# Logistic Equation Help

• May 8th 2008, 01:05 AM
aznstyles408
Logistic Equation Help
dy/dt = r(1-y/K)y - Ey

r, K, and E are constants.

This is a logistic model of fish population.

a) Show that if E < r, then there are two equilibrium points y1 = 0
and y2 = K(1 - E/r) > 0

b) Show that y = y1 is unstable and y = y2 is asymptotically stable.

c)A sustainable yield Y of the fisher is a rate at which fish can be caught
indefinitely. It is the product of the effort E and the asymptotically stable
population y2. Find Y as a function of the effort E.

d) Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym.

Any hints of helps are appreciated. I'm having a real hard time with this (Worried).
• May 8th 2008, 01:37 AM
mr fantastic
Quote:

Originally Posted by aznstyles408
dy/dt = r(1-y/K)y - Ey

r, K, and E are constants.

This is a logistic model of fish population.

a) Show that if E < r, then there are two equilibrium points y1 = 0
and y2 = K(1 - E/r) > 0

Mr F says: By definition, solve 0 = r(1-y/K)y - Ey.

b) Show that y = y1 is unstable and y = y2 is asymptotically stable.

c)A sustainable yield Y of the fisher is a rate at which fish can be caught
indefinitely. It is the product of the effort E and the asymptotically stable
population y2. Find Y as a function of the effort E.

d) Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym.

Any hints of helps are appreciated. I'm having a real hard time with this (Worried).

..
• May 8th 2008, 01:58 AM
aznstyles408
yea thanks for the help. About 2 mins after I made the post I realized that's all I had to do was make dy/dt = 0. (Headbang)

I'm still having a hard time with D though.
Where they saying determine E so that Y = EK(1-r/K) is maximized.
• May 8th 2008, 02:16 AM
mr fantastic
Quote:

Originally Posted by aznstyles408
yea thanks for the help. About 2 mins after I made the post I realized that's all I had to do was make dy/dt = 0. (Headbang)

I'm still having a hard time with D though.
Where they saying determine E so that Y = EK(1-r/K) is maximized. Mr F asks: Where has this expression for Y come from?

"A sustainable yield Y ..... is the product of the effort E and the asymptotically stable population y2."

So Y = E y2.

"y2 = K(1 - E/r)"

So clearly Y = KE(1 - E/r).

This is a parabola with a maximum turning point. It's maximum occurs halfway between it's two E-intercepts ......
• May 8th 2008, 02:37 AM
aznstyles408
The expression for Y was given in question c. Where it states that Y
is equal to the product of E and y2. So Y = Ey2. And since y2 = K(1 - r/k), we can substitute in and get Y = EK(1 - r/K). And they ask to determine E so that it maximizes Y.
• May 8th 2008, 02:42 AM
aznstyles408
thank you so much for all the help!!! (Rofl)
• May 8th 2008, 02:51 AM
aznstyles408
ahhh!! it's actually Y = EK(1 - E/r). Sorry for the mistake