System of difference equations with Mathematica

[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.

 

[jvicens]

Hello there,

Thank you for sharing your system of difference equations. It looks like a complex and interesting problem.

To answer your first question, it is possible that the system is unsolvable in Mathematica due to the product of W[n] and L[n]. However, it is also possible that there is a specific syntax or formatting issue in your function call that is causing the problem. I would suggest double-checking your syntax and possibly seeking help from other Mathematica users or forums to see if anyone has encountered a similar issue.

As for your second question, it is possible to simplify a system of difference equations into a single higher order equation in Mathematica. One approach is to use the "DifferenceRoot" function, which can represent a recurrence relation in a single higher order equation. You can also try using the "RecurrenceTable" function to generate a table of values for your system and then use the "FindSequenceFunction" function to find a single equation that fits the data.

I hope this helps and good luck with your research!