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

Answer:

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

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

Step-by-step 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

$x_1 = A \cdot x$

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

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

$x_2 = A \cdot x_1$

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

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