# Showing an equilibrium point with an inequality

• Mar 4th 2010, 01:30 PM
FoxyGrandma3000
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 $\displaystyle y_{2}$?
• Mar 4th 2010, 02:43 PM
Jester
Quote:

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 $\displaystyle E < r$.