# System of difference equations with Mathematica

1. May 16, 2012

### DAVA

Hello all,
I am attempting to characterize a molecular pathway through a system of difference equations:

W[n + 1] = W[n] - b*L[n]*W[n] + nw
H[n + 1] = (1 - dwl)*H[n] + b*W[n]*L[n]
G[n + 1] = G[n] - g*H[n] + ng
B[n + 1] = (1 - t)*B[n] - e*G[n] + nb
T[n + 1] = (1 - dd)*T[n] + t*B[n]
L[n + 1] = L[n] + f*T[n] - b*L[n]*W[n]

where b, nw, dwl, g, ng, t, e, nb, dd, and f are all constants.
n refers to the iteration.

I currently have two problems and would appreciate any input:

1) RSolve in mathematica won't solve it as soon as I include the product of W[n] and L[n] in my system of equations. Does this make the system unsolvable fundamentally or unsolvable just for Mathematica?

Below is the complete function call (I have solved a simpler system without the L[n]*W[n] this way):

Clear["Global`*"];
Clear[n];

RSolve[
{W[n + 1] == W[n] - b*L[n]*W[n] + nw,
H[n + 1] == (1 - dwl)*H[n] + b*W[n]*L[n],
G[n + 1] == G[n] - g*H[n] + ng,
B[n + 1] == (1 - t)*B[n] - e*G[n] + nb,
T[n + 1] == (1 - dd)*T[n] + t*B[n],
L[n + 1] == L[n] + f*T[n] - b*L[n]*W[n],
W[0] == 2, H[0] == 2, G[0] == 36, B[0] == 320, T[0] == 16, L[0] == 4},
{W[n], H[n], G[n], B[n], T[n], L[n]}, n
]

2) In general, not just for the system above, let's say that I want to simplify a system into a single higher order equation (e.g. W[n],W[n+1],W[n+2], etc.). Can this be done in Mathematica? My current websearch has not revealed any example.

Thank you.