Dear Matt,

I am not sure what exactly you are looking for but here's an example of a so-called SIR system that models the spreading of a disease in a population. Three interrelated variables (susceptibles, infected, recovered) are iterated.

(*equations*)

sus[i_] := sus[i] = sus[i - 1] - \[Rho] sus[i - 1] inf[i - 1];

inf[i_] := inf[i] = inf[i - 1] + \[Rho] sus[i - 1] inf[i - 1] - \[Lambda] inf[i - 1];

rec[i_] := rec[i] = rec[i - 1] + \[Lambda] inf[i - 1];

(*initial conditions and parameters*)

sus[1] = 0.95; inf[1] = 0.05; rec[1] = 0; \[Rho] = 0.2; \[Lambda] = 0.1;

(*time course*)

tcourse = Table[{sus[i], inf[i], rec[i]}, {i, 1, 100}] // AbsoluteTiming

(*Plot*)

ListPlot[Transpose[tcourse]]

Of course there are other ways such as this one (not that easy to read):

(*Parameters*)

c = 0.1; b = 0.2;

Manipulate[results = {{0.05, 0.95, 0.0}};

(*Iterations*)

For[i = 0, i <= 100, i++,

AppendTo[results, {(1 - c) results[[-1, 1]] + b*results[[-1, 1]] results[[-1, 2]], results[[-1, 2]]*(1 - b results[[-1, 1]]), results[[-1, 3]] + c results[[-1, 1]] }]];

(*Plotting*)

ListPlot[Transpose[results]],

{{c, 0.1}, 0.0, 0.9}, {{b, 0.2}, 0.0, 0.9}]

Note that in the second code the "variables" are in a different order. I hope this helps a bit.

M.