Question

You are researching the penguin population in the Falkland Islands. There are 1000 penguins in the beginning of the first year: 500 are male, and 500 are female. Each couple makes one chick every spring (assuming every penguin has a partner). Chicks that are one year old also reproduce. However, 20% of all penguins, both adults and chicks, die each year due to natural causes. You need to create a formula to determine the population of the penguins (including chicks) at the end of the 7th year.

What is the formula to determine the population of the penguins at the end of the 7th year?

Answer 1

The formula to determine the population of penguins at the end of the 7th year is:

Explanation:

With this information, we know that the initial population at end of year 0 is 1000 penguins.

The first year will born 500 chicks (50% of the population) and also 20% of the total population will die.

We can then model the population for the end of year 1 as:

As this dynamic will continue with the years, we can generalize as:

Then, the value of the population at the end of the 7th year should be:

Answer 2

The formula to determine the population of penguins at the end of the 7th year is:

`P_7=1.3^7\cdot P_0=6.275\cdot 1000=6275`

Explanation:

With this information, we know that the initial population at end of year 0 is 1000 penguins.

The first year will born 500 chicks (50% of the population) and also 20% of the total population will die.

We can then model the population for the end of year 1 as:

`P_1=P_0+0.5P_0-0.2P_0=1.3P_0`

As this dynamic will continue with the years, we can generalize as:

`P_2=1.3P_1=1.3\cdot (1.3P_0)=1.3^2P_0\\\\P_n=1.3^nP_0`

Then, the value of the population at the end of the 7th year should be:

`P_7=1.3^7\cdot P_0=6.275\cdot 1000=6275`