Logistic Equation: Finding the Limit of Population Growth

[nietzsche]

Homework Statement

A population develops according to logistic equation:
\frac{dy}{dt} = y(0.5-0.01y)

Determine the following:
\lim_{t\rightarrow\infty} y(t)

The Attempt at a Solution

By finding an equilibrium solution to the differential equation, we see that dy/dt = 0 when y = 0 or y = 50.

But does the limit exist? What if y = 0 initially?

 

[Cyosis]

You should try to solve the differential equation first.

 

[nietzsche]

But do you have to do the differential to determine the limit? Because we can see that if y=0 and t goes to infinity, y will still equal 0. and if y does not equal 0 initially, then y will go to 50.

 

[Cyosis]

You have picked two fixed values for y which indeed satisfy the differential equation, however these values aren't the only solutions. The more interesting solutions are the ones that still depend on t.

 

[nietzsche]

I'm sorry, I don't understand.

I solved the differential equation:

y(t) = 50 / (1+Ae^(-0.5t)

where A = (50-y(0)) / y(0)

so isn't it still dependent on the value of y(0)?

 

[nietzsche]

OH. so now if we take the limit, as t approaches infinity, then A will go to 0, no matter what. is that correct?

 

[Fightfish]

Yea, more correctly, Ae^(-0.5t) goes to zero

 

[nietzsche]

ah okay, thanks.

i still don't get the concept though. how can the limit of the equation be 50 if the initial population is 0?

 

[Fightfish]

Since the equation is modeling the development of a population, that would imply that y(0) cannot be zero; otherwise you have no population to begin with.

 

[nietzsche]

haha, i suppose. thanks very much cyosis and fightfish.

 

[Mark44]

OH. so now if we take the limit, as t approaches infinity, then A will go to 0, no matter what. is that correct?

No, it's not. As fightfish said, Ae^-.5t goes to 0 as t approaches infinity. This, however, does not imply that A is 0 or is approaching zero. A is a constant in the problem.

 

[nietzsche]

sorry, i'll be more specific next time...

 

[g_edgar]

>> how can the limit of the equation be 50 if the initial population is 0?

You need to object to nietsche, who said the solution is:
y(t) = 50 / (1+Ae^(-0.5t)
that is NOT the solution if y(0)=0...