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Chapter 3: Problem 20

A population with four age classes has a Leslie matrix \(L=\left[\begin{array}{llll}0 & 1 & 2 & 5 \\ 0.5 & 0 & 0 & 0 \\ 0 & 0.7 & 0 & 0 \\ 0 & 0 & 0.3 & 0\end{array}\right] .\) If the initial population vector is \(\mathbf{x}_{0}=\left[\begin{array}{l}10 \\ 10 \\ 10 \\\ 10\end{array}\right],\) compute \(\mathbf{x}_{1}, \mathbf{x}_{2},\) and \(\mathbf{x}_{3}\).

Short Answer

Expert verified
\(\mathbf{x}_{1} = \begin{bmatrix} 80 \\ 5 \\ 7 \\ 3 \end{bmatrix}\), \(\mathbf{x}_{2} = \begin{bmatrix} 34 \\ 40 \\ 3.5 \\ 2.1 \end{bmatrix}\), \(\mathbf{x}_{3} = \begin{bmatrix} 57.5 \\ 17 \\ 28 \\ 1.05 \end{bmatrix}\).

Step by step solution

01

Understand the Leslie Matrix

The Leslie matrix is a structured matrix used in population dynamics models. It describes how individuals in different age classes contribute to the population in the next age class or reproduction cycle. Here, the matrix:\[ L=\begin{bmatrix} 0 & 1 & 2 & 5 \ 0.5 & 0 & 0 & 0 \ 0 & 0.7 & 0 & 0 \ 0 & 0 & 0.3 & 0 \ \end{bmatrix} \] indicates birth rates in the first row and survival rates in the sub-diagonals.
02

Multiply Leslie Matrix by Initial Population Vector

To find the next generation or population vector \(\mathbf{x}_{1}\), multiply the Leslie matrix \(L\) by the initial population vector \(\mathbf{x}_{0}\):\[ \mathbf{x}_{1} = L \cdot \mathbf{x}_{0} = \begin{bmatrix} 0 & 1 & 2 & 5 \ 0.5 & 0 & 0 & 0 \ 0 & 0.7 & 0 & 0 \ 0 & 0 & 0.3 & 0 \end{bmatrix} \cdot \begin{bmatrix} 10 \ 10 \ 10 \ 10 \end{bmatrix}. \]Perform the matrix multiplication step-by-step.
03

Calculate \(\mathbf{x}_{1}\)

To calculate \(\mathbf{x}_{1}\):1. The first element: \(0\times10 + 1\times10 + 2\times10 + 5\times10 = 0 + 10 + 20 + 50 = 80\).2. The second element: \(0.5\times10 + 0 + 0 + 0 = 5\).3. The third element: \(0 + 0.7\times10 + 0 + 0 = 7\).4. The fourth element: \(0 + 0 + 0.3\times10 + 0 = 3\).Thus, \(\mathbf{x}_{1} = \begin{bmatrix} 80 \ 5 \ 7 \ 3 \end{bmatrix}\).
04

Calculate \(\mathbf{x}_{2}\)

Repeat the process by multiplying \(L\) with \(\mathbf{x}_{1}\):\[ \mathbf{x}_{2} = L \cdot \mathbf{x}_{1} = \begin{bmatrix} 0 & 1 & 2 & 5 \ 0.5 & 0 & 0 & 0 \ 0 & 0.7 & 0 & 0 \ 0 & 0 & 0.3 & 0 \end{bmatrix} \cdot \begin{bmatrix} 80 \ 5 \ 7 \ 3 \end{bmatrix}. \]
05

Calculate \(\mathbf{x}_{2}\) Values

Compute each element:1. First element: \(0\times80 + 1\times5 + 2\times7 + 5\times3 = 0 + 5 + 14 + 15 = 34\).2. Second element: \(0.5\times80 + 0 + 0 + 0 = 40\).3. Third element: \(0 + 0.7\times5 + 0 + 0 = 3.5\).4. Fourth element: \(0 + 0 + 0.3\times7 + 0 = 2.1\).Thus, \(\mathbf{x}_{2} = \begin{bmatrix} 34 \ 40 \ 3.5 \ 2.1 \end{bmatrix}\).
06

Calculate \(\mathbf{x}_{3}\)

