Problem 75 - Calculus for Biology and Medicine

Question

Suppose that $$ L=\left[\begin{array}{ll} 2 & 4 \\ 0.3 & 0 \end{array}\right] $$ 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 complex indicating a cyclic population structure. The biological implication suggests alternating population sizes and no stable age distribution.

Step 1: Define the Characteristic Equation

The eigenvalues of matrix \( L \) are the solutions to the characteristic equation \( \det(L - \lambda I) = 0 \), where \( \lambda \) is an eigenvalue and \( I \) is the identity matrix of the same size as \( L \).

Step 2: Calculate the Characteristic Polynomial

Compute \( L - \lambda I = \left[ \begin{array}{cc} 2-\lambda & 4 \ 0.3 & -\lambda \end{array} \right] \) and find its determinant: \( \det(L - \lambda I) = (2-\lambda)(-\lambda) - (4)(0.3) \).

Step 3: Simplify the Characteristic Polynomial

Expand the expression from Step 2 to get the characteristic polynomial: \( -\lambda^2 + 2\lambda - 1.2 \).

Step 4: Solve for Eigenvalues

Solve the quadratic equation from Step 3: \(-\lambda^2 + 2\lambda - 1.2 = 0\) using the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -1, b = 2, c = -1.2 \).

Step 5: Calculating Eigenvalues

Using the quadratic formula, the eigenvalues are \( \lambda_1 = \frac{2 + \sqrt{2^2 - 4(-1)(-1.2)}}{2(-1)} \) and \( \lambda_2 = \frac{2 - \sqrt{2^2 - 4(-1)(-1.2)}}{2(-1)} \). First, calculate the discriminant: \( 4 - 4.8 = -0.8 \). Since it is negative, the quadratic equation has complex solutions.

Step 6: Biological Interpretation of Larger Eigenvalue

In Leslie matrices, the larger eigenvalue \( \lambda_1 \) represents the intrinsic growth rate of the population. A complex eigenvalue indicates oscillations or cyclic behavior in population dynamics, which might imply alternating cycles of increase and decrease in age classes.

Step 7: Find Stable Age Distribution

Since the eigenvalues are complex, finding a real-valued stable age distribution analytically is not straightforward. Generally, stable age distributions are associated with real, positive eigenvalues.

Key Concepts

Eigenvalues

Eigenvalues are fundamental in understanding matrix transformations and have critical biological implications, especially in population dynamics using a Leslie matrix. An eigenvalue is a scalar that, when multiplied by an eigenvector, results in a directional scaling of a vector, maintaining the vector's orientation in a linear transformation. Let's break it down for you:

• The Leslie Matrix provides a model to predict changes in age-structured populations by showing how one age class contributes to the next.
• The mathematical condition for determining eigenvalues is the characteristic equation: \( \det(L - \lambda I) = 0 \), where \( L \) is our matrix and \( I \) is the identity matrix.

In this exercise, the eigenvalues of the Leslie matrix, found by solving the characteristic polynomial, are crucial for understanding the population trends. Here, complex eigenvalues would indicate oscillatory dynamics rather than steady growth, highlighting subtler changes in population age structure over time.

Characteristic Polynomial

A characteristic polynomial emerges from the elements of a matrix and is crucial for finding eigenvalues. It serves as a bridge between linear transformations (like those modeled by the Leslie matrix) and their underlying algebraic properties.

To develop the characteristic polynomial:

• Subtract \( \lambda I \) from the matrix \( L \) to form \( L - \lambda I \).
• Calculate the determinant of this new matrix to get \( \det(L - \lambda I) \).

From the original exercise, after performing the subtraction and determinant calculation, the characteristic polynomial simplifies to \( -\lambda^2 + 2\lambda - 1.2 \). Solving this equation helps us find the eigenvalues, which reflect the intrinsic properties of the population’s growth dynamics, possibly indicating stability or oscillatory behavior.

Stable Age Distribution

The concept of a stable age distribution is closely linked with the largest eigenvalue of a Leslie matrix, which is usually real and positive if the population is projected for long-term stability. A stable age distribution means that the relative proportions of individuals in different age classes remain constant as the population grows.

However, challenges arise when eigenvalues are complex. Normally, a real and dominant eigenvalue indicates smooth, exponential growth leading to a stable distribution.

With complex eigenvalues, as seen in this problem, predicting a stable age distribution analytically becomes complex since it pairs with changing cycles of population growth and decline. Yet, numerical simulations or methods might help understand these dynamics by iterating the Leslie matrix over time to observe eventual trends in the age distribution, even if these trends aren't straightforward due to oscillatory behavior.