Setting up the model wave equations

If we concentrate on a thin slice of the concentration profiles, then the rate of change of concentration in time in that slice is given by the net rate of diffusion in and the chemical rate of removal by reaction. The former is given by a term involving the second partial derivative with respect to space and the diffusion coefficient, the latter by the term k ab2. Thus, for the reactant A, we have the governing equation 

We will also assume that A and B are similar chemical species, with the same value for their diffusion coefficients, DA = DB = D (this is only a special case in the sense that the mathematics is significantly easier). 

This is an equation in the total concentration of chemicals. One solution of eqn (11.5) which satisfies the boundary conditions is that the sum of the concentrations a + b should be equal to a constant everywhere and at all times. This constant is determined by the reaction stoichiometry and gives the sensible relationship 

If this result is used to substitute for a in (11.3) we obtain a single equation for the one independent concentration, b  

This is sometimes known as the cubic Fisher equation and has been applied variously in chemical kinetics, population biology, and to the spread of infectious diseases or advantageous genes. 

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