Multiply the Leslie Matrix \(L\) with \(\mathbf{x}_{2}\):\[ \mathbf{x}_{3} = L \cdot \mathbf{x}_{2} = \begin{bmatrix} 0 & 1 & 2 & 5 \ 0.5 & 0 & 0 & 0 \ 0 & 0.7 & 0 & 0 \ 0 & 0 & 0.3 & 0 \end{bmatrix} \cdot \begin{bmatrix} 34 \ 40 \ 3.5 \ 2.1 \end{bmatrix}. \]
07

Calculate \(\mathbf{x}_{3}\) Values

Calculate for each component:1. First element: \(0\times34 + 1\times40 + 2\times3.5 + 5\times2.1 = 0 + 40 + 7 + 10.5 = 57.5\).2. Second element: \(0.5\times34 + 0 + 0 + 0 = 17\).3. Third element: \(0 + 0.7\times40 + 0 + 0 = 28\).4. Fourth element: \(0 + 0 + 0.3\times3.5 + 0 = 1.05\).Therefore, \(\mathbf{x}_{3} = \begin{bmatrix} 57.5 \ 17 \ 28 \ 1.05 \end{bmatrix}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Dynamics
Population dynamics is an area of ecology that focuses on understanding how populations of species change over time and space. It is a branch that examines both short-term fluctuations and long-term patterns of populations.
Population dynamics considers various factors:
  • Birth Rates: The rate at which new individuals are added to a population through reproduction.
  • Death or Mortality Rates: The rate of loss of individuals from a population due to death.
  • Immigration and Emigration: The movement of individuals into (immigration) or out of (emigration) a population.
In the context of mathematical modeling, population dynamics often involves understanding how these processes interact to influence population size and structure. Models like the Leslie matrix make these dynamics more predictable, allowing scientists to project future population sizes given certain assumptions about birth and survival rates.
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra used throughout various scientific disciplines including ecology and population biology. It involves the multiplication of two matrices to produce a new matrix, where each element is the result of summing the products of corresponding row and column elements.
Here鈥檚 how matrix multiplication is executed:
  • First, the number of columns in the first matrix must match the number of rows in the second matrix.
  • Second, calculate each element in the resulting matrix by taking the dot product of the row from the first matrix and the column from the second matrix.
This operation is used in Leslie matrices to project population distributions across different ages over time. Each multiplication step updates the current age class distribution to the next time step, reflecting changes through birth rates and survival probabilities.
Age-Structured Models
Age-structured models are crucial in representing populations divided into different age classes. These models help predict how populations grow and change over time.
Components of age-structured models include:
  • Leslie Matrix: This matrix captures both the fecundity (births) and survival of different age classes.
  • Age Classes: Populations are divided into distinct groups, often representing different stages of life (e.g., juvenile, adult).
  • Population Vectors: These vectors show the number of individuals existing within each age group at a given time.
The dynamic nature of these models allows researchers to simulate population changes, aiding in forecasting demographic shifts, managing species populations, and conserving ecosystems.
Eigenvalues in Population Models
Eigenvalues play a central role in interpreting Leslie matrices and forecasting long-term behavior in population models. When you apply an age-structured matrix model, you essentially iterate the Leslie matrix, leading the population distribution toward its stable age structure, often driven by the dominant eigenvalue.
Key concepts include:
  • Dominant Eigenvalue: This value provides the growth rate of the population. It determines whether a population is growing, declining, or stable over time.
  • Stable Age Distribution: Once the population's growth is determined by the eigenvalue, the population structure moves towards a stable state where the proportions of the population in each age class remain constant.
Understanding eigenvalues thus helps ecologists and biologists anticipate long-term population trends and make informed conservation and management decisions.

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Most popular questions from this chapter

Draw a graph that has the given adjacency matrix. $$\left[\begin{array}{ccccc} 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \end{array}\right]$$

If \(A\) is a \(3 \times 5\) matrix, what are the possible values of nullity \((A) ?\)

Verify Theorem 3.32 by finding the matrix of S o \(T\) (a) by direct substitution and (b) by matrix multiplication of \([S][T]\). $$T\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{l} x_{1}-x_{2} \\ x_{1}+x_{2} \end{array}\right], S\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right]=\left[\begin{array}{l} 2 y_{1} \\ -y_{2} \end{array}\right]$$

If \(A\) is a \(4 \times 2\) matrix, what are the possible values of nullity \((A) ?\)

Let \(A\) be the adjacency matrix of a graph \(G\) (a) By induction, prove that for all \(n \geq 1\), the \((i, j)\) entry of \(A^{n}\) is equal to the number of \(n\) -paths between vertices \(i\) and \(j\) (b) How do the statement and proof in part (a) have to be modified if \(G\) is a digraph?
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