Problem 78 - Calculus for Biology and Medicine

Question

Suppose that $$ L=\left[\begin{array}{ll} 0 & 5 \\ 0.9 & 0 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.

Step-by-Step Solution

Answer

The eigenvalues are \(\sqrt{4.5}\) and \(-\sqrt{4.5}\). The larger eigenvalue, \(\sqrt{4.5}\), indicates population growth. The stable age distribution is proportional to \([1, 0.9/\sqrt{4.5}]\).

Step 1: Find the Characteristic Polynomial

To find the eigenvalues of the matrix \( L \), we first need to calculate its characteristic polynomial. The characteristic polynomial of a matrix \( A \) is given by \( \det(A - \lambda I) \), where \( I \) is the identity matrix. For \( L \), this becomes: \[\det\left(\begin{bmatrix}0 - \lambda & 5 \0.9 & 0 - \lambda \end{bmatrix}\right) = 0\]Calculating the determinant:\[-\lambda(-\lambda) - (5)(0.9) = \lambda^2 - 4.5 = 0\]

Step 2: Solve the Characteristic Equation

From the characteristic polynomial, we get the equation:\[\lambda^2 - 4.5 = 0\]Solve for \( \lambda \):\[\lambda^2 = 4.5 \\lambda = \pm \sqrt{4.5}\] The eigenvalues are \( \lambda_1 = \sqrt{4.5} \) and \( \lambda_2 = -\sqrt{4.5} \).

Step 3: Interpret the Larger Eigenvalue

In biological terms, the larger eigenvalue of the Leslie matrix represents the growth rate of the population. If \( \lambda_1 = \sqrt{4.5} \) \( (\approx 2.12) \) is greater than 1, it indicates that the population is growing. This eigenvalue can be interpreted as the long-term growth rate when the population's age distribution stabilizes.

Step 4: Calculate the Stable Age Distribution

The stable age distribution corresponds to the eigenvector associated with the largest eigenvalue. For \( \lambda_1 = \sqrt{4.5} \), we solve \( (L - \lambda_1 I)v = 0 \) to find the eigenvector.Substitute \( \lambda_1 = \sqrt{4.5} \) into the matrix equation:\[\begin{bmatrix}-\sqrt{4.5} & 5 \0.9 & -\sqrt{4.5} \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\]This leads to the equations:\(-\sqrt{4.5}x + 5y = 0\) and \(0.9x - \sqrt{4.5}y = 0\).Solving these, we find that the eigenvector \( v = \begin{bmatrix} 1 \ \frac{0.9}{\sqrt{4.5}} \end{bmatrix} \). Normalize this vector to form the stable age distribution.

Key Concepts

Eigenvalues

When we talk about eigenvalues in the context of a Leslie matrix, we're referring to specific scalar values that give us crucial information about the population's dynamics. To find these, we use the characteristic polynomial of the Leslie matrix. For matrix \( L \), we form the equation \( \det(L - \lambda I) = 0 \), which simplifies to \( \lambda^2 - 4.5 = 0 \). By solving this, we arrive at two eigenvalues: \( \lambda_1 = \sqrt{4.5} \) and \( \lambda_2 = -\sqrt{4.5} \).

These eigenvalues tell us important things. Here are some insights:

• The larger eigenvalue \( \lambda_1 \) holds a special biological significance.
• It indicates the potential growth rate of the population over time.

We generally pay more attention to the larger eigenvalue here because it directly impacts whether the population grows, stays the same, or declines.

Stable Age Distribution

The stable age distribution is a key concept in population dynamics, especially when analyzing a Leslie matrix. It tells us the proportion of individuals in each age class that the population will stabilize at over time, provided the conditions remain constant. To find this distribution, we calculate the eigenvector associated with the largest eigenvalue of the Leslie matrix. For our matrix \( L \), this means solving \( (L - \lambda_1 I)v = 0 \). After substituting \( \lambda_1 = \sqrt{4.5} \), we find the eigenvector \( v \). By simplifying and normalizing this eigenvector, we derive the stable age distribution.

More specifically:

• The eigenvector \( v = \begin{bmatrix} 1 \, \frac{0.9}{\sqrt{4.5}} \end{bmatrix} \)
• This tells us that for every individual in the first age class, there are approximately \( \frac{0.9}{\sqrt{4.5}} \) in the second age class.

The idea is that over time, the population will naturally settle into this distribution pattern, assuming the birth and survival rates stay consistent.

Population Growth Rate

The population growth rate is one of the most vital metrics we can derive from the Leslie matrix using its eigenvalues. Particularly, the largest eigenvalue \( \lambda_1 \) provides insight into the potential growth trend. If \( \lambda_1 > 1 \), this indicates that the population is generally expanding. Let's break it down:

• \( \lambda_1 = \sqrt{4.5} \approx 2.12 \). Since this is greater than 1, it signals growth.
• This growth is exponential, meaning the population could more than double each generation if conditions remain stable.

The practical interpretation is that as long as the age distribution stabilizes, the population will expand at a predictable and constant rate as described by the dominant eigenvalue. This provides an essential basis for long-term population planning and management.