RSolve Lotka-Volterra: A Closed Form Solution to this Difference Equation?

Question

I would like to solve a difference equation, a Lotka-Volterra difference equation. I tried a model from http://www.stolaf.edu/people/mckelvey/envision.dir/nonDE.lotka-volt.html

Using their parameters:

a = 0.04; b = 0.0005; c = 0.2; d = 0.1;

I obtain the following:

eqs = {r[n + 1] == r[n] + a*r[n] - b*r[n]*f[n], f[n + 1] == f[n] + b*d*r[n]*f[n] - c*f[n]}
RSolve[eqs, {r[n], f[n]}, n]

(* RSolve[{r[1 + n] == 1.04 r[n] - 0.0005 f[n] r[n], 
          f[1 + n] == 0.8 f[n] + 0.00005 f[n] r[n]}, {r[n], f[n]}, n] *)

Mathematica does not give any clue, what is wrong. Can anyone help me find out where the problem is?

Answer 1

When a difference equation does not have a closed form solution, you cannot realistically expect Mathematica to find one. The exercises on the page you link to ask for the steady state, not for a closed form solution! We can find the steady state solution by letting r=r[n] and f=f[n] and then rewriting the relationships that hold at the fixed point.

ss = {r - (r + a*r - b*r*f) == 0, f - (f + b*d*r*f - c*f) == 0}

The steady state is when ss holds. This can be handled easily by Solve

Solve[ss, {f, r}]

which returns two solutions:

{{f -> 0, r -> 0}, {f -> a/b, r -> c/(b d)}}

If you wish to simulate the equation, this can be done many ways. One is to explicitly use recursion:

Clear[a, b, c, d, r, f];
a = 0.04; b = 0.0005; c = 0.2; d = 0.1;
r[n_] := r[n] = r[n - 1] + a*r[n - 1] - b*r[n - 1]*f[n - 1]; 
f[n_] := f[n] = f[n - 1] + b*d*r[n - 1]*f[n - 1] - c*f[n - 1];
r[0] := c/(b d) - 100;
f[0] := a/b;

allR = r /@ Range[1500];
allF = f /@ Range[1500];

The final two lines calculate r and f (the number of rabbits and foxes, presumably) for 1500 timesteps. The output can be easily visualized. With these particular values, it doesn't look like a very happy ecosystem.

ListPlot[{allR, allF}]