Question

How many eigenvalues are associated with a Leslie matrix?

A) One
B) Two
C) Three
D) Four

Answer 1

Four eigenvalues are associated with a Leslie matrix.

Step-by-step explanation:

A Leslie matrix is a square matrix used in population dynamics to model the age-structured populations of renewable resources, such as fish or deer. These matrices have a unique set of eigenvalues, which are the roots of the characteristic equation of the matrix. The eigenvalues provide important information about the population dynamics, such as the growth rate, stability, and extinction probability.

In general, a Leslie matrix has four eigenvalues, which can be real or complex. The four eigenvalues are typically labeled as follows:

1. The dominant eigenvalue (λ1): This is the largest eigenvalue in absolute value and represents the net reproductive rate of the population. It is also known as the spectral radius of the Leslie matrix.

2. The subdominant eigenvalue (λ2): This is the second largest eigenvalue in absolute value and represents a secondary population growth rate. It may be positive, negative, or complex depending on the population dynamics.

3. The neutral eigenvalue (λ3): This is an eigenvalue with zero real part and represents a stable equilibrium point where the population size remains constant over time. It may be complex if there is oscillatory behavior in the population dynamics.

4. The extinction eigenvalue (λ4): This is an eigenvalue with negative real part and represents a stable equilibrium point where the population size decreases over time towards extinction. It may be complex if there is oscillatory behavior in the population dynamics before extinction occurs.

The distribution of these eigenvalues provides insights into the population dynamics and can help to identify potential issues such as overexploitation, underpopulation, or oscillatory behavior that could lead to extinction. Understanding these eigenvalues is therefore critical for effective resource management and conservation strategies.

Answer 2

A Leslie matrix, commonly used in population dynamics, typically has three eigenvalues. These eigenvalues represent the population growth rates for different age classes in the population. Thus, the correct answer is option C) Three

Step-by-step explanation:

The correct answer is C) Three. A Leslie matrix is a square matrix used in population dynamics, particularly in the study of age-structured populations. It is associated with three eigenvalues. These eigenvalues represent the population growth rates for the different age classes in the population.

In more detail, the Leslie matrix is used to model the transitions of individuals between different age groups over discrete time intervals. The eigenvalues of the Leslie matrix correspond to the population growth rates associated with each age class. These eigenvalues are crucial in determining the long-term behavior of the population, such as whether it will grow, decline, or remain stable over time.

Understanding the eigenvalues of the Leslie matrix is fundamental in predicting and analyzing population dynamics, making it a valuable tool in ecological and demographic studies.

Thus, the correct answer is option C) Three