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Analysis of two variable Non-linear difference equation

  1. Jul 7, 2011 #1
    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:

    \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
     
  2. jcsd
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