Population Growth with Finite Resources:

[Scalemail Ted]

Homework Statement

Homework Equations

<see above>

The Attempt at a Solution

I'm a bit unsure how to set this up to solve for a solution. Any advice?

Its obviously a separable differential equation. But I'm unsure what it is I'm looking for. This looks different then some population examples in the book.

If I read the given equation correctly it states that the change in population (dp) in regards to time (dt) is equal to the constant rate of change (K) multiplied by the difference between the maximum population minus the population.

Then I see that my initial values are k = 0.01, Pmax = 1000, p(0) = 200, and t0 = 0.

But what do I do with this data? What am i solving for?

 

[Scalemail Ted]

I'm re-reading through the chapter now and the closest thing the original given equation looks similar to is a logistic model. Is this correlation correct?

Logistic model as per the book:

The stated expression for a logistic model is:

dp/dt = -Ap(p - p₁), p(0) = p₀

where A = k₃/2 and p₁ = (2k₁/k₃) + 1

If the original problem is a logistic model then my problem is the examples in the book give secondary and tertiary values of t and p(t). But the original problem does not.

 

[epenguin]

I do not know what "secondary and tertiary values" your last paragraph means but what to do? Solve the differential equation!

It is probably done in your book but see if you can do it. Simple application of partial fractions. (We have had several examples on these in recent days. People study and do exercises in them in algebra, they are quite easy. Then when they meet simple examples of them in differential equations they have no idea again. ).

Then see qualitatively what it looks like. Would it surprise you if at the end (t = infinity) the population was the maximum population? Would it surprise you if it wasn't? Does the graph of P against t or log t have a symmetry? Does the original equation say anything about that?