A Model of Algal Growth

This specifies the model, and it is well at this point to look back at the assumptions that have been made. They are (1) the immobility of the algae, which cuts out convective and diffusive terms, and (2,3) the validity of the growth and Beer s laws. No mention need be made of the depth of the pond if the immobility is conceded, for the solution for infinite depth is simply truncated at the finite depth. How adequate the model is depends on its purpose. For the present purpose of illustrating model building, it is admirable for predicting the total growth, it may be less accurate and, for the details of distribution, still less. The latter purposes will demand a comparison with experiment, but since we are concerned with the first we can proceed with equanimity. 

Here there are no obvious characteristic values of 7 and N they are bound up in the boundary conditions. Let 7 and N be such values, to be determined later, then (1 laN ) is a characteristic length and (1 Ibl ) a characteristic time. Thus we put 

It is possible to get a simple solution to these equations and indeed to much more general ones in which uv is replaced by F(u) G(v). The fact that 

But substituting back into the boundary conditions gives 

This model bears the imprint of its own limitations as a good model should. It is assumed that there is no movement, yet algae are growing in the upper layers of the water and presumably making the medium more dense. This suggest that sedimentation or convective roll-over should be considered. We shall not go into such generalizations here, but merely leave this example with the comment that the same equations turn up in the theory of a continuous reactor in which a belt of material is to be impregnated with a solution, whose solute is immobilized on sites in the belt. 

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