Problem 2 - Calculus for Biology and Medicine

Question

Assume that a population is divided into three age classes and that \(80 \%\) of the females age 0 and \(10 \%\) of the females age present at time \(t\) survive until time \(t+1\). Assume further that females age 1 have an average of \(1.6\) female offspring and females age 2 have an average of \(3.9\) female offspring. If, at time 0 , the population consists of 1000 females age 0,100 females age 1 , and 20 females age 2 , find the Leslie matrix and the number of females in each age class at time \(3 .\)

Step-by-Step Solution

Answer

Age 0: 21396.8, Age 1: 5888, Age 2: 159.2.

Step 1: Define the Leslie Matrix

The Leslie matrix provides a way to model the growth of a population structured by age classes. For this problem, our Leslie matrix will be a 3x3 matrix where:\[ L = \begin{bmatrix} F_1 & F_2 & F_3 \ S_1 & 0 & 0 \ 0 & S_2 & 0 \end{bmatrix} \] where:\[ F_1 = 1.6, \quad F_2 = 3.9, \quad F_3 = 0 \ S_1 = 0.8, \quad S_2 = 0.1 \] The fertility rates \( F_1, F_2, F_3 \) represents the average number of offspring per female for ages 1 and 2, and there is no reproduction for age 0 so \( F_3 = 0 \). The survival rates \( S_1 \) and \( S_2 \) correspond to the percentage of females that survive from age 0 to age 1 and from age 1 to age 2, respectively.

Step 2: Construct the Leslie Matrix

Plug the values into the Leslie matrix template: \[ L = \begin{bmatrix} 1.6 & 3.9 & 0 \ 0.8 & 0 & 0 \ 0 & 0.1 & 0 \end{bmatrix} \] This matrix represents the transitions between age classes over one time period.

Step 3: Initial Population Vector

The initial population vector, \( \mathbf{N}_0 \), represents the population at time 0:\[ \mathbf{N}_0 = \begin{bmatrix} 1000 \ 100 \ 20 \end{bmatrix} \] This vector indicates there are 1000 females aged 0, 100 aged 1, and 20 aged 2.

Step 4: Calculate Population at Time 1

The population at time 1, \( \mathbf{N}_1 \), is found by multiplying the Leslie matrix by the initial population vector:\[ \mathbf{N}_1 = L \cdot \mathbf{N}_0 = \begin{bmatrix} 1.6 & 3.9 & 0 \ 0.8 & 0 & 0 \ 0 & 0.1 & 0 \end{bmatrix} \begin{bmatrix} 1000 \ 100 \ 20 \end{bmatrix} = \begin{bmatrix} 1.6 \times 1000 + 3.9 \times 100 + 0 \times 20 \ 0.8 \times 1000 + 0 \times 100 + 0 \times 20 \ 0 \times 1000 + 0.1 \times 100 + 0 \times 20 \end{bmatrix} = \begin{bmatrix} 1990 \ 800 \ 10 \end{bmatrix} \]

Step 5: Calculate Population at Time 2

Repeat the process to calculate \( \mathbf{N}_2 \):\[ \mathbf{N}_2 = L \cdot \mathbf{N}_1 = \begin{bmatrix} 1.6 & 3.9 & 0 \ 0.8 & 0 & 0 \ 0 & 0.1 & 0 \end{bmatrix} \begin{bmatrix} 1990 \ 800 \ 10 \end{bmatrix} = \begin{bmatrix} 1.6 \times 1990 + 3.9 \times 800 + 0 \times 10 \ 0.8 \times 1990 + 0 \times 800 + 0 \times 10 \ 0 \times 1990 + 0.1 \times 800 + 0 \times 10 \end{bmatrix} = \begin{bmatrix} 7360 \ 1592 \ 80 \end{bmatrix} \]

Step 6: Calculate Population at Time 3

Finally, calculate \( \mathbf{N}_3 \):\[ \mathbf{N}_3 = L \cdot \mathbf{N}_2 = \begin{bmatrix} 1.6 & 3.9 & 0 \ 0.8 & 0 & 0 \ 0 & 0.1 & 0 \end{bmatrix} \begin{bmatrix} 7360 \ 1592 \ 80 \end{bmatrix} = \begin{bmatrix} 1.6 \times 7360 + 3.9 \times 1592 + 0 \times 80 \ 0.8 \times 7360 + 0 \times 1592 + 0 \times 80 \ 0 \times 7360 + 0.1 \times 1592 + 0 \times 80 \end{bmatrix} = \begin{bmatrix} 21396.8 \ 5888 \ 159.2 \end{bmatrix} \]

Key Concepts

Population Modeling

Population modeling is a mathematical method to understand and predict the changes in population sizes over time. Through these models, scientists and researchers can analyze how different factors like birth rates, death rates, and age structure impact the growth or decline of a population. In this context, the Leslie matrix is a prominent tool used to model the dynamics of structured populations, specifically those divided into distinct age classes.

Using a matrix format, population modeling helps to capture:

• How many individuals are expected to survive to the next age class.
• The number of new individuals added to the population through reproduction.
• The progression of individuals through successive age classes over time.

Population models like the Leslie matrix offer a framework for simulating potential future scenarios in population dynamics and assessing the impacts of external factors such as changing environments or resource availability.

Age Class Structure

Age class structure refers to the division of a population into groups based on age. This classification allows for a more accurate analysis of how different segments within a population contribute to its overall growth or decline. For instance, in a population divided into three age classes—young, middle-age, and old—each class can have distinct survival and reproduction characteristics.

The Leslie matrix uses such an age class structure to monitor the transition of individuals from one age class to the next over time. Here's how it generally works:

• Individuals in the youngest age class mature to become part of the next age class over the specified time interval.
• Individuals in middle age classes can contribute to the population growth through reproduction.
• The oldest age class usually indicates no further age advancement or reproduction.

By understanding the age class structure, one can predict patterns within the population, making it easier to make informed decisions related to conservation, resource management, or urban planning.

Survival and Fertility Rates

Survival and fertility rates are crucial components of population modeling, significantly influencing population dynamics. In an age-structured model like the Leslie matrix, these rates determine the population's growth rate and its composition.

Survival rates are the probability that an individual in a given age class will survive to the next age class. For instance, if a survival rate is given as 80%, it means that 80 out of every 100 individuals are expected to move to the next age class.

• These rates can be affected by factors such as environmental conditions, predators, and disease.
• Accurate survival rates help predict the number of individuals that will mature over time.

Fertility rates indicate the average number of offspring produced by females in a particular age class. These rates decide the rate at which new individuals are added to the population.

• Fertility is often higher in middle-age classes and decreases in older age classes.
• Changes in fertility rates can signal shifts in population growth trends, enabling adjustments in resource allocation or management strategies.

Both survival and fertility rates are fundamental in calculating future population sizes and understanding the potential impacts of demographic changes.