Logistic Growth DE

[suko123]

In 1970 the population of alligators on the Kennedy Space Center grounds was estimated to be 300. In 1975, the population of the alligators on the Kennedy Space Center grounds was estimated to be 1200. In 1980, the population had grown to an estimated 1500. Use a logistic model to estimate the alligator population on the Kennedy Center grounds in the year 2005.

This is what I did, but I am not sure if this is correct and please please correct me on what I did wrong if I did anything wrong.
P(t) = p*e^(k*t)
1200 = 300e^(5k)
k = ln(4)/5

1500 = 1200e^(5k)
k = ln(5/4)/5

P(35) = 300e^(ln(4)/5 + ln(5/4)/5)*35

Solution: 23,437,500

 

[stapel]

In 1970 the population of alligators on the Kennedy Space Center grounds was estimated to be 300. In 1975, the population of the alligators on the Kennedy Space Center grounds was estimated to be 1200. In 1980, the population had grown to an estimated 1500. Use a logistic model to estimate the alligator population on the Kennedy Center grounds in the year 2005.

This is what I did, but I am not sure if this is correct and please please correct me on what I did wrong if I did anything wrong.
P(t) = p*e^(k*t)
1200 = 300e^(5k)
k = ln(4)/5

1500 = 1200e^(5k)
k = ln(5/4)/5

P(35) = 300e^(ln(4)/5 + ln(5/4)/5)*35

How did you arrive at this equation? What "model" have you been given?

Solution: 23,437,500

Does this value seem reasonable to you? :shock:

Look at the known data. In five years, the population grew by a factor of four; the new population was four times the previous population. In another five years, the population grew by only one-quarter; the new population was only 5/4 times the previous population. The growth would have had to have returned to the previous levels (times four every five years) to arrive at your number. Does that sound like "logistic growth"?

 

[HallsofIvy]

You title this "logistic growth DE" but there is no differential equation in what you wrote.

In 1970 the population of alligators on the Kennedy Space Center grounds was estimated to be 300. In 1975, the population of the alligators on the Kennedy Space Center grounds was estimated to be 1200. In 1980, the population had grown to an estimated 1500. Use a logistic model to estimate the alligator population on the Kennedy Center grounds in the year 2005.

Do you know what the "logistic model" is? A "logistic DE" is a differential equation of the form dy/dx= Ay(B- y). It often applies to growth problems where there is a constraint on the size of the population. Generally, the rate of growth of a population is proportional to the size of the population- that is the "Ay" part. But often there is something like a food or space constraint that prevents the population from going past some upper bound. That is the "B- y" part. B is the upper bound.

To solve dy/dx= Ay(B- y), write it as \(\displaystyle \frac{dy}{y(B- y)}= A dx\) and integrate both sides. You can use "partial fractions" on the left. That will give you a formula involving A, B, and the "constant of integration". Use the information about 1970, 1975, and 1980 to determine those three constants.