The Malthusian law for population growth have the folowing form:
[tex]P(t)=P_{0}e^{rt}[/tex]the population of alligators in
where [tex]P_{0}[/tex] is the initial population size , r is the rate of growth and t is time.
By substracting 1980 from t we could say that our [tex]P_{0}[/tex] is 1700 (the population of alligators in 1980):
[tex]P(1980)=P_{0} e^{r(1980-1980)}\\P(1980)=P_{0} e^{0}\\P(1980)=P_{0} =1700[/tex]
and that let us with:
[tex]P(t)=1700e^{r(t-1980)}[/tex]
Now we can solve for the 2005 population of 7000 alligator:
[tex]P(2005)=1700e^{r(2005-1980)}[/tex]
By knowing the definition of exponential we can solve:
[tex]7000=1700e^{r(2005-1980)}\\7000/1700=e^{r(2005-1980)}\\ln (7000/1700)=r(25)\\\frac{ln (7000/1700)}{25} =r\\r=0.0566[/tex]
now we can estimate by replacing t with 2020
[tex]P(2020)=1700e^{0.0566(2020-1980)}[/tex]
[tex]P(2020)=16364[/tex]
So the approximate population in 2020 is 16364 alligators in that region.
