Question

In 1980 the population of alligators in a particular region was estimated to be 1700. In 2005 the population had grown to an estimated 7000.
Using the Malthusian law for population​ growth, estimate the alligator population in this region in the year 2020.

Answer 1

The Malthusian law for population growth have the folowing form:

$P(t)=P_{0}e^{rt}$the population of alligators in

where $P_{0}$ 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 $P_{0}$ is 1700 (the population of alligators in 1980):

$P(1980)=P_{0} e^{r(1980-1980)}\\P(1980)=P_{0} e^{0}\\P(1980)=P_{0} =1700$

and that let us with:

$P(t)=1700e^{r(t-1980)}$

Now we can solve for the 2005 population of 7000 alligator:

$P(2005)=1700e^{r(2005-1980)}$

By knowing the definition of exponential we can solve:

$7000=1700e^{r(2005-1980)}\\7000/1700=e^{r(2005-1980)}\\ln (7000/1700)=r(25)\\\frac{ln (7000/1700)}{25} =r\\r=0.0566$

now we can estimate by replacing t with 2020

$P(2020)=1700e^{0.0566(2020-1980)}$

$P(2020)=16364$

So the approximate population in 2020 is 16364 alligators in that region.