# Analysis of two variable Non-linear difference equation

1. Jul 7, 2011

### cuak2000

Hey everyone,

My understanding of difference equation is very limited and, after consulting a few books, I decided to post this here to get some ideas or at least some references on where to look.

I have a symmetric first order difference equation system in two variables, X and Y.
Though convergence might be interesting too, my main question with this system is the following:
Given that initially X(0) > Y(0) under what parameter values will Y catch up (or not) on X?

Just to illustrate here is my system, though I'm not sure this will be the final form:

X_{t+1} = \frac{ d*e^{bX_t} (1+m)p_{sub}(a - \frac{e^{bX_t} }{e^{bX_t} + e^{bY_t} } )^{\lambda} \, X_t }{ (1+g+m)p_{r}( e^{bX_t} + e^{bY_t}) }

and

Y_{t+1} = \frac{ d*e^{bY_t} (1+m)p_{sub}(a - \frac{e^{bY_t} }{e^{bY_t} + e^{bX_t} } )^{\lambda} \, Y_t }{ (1+g+m)p_{r}( e^{bY_t} + e^{bX_t}) }

Where d , m, p_sub , a, b, g and \lambda are positive parameters (I'll probably fix 5 numerically and analyse the system varying \lambda and d ).

My first attempt was to use the ratio of change X_{t+1} / X_t .
If X(0) > Y(0) , then if X_{t+1} / X_t > Y_{t+1} / Y_t , Y will never catch up on X. On the other hand, if Y grows faster than X and the difference doesn't go to zero, then it might be reasonable to speak about convergence. I haven't figured out what the precise requirements would be.

Like I said, this system might not be exactly what I will use (though it will look similar), so any specific or general ideas are welcome.
Thanks!

c