A Leslie matrix for a population with the usual age groups, newborns (aged 0-1), 1-year-olds (aged 1-2), etc. is given below. 0 2 0 L= 0 0 0.65 0 0.5 0 Assume that there are 60 newborns, 30 one-year-olds and 25 two-year-olds, and that the Leslie matrix has 70 eigenvalue c = 0.85 and eigenvector 25 L25 a) The initial population X(0 Select an answer In eigenvector of c= 0.85. is is not b) Select which of the following is correct way to compute X(13), the population at time 13. 2 0 70 O X(13) = 0 0.65 0 0 0.5 30 25 25 0 13 es 0 2 0 30 O X(13) = 0.65 0 0 . 60 0 0.5 0 25 60 O X(13) = 0.8513 30 25 13 0 2 0 70 O X(13) = 0 0 25 0.65 0 0.5 0 25 13 2 0 60 0 0.65 O X(13) = 0 0 30 25 0 0.5 0

(a) Yes, the initial population can be represented as an eigenvector of c=0.85. Given that the Leslie matrix L has eigenvalue c=0.85 and eigenvector X(25) = [25, 30, 25], the initial population can be represented as X(0) = [60, 30, 25]. This means that the population is distributed with 60 newborns, 30 one-year-olds, and 25-two-year-olds.

(b) To compute X(13), the population at time 13, we can use the formula X(13) = L^13 * X(0), where L^13 denotes the Leslie matrix raised to the power of 13. Multiplying L by itself 13 times allows us to calculate the population distribution at the 13th time period. Using the given options, the correct way to compute X(13) is X(13) = [0, 0.65, 0.5] * [60, 30, 25] = [39, 39, 28.75].

Therefore, the population at time 13 consists of 39 newborns, 39 one-year-olds, and approximately 28.75-two-year-olds.

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