Problem 11 - Calculus for Biology and Medicine

Question

Assume that the Leslie matrix is $$ L=\left[\begin{array}{ll} 0 & 2 \\ 0.6 & 0 \end{array}\right] $$ Suppose that, at time \(t=0, N_{0}(0)=5\) and \(N_{1}(0)=1\). Find the population vectors for \(t=0,1,2, \ldots, 10 .\) Compute the successive ratios $$ q_{0}(t)=\frac{N_{0}(t)}{N_{0}(t-1)} \text { and } q_{1}(t)=\frac{N_{1}(t)}{N_{1}(t-1)} $$ for \(t=1,2, \ldots, 10 .\) Do \(q_{0}(t)\) and \(q_{1}(t)\) converge? Compute the fraction of females age 0 for \(t=0,1, \ldots, 10 .\) Describe the longterm behavior of \(q_{0}(t)\).

Step-by-Step Solution

Answer

The successive ratios \( q_0(t) \) and \( q_1(t) \) converge to the dominant eigenvalue. The fraction of age 0 females can be computed for each \( t \).

Step 1: Initial Population Vector

The initial population vector at time \( t=0 \) is given by \( \mathbf{N}(0) = \begin{bmatrix} N_0(0) \ N_1(0) \end{bmatrix} = \begin{bmatrix} 5 \ 1 \end{bmatrix} \).

Step 2: Computation of Population Vectors

To find the population vector at time \( t=1 \), we use the Leslie matrix \( L \) to compute \( \mathbf{N}(1) = L \cdot \mathbf{N}(0) = \begin{bmatrix} 0 & 2 \ 0.6 & 0 \end{bmatrix} \cdot \begin{bmatrix} 5 \ 1 \end{bmatrix} = \begin{bmatrix} 2 \ 3 \end{bmatrix} \).Repeat this process to find \( \mathbf{N}(t) \) for \( t=2 \) to \( t=10 \).

Step 3: Calculate Successive Ratios

Find the ratios \( q_0(t) = \frac{N_0(t)}{N_0(t-1)} \) and \( q_1(t) = \frac{N_1(t)}{N_1(t-1)} \) for each \( t = 1, 2, \ldots, 10 \) using the population vectors calculated in Step 2.

Step 4: Check for Convergence

Monitor the ratios \( q_0(t) \) and \( q_1(t) \) for \( t=1 \) to \( 10 \) to determine if they converge to specific values. This involves checking if the ratios approach a constant value as \( t \) increases.

Step 5: Compute Fractions of Age 0 Females

For each \( t = 0, 1, \ldots, 10 \), compute the fraction of females age 0 using the formula \( \frac{N_0(t)}{N_0(t) + N_1(t)} \).

Step 6: Longterm Behavior of \( q_0(t) \)

Based on the convergence behavior from Step 4, describe the longterm behavior of \( q_0(t) \). Typically, the Leslie matrix model will show that this ratio converges to an eigenvalue of the matrix, indicating stable population growth.

Key Concepts

Population Vector

In the context of the Leslie matrix model, a population vector represents the distribution of a population across different age classes at a given time. The initial population vector, denoted as \( \mathbf{N}(0) \), is simply an ordered pair of numbers that shows how many individuals are in each age category at the start of the observation period.
In our example, the population vector is \( \mathbf{N}(0) = \begin{bmatrix} 5 \, 1 \end{bmatrix} \). This means that at time \( t = 0 \), there are 5 females aged 0 and 1 female aged 1.
The development of the population over time is predicted by applying the Leslie matrix to the current population vector, effectively updating it to \( \mathbf{N}(t+1) \). Through repeated application of this method, we can predict the population distribution at future times \( t = 1 \), \( t = 2 \), and so on.

• Use the initial population vector to calculate future population vectors by multiplying with the Leslie matrix.
• The population vector at each time \( t \) helps in analyzing the population structure over a series of time intervals.

Successive Ratios

Successive ratios are derived from population vectors and give insights into the population's growth trends. They are calculated for each age group at each time step and can signal whether a population is stabilizing or changing dynamics.
For instance, the ratio \( q_0(t) \) is calculated by \( \frac{N_0(t)}{N_0(t-1)} \), representing the growth or decline of the group aged 0 between time \( t-1 \) and time \( t \). Similarly, \( q_1(t) \) is \( \frac{N_1(t)}{N_1(t-1)} \).
The point is to identify patterns like convergence, which indicates the system is reaching a steady state, often related to eigenvalues of the Leslie matrix. If these ratios stabilize over time for large \( t \), the population might be approaching a stable growth rate or population structure.

• Successive ratios highlight momentary growth patterns.
• Convergence of these ratios can signify stable demographic conditions over the long term.

Eigenvalue

Eigenvalues are critical in predicting the long-term behavior of dynamics governed by the Leslie matrix. Specifically, in population studies, the dominant eigenvalue often represents the steady-state growth rate of the entire population.
When examining successive ratios or long-term distributions, convergence to a specific value often is explained by this leading eigenvalue. The behavior of \( q_0(t) \) or similar ratios ultimately reflects the eigenvalue because, over long durations, these ratios converge to it. In essence, eigenvalues provide a mathematical foundation for understanding when and how a population achieves stability.

• The dominant eigenvalue indicates potential long-term stable growth rates.
• It helps explain convergence in successive ratios and population dynamics.

Females Age Distribution

The distribution of females across different age classes helps in understanding demographic dynamics. Specifically, it can highlight whether younger or older age groups are becoming more predominant over time.
Using fractions, we compute the proportion of females aged 0 relative to the entire population, given by \( \frac{N_0(t)}{N_0(t) + N_1(t)} \). Analyzing this fraction over time reveals shifts in age distribution and aids in predicting population renewal rates.
Consistently high fractions of age 0 females suggest a robust increase in new births or youth predominance. Conversely, a decrease would imply aging or reduced birth rates. This kind of analysis serves as an early warning or predictive indicator of future population changes.

• Fractions of age groups provide insight into population renewal and stability.
• Monitoring changes in age distribution helps anticipate demographic transitions.