Problem 6
Question
Assume the given Leslie matrix L. Determine the number of age classes in the population, the fraction of one-year-olds present at time \(t\) that survive to time \(t+1\), and the average number of female offspring of a two-year-old female. $$L=\left[\begin{array}{lll}0 & 5 & 0 \\ 0.8 & 0 & 0 \\ 0 & 0.4 & 0\end{array}\right]$$
Step-by-Step Solution
Verified Answer
There are three age classes, 80% of one-year-olds survive, and two-year-olds have 5 offspring on average.
1Step 1: Identify Age Classes
The Leslie matrix \(L\) is a 3x3 matrix, suggesting there are three age classes in the population. The rows and columns of \(L\) correspond to each of these age classes, where each class represents a specific age group.
2Step 2: Determine Survival Fraction of One-Year-Olds
The survival rate of one-year-olds from time \(t\) to time \(t+1\) is found in the matrix at row 2, column 1. This element of the matrix is 0.8, indicating that 80% of one-year-olds survive to the next year.
3Step 3: Calculate Average Offspring of Two-Year-Olds
The average number of female offspring produced by two-year-old females corresponds to the element in the matrix at row 1, column 2. This value is 5, meaning that on average, a two-year-old female will have 5 female offspring.
Key Concepts
Age ClassesSurvival FractionAverage Offspring
Age Classes
The Leslie Matrix provides a structured way to understand the distribution of different age groups within a population. In this context, "age classes" refer to the distinct groups or categories within a population defined by age. For a 3x3 Leslie matrix like the one provided, there are clearly three age classes. Each row and column in this matrix represents an age class, giving us insight into how individuals in each class contribute to population dynamics.
Understanding age classes is crucial because it allows us to model and predict the population growth or decline over time. By keeping track of each age class, we can see how individuals transition between different life stages and how their life strategies influence overall population trends.
Key points to remember:
Understanding age classes is crucial because it allows us to model and predict the population growth or decline over time. By keeping track of each age class, we can see how individuals transition between different life stages and how their life strategies influence overall population trends.
Key points to remember:
- An age class generally represents a group of individuals within a specific age range.
- In the Leslie matrix, each age class forms a row and column, capturing both its survival and reproductive contributions.
- Knowing the number of age classes helps us understand how complex the population structure might be.
Survival Fraction
The "survival fraction" indicates the proportion of individuals from a specific age class that survive to the next time period. In a Leslie matrix, this information is usually found in the sub-diagonal elements. For our case, the entry at row 2, column 1 is a key representation of survival fraction for one-year-olds.
This entry, which is 0.8, tells us that 80% of the individuals in this age class will survive from time \( t \) to time \( t+1 \). This is an essential factor in population dynamics as it affects the growth rate of the population.
To comprehend survival fractions more thoroughly, consider:
This entry, which is 0.8, tells us that 80% of the individuals in this age class will survive from time \( t \) to time \( t+1 \). This is an essential factor in population dynamics as it affects the growth rate of the population.
To comprehend survival fractions more thoroughly, consider:
- The survival fraction helps in determining how many individuals from a specific age class make it to the next age class.
- Large survival fractions indicate high chances of individuals surviving, which may lead to population growth.
- Understanding these fractions enables better planning and management, especially in conservation biology, to ensure sustainable population levels.
Average Offspring
The "average offspring" refers to the mean number of offspring produced by an organism at a certain age. In our Leslie matrix, this is highlighted by the matrix element that quantifies the average number of female offspring a two-year-old female produces.
For our example, the element at row 1, column 2 is 5, which suggests that a two-year-old female, on average, gives birth to 5 offspring. This figure is crucial for understanding the reproductive contribution of each age class to the overall population.
For our example, the element at row 1, column 2 is 5, which suggests that a two-year-old female, on average, gives birth to 5 offspring. This figure is crucial for understanding the reproductive contribution of each age class to the overall population.
- Average offspring figures help us predict future population sizes and dynamics.
- They are instrumental in calculating the reproductive value of different age classes within the population.
- This information can aid in making predictions about the population's health and its potential for growth.
Other exercises in this chapter
Problem 5
Determine \(c\) such that $$ \begin{array}{l} 2 x-3 y=5 \\ 4 x-6 y=c \end{array} $$ has (a) infinitely many solutions and (b) no solutions. (c) Is it possible t
View solution Problem 6
Let \(A=(1,3,-2)\) and \(B=(0,-1,-1)\). Find the vector representation of \(\overrightarrow{A B}\).
View solution Problem 6
Represent each given vector \(\mathrm{x}=\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]\) in the \(x_{1}-x_{2}\) plane, and determine its length and th
View solution Problem 6
Let $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\ 2 & 3 & -1 \\ 0 & -2 & 0\end{array}\right] ,\boldsymbol{B}=\left[\begin{array}{rrr}\mathbf{1} & \mathbf{- 1} & \mat
View solution