to be inert so there are three concentrations variables for U*, U, and V (compared to two for the original model). In order to store free energy in U* the first reaction has an equilibrium shifted to the left, with rate constants k1+ = 0.0001 and k1– = 0.02 in forward and backward directions, respectively. The autocatalytic reaction has rate constants k2+ = 1 and k2­– = 10-9 (practically unidirectional reaction), while the decay into the inert substans F has rate constant k3+ = 0.07. As a basic parameter setting we use: flow rate f = 5, and diffusion constants DV=0.05, DU=0.1, DU*=0, but due to the first reaction going in both directions there is a ”diffusion like” effect also on U*. The inflow has a concentration composition u*in = 200, uin = 0.5707, and vin=0.123.

 

This reaction scheme is embedded in a flow reactor system similar to the above mentioned fan reactor, see Fig. 1. An advantage with this type of reaction-diffusion-advection dynamics is that characteristics of the system may gradually change along the path of the flow since the local free energy is slowly decreasing. In the fan reactor there is also the effect of decreasing flow velocity as the width of the reactor increases along the flow direction. This allows for exploration of a large part of parameter space within a single experiment.

 

A similar extended G-S model in a reaction-diffusion-advection system was considered by von Haeften and Izús (2003), but they did not allow backward reactions and they allowed for a continuous inflow from the precursor along the reactor. Here, we emphasize that the driving chemical inflow is located at one end and the outlet at the other — no chemical exchange takes place along the reactor.