(a) Consider a population with \(m\) age classes, and let \(p_{n}^{1}, p_{=1}^{2}, \ldots, p_{n}^{m}\) be the numbers of individuals within each class such that \(p_{n}^{0}\) is the number of newborns and \(p_{n}^{m}\) is the number of oldest individuals. Define $$ \begin{aligned} \alpha_{1}, \ldots, \alpha_{4}, \ldots, \alpha_{m}=& \text { number of births from individuals of a given } \\ & \text { age class, } \\ \sigma_{1}, \ldots, \sigma_{k}, \ldots, \sigma_{m-1}=& \text { fraction of } k \text { year olds that survive to be } \\ & k+1 \text { year olds. } \end{aligned} $$ Show that the system can be described by the following matrix equation: $$ \mathbf{P}_{n+1}=\mathbf{A} \mathbf{P}_{n} $$ where $$ \mathbf{P}=\left(\begin{array}{c} p^{1} \\ p^{2} \\ \vdots \\ p^{m} \end{array}\right), \quad A=\left(\begin{array}{cccc} \alpha_{1} & \alpha_{2} & \cdots & \alpha_{m} \\ \sigma_{1} & 0 & \cdots & 0 \\ 0 & \sigma_{2} & \cdots & \\ 0 & 0 & \cdots & \sigma_{m-1} 0 \end{array}\right). $$ A is called a Leslie matrix. (See references on demography for a summary of special properties of such matrices.) Note: For biological realism, we assume at least one \(\alpha_{i}>0\) and all \(\sigma_{i}>0\).) (b) The characteristic equation of a Leslie matrix is $$ p_{n}(\lambda)=\operatorname{det}(\lambda \mathbf{I}-\mathbf{A})=0 $$ Show that this leads to the equation $$ \lambda^{n}-\alpha_{1} \lambda^{n-1}-\alpha_{2} \sigma_{1} \lambda^{n-2}-\alpha_{3} \sigma_{1} \sigma_{2} \lambda^{n-3}-\cdots-\alpha_{n} \sigma_{1} \sigma_{2} \cdots \sigma_{n-1}=0 $$ (c) A Leslie matrix has a unique positive eigenvalue \(\lambda^{*}\). To prove this assertion, define the function $$ f(\lambda)=1-\frac{p_{n}(\lambda)}{\lambda^{n}} $$ Show that \(f(\lambda)\) is a monotone decreasing function with \(f(\lambda) \rightarrow+\infty\) for \(\lambda \rightarrow 0, f(\lambda) \rightarrow 0\) for \(\lambda \rightarrow \infty\). Conclude that there is a unique value of \(\lambda\) \(\lambda^{*}\) such that \(f\left(\lambda^{*}\right)=1\), and use this observation in proving the assertion. (Note: This can also be easily proved by Descartes' Rule of Signs.) (d) Suppose that \(\mathbf{v}^{*}\) is the eigenvector corresponding to \(\lambda^{*}\). Further suppose that \(\left|\lambda^{*}\right|\) is strictly greater than \(|\lambda|\) for any other eigenvalue \(\lambda\) of the Leslie matrix. Reason that successive generations will eventually produce an age distribution in which the ages are proportional to elements of \(\mathrm{v}^{*}\). This is called a stable age distribution. (Note: Some confusion in the literature stems from the fact that Leslie matrices are formulated for systems in which a census of the population is taken after births occur. Compare with the plant-seed example, which was done in this way in problem 19 but in another way in Section 1.2. A good reference on this point is M. R. Cullen (1985), Linear Models in Biology, Halstead Press, New York.

Step by step solution

01

Define the Population Vector

Let \(\mathbf{P}_{n}=\left(\begin{array}{c} p_{n}^{1} \ p_{n}^{2} \ \vdots \ p_{n}^{m} \end{array}\right)\) be the population vector at time step \(n\).

02

Transition Matrix Explanation

The transition matrix \(\mathbf{A}=\left(\begin{array}{cccc} \alpha_{1} & \alpha_{2} & \cdots & \alpha_{m} \ \sigma_{1} & 0 & \cdots & 0 \ 0 & \sigma_{2} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & \sigma_{m-1} \end{array}\right)\) describes the birthrate and survivorship.

03

Matrix Equation Derivation

To find the population vector at time step \(n+1\), \(\mathbf{P}_{n+1}\), we use the relationship: \(\mathbf{P}_{n+1} = \mathbf{A} \mathbf{P}_{n}\).

04

Characteristic Equation Setup

The characteristic equation is derived from the determinant equation \(\operatorname{det}(\lambda \mathbf{I} - \mathbf{A}) = 0\).

