The growth of a giraffe population on an island follows a logistical growth model. The maxium giraffe population on the island is estimated at 1000. The population was first measured in 1970 at 125. Then in 1975 the population was 158
a) Find the rate of change in the population in 1990 by first writing a function P(t) that models the population t years after 1970, then taking the derivative of that function
b.) Find the rate of change in the population in 1990 by using the logistic differential equation [/B]
P(t)= M/ 1+ Ae-kt Where M is the maximum capactiy and
The Attempt at a Solution
Well part A is what is really stumping me, I'm not completely sure how to write a function for this... and it's making me feel silly. I've been scouring my textbook and they have know written examples, or exercises to give me a hint. My first thought was to just find the slope given the two points, but the more I thought about it it doesn't make sense because that would indicate exponential growth, which isn't the case for the problem. I need some hints, about that part, I'm sure i can derive it by myself.
For part b I think I did it right, but maybe it's not really the rate of change. I did as follows, with P0=125 and P5=158:
P(t)=1000/(1+7e-kt the 7 coming from the second equation I wrote to solve for "A"
I then used that fact that in 1975 the population was 158 so:
k is approximately 298.38
To me this seems like a reasonable population size in 1990, but I don't think it necessarily reflects a rate of change. I know I am missing some important part but I can't figure it out so far
Any help is appreciated! Thanks in advance!