Problem 1 - Calculus for Biology and Medicine

Question

Assume that a population is divided into three age classes and that \(20 \%\) of the females age 0 and \(70 \%\) of the females age present at time \(t\) survive to time \(t+1\). Assume further that females age 1 have an average of \(2.4\) female offspring and females age 2 have an average of \(1.3\) female offspring. If, at time 0 , the population consists of 2000 females age 0,800 females age 1 . and 200 females age 2, find the Leslie matrix and the number of females in each age class at time \(2 .\)

Step-by-Step Solution

Answer

The Leslie matrix is \( \begin{bmatrix} 0 & 2.4 & 1.3 \\ 0.2 & 0 & 0 \\ 0 & 0.7 & 0 \end{bmatrix} \). At time 2, the population is \( \begin{bmatrix} 1712 \\ 506 \\ 280 \end{bmatrix} \).

Step 1: Understand the Problem

We have a population of females divided into three age groups: 0, 1, and 2. We know the survival rate and the average offspring produced by each age group. We need to find the Leslie matrix that models the population dynamics and use this matrix to calculate the number of females in each age group at time 2.

Step 2: Construct the Leslie Matrix

The Leslie matrix is a structured matrix used in population dynamics that contains fertility rates in the first row and survival rates on the sub-diagonal. For our problem, this matrix is a 3x3 matrix:\[ \begin{bmatrix} 0 & 2.4 & 1.3 \ 0.2 & 0 & 0 \ 0 & 0.7 & 0 \end{bmatrix} \] Here the first row contains the fertility rates of the age groups, the second row shows that 20% of age 0 females survive to age 1, and the third row shows that 70% of age 1 females survive to age 2.

Step 3: Define the Initial Population Vector

The initial population, specified as a vector \( \mathbf{n}_0 \), is composed of the number of females in each age group at time 0:\[ \mathbf{n}_0 = \begin{bmatrix} 2000 \ 800 \ 200 \end{bmatrix} \] This vector will be used to calculate the future population vector using the Leslie matrix.

Step 4: Calculate the Population at Time 1

Multiply the Leslie matrix by the initial population vector to find the population at time 1, denoted as \( \mathbf{n}_1 \):\[ \mathbf{n}_1 = \begin{bmatrix} 0 & 2.4 & 1.3 \ 0.2 & 0 & 0 \ 0 & 0.7 & 0 \end{bmatrix} \times \begin{bmatrix} 2000 \ 800 \ 200 \end{bmatrix} = \begin{bmatrix} 2.4 \times 800 + 1.3 \times 200 \ 0.2 \times 2000 \ 0.7 \times 800 \end{bmatrix} \]This results in \( \mathbf{n}_1 = \begin{bmatrix} 2530 \ 400 \ 560 \end{bmatrix} \).

Step 5: Calculate the Population at Time 2

Multiply the Leslie matrix by the population vector at time 1 to find the population at time 2, denoted as \( \mathbf{n}_2 \):\[ \mathbf{n}_2 = \begin{bmatrix} 0 & 2.4 & 1.3 \ 0.2 & 0 & 0 \ 0 & 0.7 & 0 \end{bmatrix} \times \begin{bmatrix} 2530 \ 400 \ 560 \end{bmatrix} = \begin{bmatrix} 2.4 \times 400 + 1.3 \times 560 \ 0.2 \times 2530 \ 0.7 \times 400 \end{bmatrix} \]This results in \( \mathbf{n}_2 = \begin{bmatrix} 1712 \ 506 \ 280 \end{bmatrix} \). This means that at time 2, there are approximately 1712 females age 0, 506 females age 1, and 280 females age 2.

Key Concepts

Understanding Population Dynamics

Population dynamics is the study of how populations of living organisms change over time, including changes in their size, structure, and distribution. It involves examining how factors such as birth rates, death rates, immigration, and emigration influence these changes.

In the context of the exercise, population dynamics focus on understanding how the population of females in different age classes changes over time based on specific survival and fertility rates. By using the Leslie matrix, a mathematical model that predicts future population changes, we can simulate how the population will evolve over observed time intervals. The matrix allows us to apply certain assumptions regarding the birth and survival probabilities to see their effects on population changes.

Key aspects of population dynamics include:

• Tracking the population size and age distribution over time.
• Understanding the impact of various rates like survival and fertility on population changes.
• Using tools like the Leslie matrix to predict future population compositions.

Each of these elements helps us better understand and predict how a particular species or group will fare under certain conditions, ultimately offering insights into species conservation, management strategies, and ecological impacts.

Deciphering Survival Rate

The survival rate is the probability that an individual will continue to live from one time period to the next. This concept is integral to predicting population dynamics because it directly affects how many individuals from each age group will remain in the population to reproduce or continue to age.

In the given exercise, the survival rate is depicted in the Leslie matrix on the sub-diagonal. This means:

• 20% of females age 0 survive to become age 1.
• 70% of females age 1 survive to become age 2.

These percentages help determine how many individuals will continue to the next life stage, thereby influencing the future composition of the population.

By understanding the survival rate, researchers and ecologists can:

• Predict how long individuals will likely stay in the population.
• Estimate future population sizes and age structure.
• Formulate strategies to enhance the survival of vulnerable age groups.

Incorporating accurate survival rates into population models is crucial for realistic predictions and effective management strategies.

Exploring Fertility Rate

Fertility rate is the average number of offspring produced by an individual in a given age group during a specific time period. It plays a crucial role in determining the growth rate of a population since it dictates how many new individuals enter the population.

In the exercise's Leslie matrix, fertility rates are featured in the first row, indicating how many offspring females in each age group contribute to the next generation:

• Females age 1 produce 2.4 female offspring on average.
• Females age 2 produce 1.3 female offspring on average.

These numbers are critical in understanding the reproductive capacity of the population and its potential growth or decline.

Fertility rate analysis helps in:

• Anticipating future population sizes and growth trends.
• Designing conservation programs to boost reproduction in endangered species.
• Understanding the reproductive behaviors that contribute to population sustainability or growth.

By clearly analyzing fertility rates, researchers can gain insights into how to support and manage populations effectively, ensuring their long-term viability.