Thread: Showing an equilibrium point with an inequality

Given $\displaystyle \frac{dy}{dt}=r(1-\frac{y}{k})y-Ey $

if $\displaystyle E<r $ show that there are two equilibrium points given by $\displaystyle y_{1}=0 $ and $\displaystyle y_{2}=k(1-\frac{E}{r})>0 $

The demonstration of $\displaystyle y_{1} $ is plain to see, but how do you go about proving the equilibrium point at $\displaystyle y_{2} $?

Originally Posted by FoxyGrandma3000

Given $\displaystyle \frac{dy}{dt}=r(1-\frac{y}{k})y-Ey $

if $\displaystyle E<r $ show that there are two equilibrium points given by $\displaystyle y_{1}=0 $ and $\displaystyle y_{2}=k(1-\frac{E}{r})>0 $

The demonstration of $\displaystyle y_{1} $ is plain to see, but how do you go about proving the equilibrium point at $\displaystyle y_{2} $?

I believe that you'll have two distinct eq. points provided that $\displaystyle E \ne r$. From your ODE

$\displaystyle
\frac{dy}{dx} = y\left(r - \frac{ry}{k} - E \right)
$

$\displaystyle y = 0$ is one, the other is found by setting $\displaystyle r - \frac{ry}{k} - E = $0 and solving for $\displaystyle y$. If you want the second eq. point to be positive then you would need $\displaystyle E < r$.