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Math Help - Showing an equilibrium point with an inequality

  1. #1
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    Showing an equilibrium point with an inequality

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

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

    The demonstration of  y_{1} is plain to see, but how do you go about proving the equilibrium point at  y_{2} ?
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  2. #2
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    Quote Originally Posted by FoxyGrandma3000 View Post
    Given  \frac{dy}{dt}=r(1-\frac{y}{k})y-Ey

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

    The demonstration of  y_{1} is plain to see, but how do you go about proving the equilibrium point at  y_{2} ?
    I believe that you'll have two distinct eq. points provided that E \ne r. From your ODE

     <br />
\frac{dy}{dx} = y\left(r - \frac{ry}{k} - E \right) <br />

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