Problem 8 - Calculus for Biology and Medicine

Question

Assume the given Leslie matrix \(L\) Determine the number of age classes in the population. What fraction of two-year-olds present at time t survive until time \(t+1\). Determine the average number of female offspring of a one-year-old female. $$L=\left[\begin{array}{lll}0 & 4.2 & 3.7 \\ 0.7 & 0 & 0 \\ 0 & 0.1 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The population has 3 age classes. Two-year-olds have a survival fraction of 0 until time \( t+1 \), and a one-year-old female has an average of 4.2 female offspring.

Step 1: Identify the Age Classes

A Leslie matrix displays age-specific fertility and survival rates across different age classes. Here, the matrix provided is a 3x3 matrix, which indicates that there are 3 age classes within this population. Therefore, the population is divided into three age groups.

Step 2: Determine Survival Fraction of Two-Year-Olds

The value in the Leslie matrix for survival of two-year-olds until time \( t+1 \) is located in the second row, third column. In the given matrix \( L \), this value is 0. This means that no two-year-olds survive to the next time period under the matrix assumptions.

Step 3: Average Number of Female Offspring of a One-Year-Old

The top row of the Leslie matrix represents fertility rates for different age groups. Specifically, the element in the top row, second column indicates the average number of female offspring produced by a one-year-old female. In matrix \( L \), this value is 4.2, meaning one-year-old females have an average of 4.2 female offspring.

Key Concepts

Age Classes

In the context of a Leslie matrix, age classes are distinct age groups within a population that are used to predict future changes in population dynamics. The Leslie matrix summarizes the survival and fertility rates of these age classes.

For the given matrix:

• The matrix is 3x3, indicating there are three age classes.
• Each row and column corresponds to a specific age group.
• The populations are divided, typically focusing on females for reproduction.

Thus, age classes help us understand population growth and dynamics in structured models like the Leslie matrix.

Survival Fraction

In population dynamics, the survival fraction is the probability that an individual of a particular age class will survive to the next age class. It is crucial for predicting population sustainability over time.

Looking at our matrix:

• Survival fractions are found below the main diagonal of the Leslie matrix.
• The value represents the fraction of individuals who survive to the next year.
• For two-year-olds, the survival fraction in the matrix is 0.

This means in the model assumptions, no two-year-olds survive to the next time period. This information is essential for checking any potential population decline over time.

Fertility Rates

Fertility rates in a Leslie matrix are the average number of offspring produced by individuals in a given age class. This value is found in the top row of the matrix, indicating reproduction rates for each group.

From our example matrix:

• The first element in the top row represents offspring of 0-year-olds, typically 0, since they are yet to reproduce immediately after birth.
• Second column, top row represents one-year-olds, here it is 4.2 - meaning each one-year-old produces an average of 4.2 females.
• The third position usually indicates the rate for two-year-olds.

These fertility rates direct our understanding of how each age group contributes to the population's growth and continuation.