Does this limit exist?

1. Aug 12, 2009

nietzsche

1. The problem statement, all variables and given/known data

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

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

3. 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?

2. Aug 12, 2009

Cyosis

You should try to solve the differential equation first.

3. Aug 12, 2009

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.

4. Aug 12, 2009

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.

5. Aug 12, 2009

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)?

Last edited: Aug 12, 2009
6. Aug 12, 2009

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?

7. Aug 12, 2009

Fightfish

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

8. Aug 12, 2009

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?

9. Aug 12, 2009

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.

10. Aug 12, 2009

nietzsche

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

11. Aug 12, 2009

Staff: Mentor

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.

12. Aug 12, 2009

nietzsche

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

13. Aug 12, 2009

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...