In “Mathematical Models in Biology”, Leah Edelstein-Keshet presents a model describing the number of circulating red blood cells (RBC’s). It assumes that the spleen filters out and destroys a fraction of the cells daily while the bone marrow produces a amount proportional to the number lost on the previous day:

where

number of RBC’s in circulation on day ,

number of RBC’s produced by marrow on day ,

fraction of RBC’s removed by the spleen,

numer of RBC’s produced per number lost.

What would be the cell count on the day?

Observe first that (1) is equivalent to

where

Let ,

Substituting (3) into (2) yields

We proceed to solve the following initial-value problem using ‘*solve_rec*‘ (see “Solving Difference Equations using Omega CAS Explorer“):

Evaluate the solution with , we have

Plotting (5) by ‘plot2d(4/3 + (-1)^(n+1)*2^(-n)/3, [n, 0, 10], WEB_IMAGE)’ fails (see Fig. 1) since plot2d treats (5) as a continuous function whose domain includes number such as .

Fig. 1

Instead, a discrete plot is needed:

Fig. 2

From Fig. 2 we see that converges to a value between and . In fact,