where , , and . Including diffusion, we get from this model the following dynamical equations:
where
and
In writing Eqs. (94)-(97) we are assuming that the species V and A are fed into the system and this happens at the same rate at which all of the species are removed ( ).
In this Section we apply the methods of Sec. V of the paper to Eqs. (86)-(89) to obtain the equations describing the slow time dynamics for this model. In order to make this calculation more specific, we consider the parameter values that are used in the experiments when replicating spot pattern is observed: , , , , , . It is not completely clear what the values of and actually are inside the gel where the reaction takes place. We will consider and . Inspired by the reduction of the ODE's performed by Gáspár and Showalter, we seek a reduction of the PDE's (86)-(89) in which the variables a and z are eliminated in favor of u and v. To this end we follow the steps of Sec. V of the paper considering , , and . We find:
Since the Z species is iodine, which binds to the gel, its diffusion coefficient may be neglected. For the sake of simplicity, we will also neglect the diffusion term of A. Inserting the expansions and in Eqs. (86)-(87) we find:
where we have defined , , , , , and . Now, the whole calculation is consistent provided that and . As we may see from Eqs. (99) and (101), both and contain terms that are proportional to and . These time derivatives may get large given that Eqs. (102)-(103) contain terms which are proportional to . However, the existence of more than two timescales is of help in this case. Assuming that the right-hand-side of Eqs. (99), (101), (102) and (103) are dominated by the terms proportional to , we may rewrite the conditions and as:
For the parameter values we are considering it is and . Thus, provided that v/u does not become too large, the conditions (105) and (106) are satisfied and the calculation is self-consistent.