Hey everyone,(adsbygoogle = window.adsbygoogle || []).push({});

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:

\begin{equation}

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}) }

\end{equation}

and

\begin{equation}

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}) }

\end{equation}

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Analysis of two variable Non-linear difference equation

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**