Question

A population has the following characteristics.(a) A total of 25% of the population survives the first year. Of that 25%, 75% survives the second year. The maximum life span is 3 years.(b) The average number of offspring for each member of the population is 3 the first year, 5 the second year, and 3 the third year.The population now consists of 144 members in each of the three age classes. How many members will there be in each age class in 1 year?0 ≤ age ≤ 1 = 1 ≤ age ≤ 2 = 2 ≤ age ≤ 3 = In 2 years?0 ≤ age ≤ 1 = 1 ≤ age ≤ 2 = 2 ≤ age ≤ 3 =

Answer 1

To answer this question, we'll perform calculations based on the provided survival and reproduction rates, without reference to any code. Here's how you can calculate the population for each age class after 1 year and then after 2 years:

### After 1 Year:
1. Newborns (0 ≤ age ≤ 1):
Each member of the current population contributes to the number of newborns through reproduction. We have 144 individuals in each of the three age classes (0-1, 1-2, 2-3 years old), each having reproduction rates of 3, 5, and 3, respectively, during their first, second, and third years.
Total newborns = (144 * 3) + (144 * 5) + (144 * 3) = 432 + 720 + 432 = 1584

2. Age 1 to 2 (1 ≤ age ≤ 2):
Only 25% of the population aged 0 to 1 survives to enter the 1 to 2 age class.
Survivors from age class 0-1 to 1-2 = 144 (starting number of age 0-1) * 0.25 = 36

3. Age 2 to 3 (2 ≤ age ≤ 3):
Only 75% of the population aged 1 to 2 survives to enter the 2 to 3 age class.
Survivors from age class 1-2 to 2-3 = 144 (starting number of age 1-2) * 0.75 = 108

Now, the population in each age class after 1 year is as follows:
0 ≤ age ≤ 1: 1584 members
1 ≤ age ≤ 2: 36 members
2 ≤ age ≤ 3: 108 members

### After 2 Years:
Repeating the same process for the next year using the numbers we obtained for Year 1:

1. Newborns (0 ≤ age ≤ 1):
Total newborns = (1584 * 3) + (36 * 5) + (108 * 3) = 4752 + 180 + 324 = 5256

2. Age 1 to 2 (1 ≤ age ≤ 2):
Survivors from age class 0-1 to 1-2 = 1584 (number of age 0-1 after 1 year) * 0.25 = 396

3. Age 2 to 3 (2 ≤ age ≤ 3):
Survivors from age class 1-2 to 2-3 = 36 (number of age 1-2 after 1 year) * 0.75 = 27

Now, the population in each age class after 2 years is as follows:
0 ≤ age ≤ 1: 5256 members
1 ≤ age ≤ 2: 396 members
2 ≤ age ≤ 3: 27 members

To summarize:
- After 1 year, there will be 1584 members aged 0-1, 36 members aged 1-2, and 108 members aged 2-3.
- After 2 years, there will be 5256 members aged 0-1, 396 members aged 1-2, and 27 members aged 2-3.

Answer 2

After 1st year, the age distribution will be

`x_1 = \left[\begin{array}{ccc}1584\\36\\108\end{array}\right]`

After 2nd year, the age distribution will be

`x_2 = \left[\begin{array}{ccc}5256\\396\\27\end{array}\right]`

Explanation:

A population has the following characteristics.

A total of 25% of the population survives the first year. Of that 25%, 75% survives the second year.

The average number of offspring for each member of the population is 3 the first year, 5 the second year, and 3 the third year.

From the above information, we can construct a transition age matrix.

`A = \left[\begin{array}{ccc}3&5&3\\0.25&0&0\\0&0.75&0\end{array}\right]`

The population now consists of 144 members in each of the three age classes.

From the above information, we can construct the current age matrix.

`x = \left[\begin{array}{ccc}144\\144\\144\end{array}\right]`

How many members will there be in each age class in 1 year?

After 1st year, the age distribution will be

`x_1 = A \cdot x`

`x_1 = \left[\begin{array}{ccc}3&5&3\\0.25&0&0\\0&0.75&0\end{array}\right] * \left[\begin{array}{ccc}144\\144\\144\end{array}\right]`

The matrix multiplication is possible since the number of columns of first matrix is equal to the number of rows of second matrix.

`x_1 = \left[\begin{array}{ccc}1584\\36\\108\end{array}\right]`

After 2nd year, the age distribution will be

`x_2 = A \cdot x_1`

`x_2 = \left[\begin{array}{ccc}3&5&3\\0.25&0&0\\0&0.75&0\end{array}\right] * \left[\begin{array}{ccc}1584\\36\\108\end{array}\right]`

`x_2 = \left[\begin{array}{ccc}5256\\396\\27\end{array}\right]`