# Population Growth problem using Malthusian Law and the Logistic Model

In 1980 the population of alligators on the Kennedy Space Center grounds was estimated to be $$1500$$. In 2006 the population had grown to an estimated $$6000$$.

(A) Using the Malthusian Law for population growth, estimate the population of alligators on the grounds in 2020.

(B) Suppose we have the additional estimate that in 1993, the population was $$4100$$. Using the logistic model, estimate the population in 2020.

So the Malthusian Law gives the following equation:

$$\frac{dN}{dt}=rN \Rightarrow=N(t)=N_0e^{rt}$$

I found $$r$$ by doing the following:

$$6000=1500e^{r(2006)} \Rightarrow \frac{\ln(4)}{2006}=.000691 = r$$

From here I calculated the growth in 2020 like so:

$$N(2020)=1500e^{.000691(2020)}=6057$$ Alligators

That, I believe, takes care of part A.

But for part B, I'm very confused. I know the Logistic Model gives:

$$\frac{dN}{dt}=rN(\left(1-\frac{N}{k}\right)$$, but since I don't have $$k$$ I'm not sure how to solve this. What I did, which doesn't seem right, was algebraically solve for $$k$$ using the aforementioned equation and getting:

$$k=1-N$$ which means my $$k$$ (or maximum carry capacity) is $$4099$$

What am I doing wrong here?

• – Moo Mar 12 '16 at 22:15