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MAT244-2014F => MAT244 Math--Tests => FE => Topic started by: Victor Ivrii on December 08, 2014, 04:16:29 PM

Title: FE5
Post by: Victor Ivrii on December 08, 2014, 04:16:29 PM
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'_t = x (5-2x-3y)\, , \\
&y'_t = y (5-3x-2y)
\end{aligned}\right.
\end{equation*}

(a) describe the locations of all critical points,

(b) classify their types (including whatever relevant: stability, orientation, etc.),

(c) sketch the phase portraits near the critical points,

(d) sketch the phase portrait of this system of ODEs.



Solution

(a) Solving $x (5-2x-3y=0$, $y (5-3x-2y )$ we have 4 cases $x=y=0$, $x=5-3x-2y=0$, $y=5-2x-3y=0$ and $5-3x-2y =5-2x-3=0$ giving us 4 points $(0,0)$, $(0,\frac{5}{2})$, $(\frac{5}{2},0)$ and $(1,1)$.

(b) Let  $f= x (5-2x-3y)=5x-2x^2-3xy$, $g=y (5-3x-2y)=5y-2y^2-3xy$. Then $f_x=5-4x-3y$, $f_y=-3x$, $g_x=-3y$, $g_y=5-4y-3x$.



(c-d) Plotting


Remark This is ``two competing species'' system.