05

Matrix Determinant Calculation

The determinant \(\operatorname{det}(\lambda \mathbf{I} - \mathbf{A})\) yields the polynomial \(\lambda^{n} - \alpha_{1} \lambda^{n-1} - \alpha_{2} \sigma_{1} \lambda^{n-2} - \alpha_{3} \sigma_{1} \sigma_{2} \lambda^{n-3} - \cdots - \alpha_{n} \sigma_{1} \sigma_{2} \cdots \sigma_{n-1} = 0\).

06

Function Definition

Define the function \(f(\lambda) = 1 - \frac{p_{n}(\lambda)}{\lambda^{n}}\) to further analyze the unique positive eigenvalue.

07

Monotonicity and Limits

Show that \(f(\lambda)\) is a monotone decreasing function. As \(\lambda \rightarrow 0, f(\lambda) \rightarrow + \infty\), and as \(\lambda \rightarrow \infty, f(\lambda) \rightarrow 0\).

08

Unique Eigenvalue Conclusion

Conclude that there is a unique \(\lambda^{*}\) such that \(f(\lambda^{*}) = 1\).

09

Stable Age Distribution

Given that \(\left| \lambda^{*} \right| > \left| \lambda \right| \) for any other eigenvalue \((\lambda)\) of the Leslie matrix, reason that successive generations will produce an age distribution proportional to the eigenvector \((\mathbf{v}^{*})\).

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Key Concepts

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

Age-structured Population Dynamics

Age-structured population dynamics is an approach used in ecological and biological studies to model populations by age classes. This means breaking down the entire population into distinct groups based on age.
In our problem, we have age classes such as newborns, young individuals, and older individuals. The population within each age class at time step n is represented by the population vector \(\bf{P}_n \). This vector contains the number of individuals in each age class.
The dynamics of the population are described by the Leslie matrix, denoted as \(\bf{A} \). This matrix helps project the population into future time steps. It takes into account factors such as birth rates (\(\bf{\alpha} \)) from individuals in each age class and the survival rates (\(\bf{\sigma} \)) of individuals as they age.
This model is useful in predicting how the population will change over time and can be used for various applications, including conservation efforts, pest control, and understanding the impact of environmental changes on populations.

Eigenvalues and Eigenvectors

When dealing with matrices, eigenvalues and eigenvectors are key concepts. In the context of the Leslie matrix, they help understand long-term population dynamics.
Eigenvalues are special numbers associated with a matrix that provide insight into the stability and growth rate of populations. When you solve the characteristic equation det(\(\lambda \bf{I} - \bf{A} \)) = 0, you're finding the eigenvalues \(\bf{\lambda} \) of the Leslie matrix. These are solutions to a polynomial equation derived from the matrix.
Corresponding to each eigenvalue \(\bf{\lambda^*} \) is an eigenvector \(\bf{v^*} \). This eigenvector represents a stable age distribution, meaning that as generations pass, the proportion of the population in each age class becomes constant, matching the elements of \(\bf{v^*} \).
Typically, in a Leslie matrix, one eigenvalue is dominant (the largest in absolute magnitude), denoted as \(\bf{\lambda^*} \). This dominant eigenvalue is crucial since it dictates the long-term growth rate of the population. If \(\bf{\lambda^*} \) is greater than one, the population grows; if it's less than one, the population declines.

Characteristic Equation

The characteristic equation of a matrix is a polynomial equation obtained by subtracting \(\bf{\lambda} \) times the identity matrix \(\bf{I} \) from \(\bf{A} \) and then finding the determinant. For the Leslie matrix, the characteristic equation is written as: det(\(\lambda \bf{I} - \bf{A} \)) = 0.
Solving this equation gives us the eigenvalues \(\bf{\lambda} \). Its form for an \( n \)-age-class Leslie matrix looks like: \(\bf{\lambda^n - \alpha_1 \lambda^{n-1} - \alpha_2 \sigma_1 \lambda^{n-2} - ... - \alpha_n \sigma_1 \sigma_2 ... \sigma_{n-1} = 0} \).
The terms in the equation represent a combination of birth rates (\(\bf{\alpha} \)) and survival rates (\(\bf{\sigma} \)). The solution of this polynomial gives us the eigenvalues, which are crucial for analyzing the population's long-term behavior.
A unique positive eigenvalue \(\bf{\lambda^*} \) often determines the population's growth or decline rate. To further analyze \(\bf{\lambda} \), we use the function: \(\bf{f(\lambda) = 1 - \frac{p_n(\lambda)}{\lambda^n}} \). Since \(\bf{f(\lambda)} \) is a monotone decreasing function, and given its behavior at limits, there is a unique \(\bf{\lambda^*} \) that satisfies \(\bf{f(\lambda^*) = 1} \). This confirms \(\bf{\lambda^*} \) as a unique positive eigenvalue.