I only get 3 roots for your system. What makes you think there is a fourth?
I'm trying to evaluate this system, by finding and categorizing 4 equilibria points as stable, asymptotically stable, or unstable (nodes, saddles, foci, or centers). Then I have to determine if this describes a species coexistence or competitive exclusion scenario.
I have found three of the 4 equil. points, and evaluated them in a Jacobi matrix. The fourth (positive equil. point) seems to not exist....as I solve the system to find that -0.5=0, which is obviously false.
I'm afraid I may have made a mistake somewhere. Can I get a push in the right direction?
let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3 }y_{n}+\lambda_{3}z_{n}}{A_{3}+B_{3}x_{n}+c_{3}y_{ n}+D_{3}z_{n}}{}\end{cases}$
$n=0,1,....$
with nonnegative paramaters and with nonnegative initial conditions such that the denominators are always positive.
how i could show that :for every positive solution of the difference equation:
$z_{n+1}= \frac{\alpha}{1+\prod_{i=0}^{k}z_{n-i}},n=0,1,.... $ has a finit limit ?
Thank you for any replies or any help.