# Thread: {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations

1. ## {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations

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?

2. ## Re: {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations

I only get 3 roots for your system. What makes you think there is a fourth?

3. ## Re: {x'=x(1-x-y), y'=y(1.5-x-y)} System of Differential equations

Hi !
The general solution of the EDOs system is shown in attachment.

4. ## Rational dynamical system with nonnegative paramaters

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.