Problem 75 - Calculus for Biology and Medicine

Question

Assume the given Leslie matrix \(L .\) Determine the number of age classes in the population, the fraction of oneyear-olds that survive until the end of the next breeding season, and the average number of female offspring of a two- year-old female. $$ L=\left[\begin{array}{llll} 2 & 3 & 2 & 1 \\ 0.4 & 0 & 0 & 0 \\ 0 & 0.6 & 0 & 0 \\ 0 & 0 & 0.8 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

There are 4 age classes. The survival fraction for one-year-olds is 0.4. A two-year-old female has, on average, 2 female offspring.

Step 1: Identify Age Classes

A Leslie matrix is a square matrix used to model the dynamics of a structured population, where each row (or column) corresponds to an age class. The size of the matrix, which is 4x4 in this case, indicates there are 4 age classes in the population.

Step 2: Locate Survival Fractions in Leslie Matrix

Survival fractions of different age classes are represented by the sub-diagonal elements in the Leslie matrix. To find the fraction of one-year-olds that survive until the end of the next breeding season, look at the element in the second row and first column of the matrix, which is 0.4.

Step 3: Identify Offspring Information for Specific Ages

In the Leslie matrix, the top row represents fecundity rates, or the average number of female offspring that females of each age class produce. For a two-year-old female, check the value in the top row and third column, which is 2.

Key Concepts

Population Dynamics

Population dynamics is the study of how populations of organisms change over time and space. It involves understanding the factors that influence population growth, stability, and decline. In the context of the Leslie Matrix, population dynamics focuses on structured populations where individuals are grouped into age classes. Each age group may have different survival and reproduction rates, which can significantly affect the overall population.
This understanding is crucial for predicting future population trends and managing wildlife and conservation efforts.

Age Class

Age class refers to the categorization of a population into different stages based on age. In ecology, these categories help in analyzing how age affects the survival and reproduction of a species.
In a Leslie matrix, each row and column corresponds to a specific age class. For example, a 4x4 Leslie matrix, like the one given, indicates four distinct age groups. Each group may have different characteristics affecting their chances of survival and reproduction.
The concept of age class is vital in the study of population dynamics as it helps identify how different age groups contribute to the population's future.

Survival Fractions

Survival fractions are a key component of the Leslie Matrix, representing the probability of individuals in a specific age class surviving to the next. They are located in the sub-diagonal elements of the Leslie Matrix.
An example extracted from the provided matrix would be the survival fraction of one-year-olds. This is found in the second row, first column, which is 0.4. This indicates that 40% of one-year-olds survive until the next breeding season.

• Survival fractions are critical for predicting future population size.
• They tell us about specific age-related mortality rates within the population.

Fecundity Rates

Fecundity rates refer to the average number of female offspring produced by a female of a given age class in one breeding season. In the Leslie Matrix, these rates are found in the first row.
For instance, in the matrix from our exercise, the fecundity rate for a two-year-old female is depicted in the first row and third column, with a value of 2. This means, on average, a two-year-old will produce two female offspring per breeding season.

• High fecundity rates can lead to rapid population growth.
• Understanding these rates helps in determining the reproductive potential of different age classes.