Problem 71 - Calculus for Biology and Medicine

Question

Suppose that breeding occurs once a year and that a census is taken at the end of each breeding season. Assume that a population is divided into three age classes and that \(20 \%\) of the females age 0 and \(70 \%\) of the females age 1 survive until the end of the next breeding season. Assume further that females age 1 have an average of \(3.2\) female offspring and females age 2 have an average of \(1.7\) 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 age distribution at time 2 .

Step-by-Step Solution

Answer

At time 2, the age distribution is 2160 in age 0, 632 in age 1, and 280 in age 2.

Step 1: Identify Survival and Fertility Rates

We start by establishing the survival rates and fertility rates for each age class. - Survival rate from age 0 to 1: 0.2 (20%) - Survival rate from age 1 to 2: 0.7 (70%) - Females age 1 have 3.2 offspring. - Females age 2 have 1.7 offspring.

Step 2: Establish the Leslie Matrix

The Leslie matrix is used to project the age distribution of a population into the future. Its general form is:\[L = \begin{pmatrix} F_0 & F_1 & F_2 \ P_0 & 0 & 0 \ 0 & P_1 & 0 \end{pmatrix}\]Where:- \(F_0\) is the fertility of age 0 (zero since they do not reproduce).- \(F_1 = 3.2\)- \(F_2 = 1.7\)- \(P_0 = 0.2\) (survival from age 0 to 1)- \(P_1 = 0.7\) (survival from age 1 to 2)Thus, the Leslie matrix for this problem is: \[L = \begin{pmatrix} 0 & 3.2 & 1.7 \ 0.2 & 0 & 0 \ 0 & 0.7 & 0 \end{pmatrix}\]

Step 3: Define the Initial Population Vector

The initial population vector \( \mathbf{v_0} \) represents the number of females in each age class at time 0.\[\mathbf{v_0} = \begin{pmatrix} 2000 \ 800 \ 200 \end{pmatrix}\]

Step 4: Compute the Population at Time 1

To find the population distribution at time 1, multiply the Leslie matrix \(L\) by the initial population vector \(\mathbf{v_0}\):\[\mathbf{v_1} = L \cdot \mathbf{v_0} = \begin{pmatrix} 0 & 3.2 & 1.7 \ 0.2 & 0 & 0 \ 0 & 0.7 & 0 \end{pmatrix} \cdot \begin{pmatrix} 2000 \ 800 \ 200 \end{pmatrix} = \begin{pmatrix} 3160 \ 400 \ 560 \end{pmatrix}\]This gives us 3160 individuals in age 0, 400 individuals in age 1, and 560 individuals in age 2.

Step 5: Compute the Population at Time 2

Repeat the multiplication process to find the population at time 2 using the vector from time 1:\[\mathbf{v_2} = L \cdot \mathbf{v_1} = \begin{pmatrix} 0 & 3.2 & 1.7 \ 0.2 & 0 & 0 \ 0 & 0.7 & 0 \end{pmatrix} \cdot \begin{pmatrix} 3160 \ 400 \ 560 \end{pmatrix} = \begin{pmatrix} 2160 \ 632 \ 280 \end{pmatrix}\]At time 2, we have 2160 individuals in age 0, 632 individuals in age 1, and 280 individuals in age 2.

Key Concepts

Population Dynamics

Population dynamics is the study of how and why populations change over time. It is an essential part of ecology and helps us understand the interactions between organisms and their environment. In the context of this exercise, population dynamics involves examining the changing sizes of different age groups in a population of females across breeding seasons. This change is influenced by various factors including survival rates, which determine how many individuals live from one age to the next, and fertility rates, which indicate how many offspring are produced. Understanding population dynamics helps ecologists predict future population sizes, assess the health of a species, and make informed decisions for conservation and management. Monitoring these changes can indicate if a species is at risk of decline or if a population is growing rapidly, which may result in issues such as overpopulation.

Age Distribution

Age distribution refers to how individuals in a population are spread across different age classes. It allows us to see not only how many individuals there are in the population, but also their ages, which is crucial for managing species and ecosystems effectively. In this exercise, we deal with three age classes: age 0 (newborns), age 1 (intermediate aged), and age 2 (oldest). By looking at the initial population vector, we identify how many individuals exist in each class at the outset. Changes in age distribution can provide insights into the long-term viability of a population. For instance, a population with a skewed age distribution towards older individuals might face a decline if reproduction rates do not compensate for aging individuals. On the other hand, a younger age distribution suggests potential for growth.

Survival and Fertility Rates

Survival and fertility rates are key components in modeling population dynamics. They inform us about the life history traits of a species and are vital for building models such as the Leslie matrix. - **Survival Rate**: The probability of individuals surviving from one age class to the next. For example, in this exercise, the survival rate from age 0 to 1 is 0.2, meaning 20% of individuals survive. For age 1 to 2, it is 0.7, meaning 70% survive. - **Fertility Rate**: The average number of offspring produced by individuals in a particular age class. In our example, females aged 1 produce an average of 3.2 offspring, while those in age 2 have 1.7. These rates allow us to predict how populations might grow or shrink over time. They are not static and can be influenced by environmental changes, thus requiring regular updates for accurate models.

Linear Algebra in Biology

Linear algebra provides powerful tools that can be applied in biological contexts, particularly in modeling population dynamics. The Leslie matrix is a great example of how linear algebra can translate biological processes into a mathematical framework. The Leslie matrix is specifically designed for age-structured populations. It allows us to predict future age distributions by using survival and fertility rates, organized in a matrix format. By multiplying this matrix by a population vector, which contains the number of individuals in each age class, predictions about future populations can be made effortlessly. Linear algebra simplifies the complexity involved in population studies, enabling biologists to simulate various scenarios, such as changes in birth or survival rates, and see potential impacts without needing extensive observations. This helps in strategic planning and conservation efforts, showcasing the intersection of mathematics and biology.