Problem 69 - Calculus for Biology and Medicine

Question

In Problems , graph the line \(\frac{N_{t}}{N_{t+1}}=\frac{1}{R}\) in the \(N_{t}-\frac{N_{t}}{N_{t+1}}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, \frac{N_{t}}{N_{t+1}}\right), t=0\), 1, 2, for the given value of \(N_{0}\). Find the parent-offspring ratio. $$ R=3, N_{0}=2 $$

Step-by-Step Solution

Answer

The line is horizontal at \( \frac{1}{3} \), and points are at \((2, \frac{1}{3}),(6, \frac{1}{3}),(18, \frac{1}{3})\). The parent-offspring ratio is \( \frac{1}{3} \).

Step 1: Understand the Equation

The given problem requires you to graph the equation \( \frac{N_{t}}{N_{t+1}} = \frac{1}{R} \) for \( R = 3 \). This is a relationship between the population at time \( t \) and time \( t+1 \). It inversely depends on \( R \), a reproduction rate or growth factor.

Step 2: Set Up the Graphing Axis

We are asked to graph the line in the \( N_t - \frac{N_t}{N_{t+1}} \) plane. Here, \( N_t \) is the horizontal axis and \( \frac{N_t}{N_{t+1}} \) is the vertical axis. We set \( \frac{N_t}{N_{t+1}} = \frac{1}{3} \), meaning the vertical value will be constant at \( \frac{1}{3} \) for any point \( N_t \).

Step 3: Graph the Line

On a graph, draw a horizontal line parallel to the x-axis that intersects the y-axis at \( \frac{1}{3} \). This line represents all points where \( \frac{N_t}{N_{t+1}} = \frac{1}{3} \).

Step 4: Calculate Points for t = 0, 1, 2

Given \( N_0 = 2 \):- For \( t = 0 \), \( \frac{N_{0}}{N_{1}} = \frac{1}{3} \). Solving for \( N_1 \), we get \( N_1 = 6 \).- For \( t = 1 \), \( \frac{N_{1}}{N_{2}} = \frac{1}{3} \). Solving for \( N_2 \), we get \( N_2 = 18 \).- For \( t = 2 \), \( \frac{N_{2}}{N_{3}} = \frac{1}{3} \). Solving for \( N_3 \), we get \( N_3 = 54 \).

Step 5: Plot the Points on the Graph

Plot the points \( (2, \frac{1}{3}), (6, \frac{1}{3}), (18, \frac{1}{3}) \) on the graph along the horizontal line at \( \frac{N_t}{N_{t+1}} = \frac{1}{3} \). These points represent the population at \( t = 0, 1, 2 \), respectively.

Step 6: Determine the Parent-Offspring Ratio

The parent-offspring ratio is given by \( \frac{N_{t}}{N_{t+1}} \), which in this case is constant and equals \( \frac{1}{3} \). This means every parent in one generation corresponds to 3 offspring in the next generation given \( R = 3 \).

Key Concepts

Understanding Ratio Equations

Ratio equations play a crucial role in population dynamics, providing insights into how populations change over time. In a ratio equation like \( \frac{N_{t}}{N_{t+1}} = \frac{1}{R} \), we are describing the relationship between the current population size at time \( t \), denoted as \( N_t \), and the population at the next time point \( t+1 \), represented by \( N_{t+1} \). The ratio \( \frac{1}{R} \) reflects the influence of the growth factor \( R \), which quantitatively describes how each generation relates to the following one.

When \( R = 3 \), the equation \( \frac{N_{t}}{N_{t+1}} = \frac{1}{3} \) signifies that each segment of the initial population will triple in the next generation. Thus, this constant ratio helps predict future population sizes based solely on the current size. By understanding and utilizing these ratio equations, we can forecast long-term trends in population growth.

Exploring Population Modeling

Population modeling is an essential aspect of studying population dynamics. It involves creating mathematical models that represent population changes over time.

In our example, the model expressed by \( \frac{N_{t}}{N_{t+1}} = \frac{1}{R} \) serves as a simplistic linear model to understand how populations evolve.\( N_t \) and \( N_{t+1} \) denote population size at two successive time steps. While this model assumes constant factors over time, it offers a means to visualize and predict growth patterns in different scenarios provided the reproductive rate is known.

Improvements to this model often involve incorporating more complex relationships or variables, such as natural disasters, constraints in resources, or diseases. Despite its simplicity, this basic model highlights the foundational aspects of population dynamics: birth, growth, and eventual sustainability.

Delving into Growth Factor Analysis

Growth factor analysis is fundamental in understanding how populations expand or contract. In the context of this exercise, the growth factor \( R \) directly influences the population's size at each time step.

• For a growth factor \( R > 1 \), the population is expected to grow exponentially over time.
• A growth factor \( R = 1 \) indicates a stable population where each generation is the same size as the previous one.
• If \( R < 1 \), the population is expected to decline, which might suggest factors like increased mortality, limited resources, or environmental changes.

When calculating specific generations as seen in the exercise, one must understand how this factor translates into real-world applications. For instance, if the initial population \( N_0 \) is 2 with \( R = 3 \), each successive generation will continue multiplying by 3, illustrating rapid growth.
Analyzing growth factors allows us to not only predict future populations but also to create strategic interventions when population control or conservation is necessary.