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?
Best Answer
(A) - Maybey the use of t = 2006 is not a great idea, why not the difference ( t = 2006 - 1980), so you have a simpler time scale. I don't think with your method t = 2020 will give you the population at year 2020...
(B) - Not sure what you did here, but I hope k is a constant?! Some more details on your calculations would help. Here you can find the steps in an analytic solution of the logistic equation. Hope this gets you going.