Problem 43 - Calculus for Biology and Medicine

Question

\mathrm{\\{} I n ~ P r o b l e m s ~ 4 3 - 5 0 , ~ g r a p h ~ t h e ~ l i n e ~ \(\boldsymbol{N}_{t+1}=\boldsymbol{R} N_{t}\) in the \(\boldsymbol{N}_{t}-\boldsymbol{N}_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0,1\), and 2, for the given value of \(N_{0}\) $$ R=2, N_{0}=2 $$

Step-by-Step Solution

Answer

Plot the line through points (2, 4), (4, 8), and (8, 16).

Step 1: Understanding the Equation

The given equation is \(N_{t+1}=R \cdot N_{t}\). In this context, \(N_t\) represents the population at time \(t\), and \(N_{t+1}\) is the population at the next time step. \(R\) is the growth rate. We are tasked with finding \(N_{t+1}\) for given \(N_t\) values.

Step 2: Applying Initial Values

Given that \(R=2\) and \(N_{0}=2\), we start by calculating for \(t=0\). Substitute \(t=0\) and \(N_0=2\) into the equation: \[N_{1} = 2 \cdot 2 = 4.\]

Step 3: Calculating for t=1

Now, use the result from \(t=0\) to compute \(t=1\). We have \(N_1=4\) from the previous step. Substitute \(N_1\) into the equation: \[N_{2} = 2 \cdot 4 = 8.\]

Step 4: Calculating for t=2

Use earlier results to compute \(t=2\). With \(N_2=8\), substitute into the equation: \[N_{3} = 2 \cdot 8 = 16.\]

Step 5: Plotting the Points

Now, plot the points (\(N_0\), \(N_1\)), (\(N_1\), \(N_2\)), and (\(N_2\), \(N_3\)) in the \(N_t-N_{t+1}\) plane. These points are: \((2, 4)\), \((4, 8)\) and \((8, 16)\).

Step 6: Drawing the Line

Connect the plotted points to form a line. Since the relationship \(N_{t+1}=2N_{t}\) is linear, the points should align straight.

Key Concepts

Understanding Growth Rate in Population Dynamics

In the context of population dynamics, the growth rate is a crucial factor. It represents how quickly a population increases over time. For this exercise, the growth rate is indicated by the variable \(R\). Here, \(R = 2\), which signifies doubling the population with each time step. This means that at each new time interval, the population is twice what it was in the previous period.

The growth rate is especially important when modeling how populations evolve. A higher growth rate results in more rapid increases. Small changes in \(R\) can lead to significant differences in outcomes, making it a powerful predictor in models.

• Exponential Growth: When \(R > 1\), the population grows exponentially.
• Steady State: If \(R = 1\), the population remains constant.
• Decay: When \(R < 1\), the population declines over time.

Understanding the impact of the growth rate helps in predicting future population sizes and planning for resources accordingly. This exercise showcases how a growth rate of 2 exponentially increases the population in a simple model.

Graphing the Population Data

Graphing is a powerful tool that provides a visual representation of data, making patterns and trends easier to see. In this problem, plotting points on the \(N_t-N_{t+1}\) plane helps visualize the relationship specified by the equation \(N_{t+1} = R \cdot N_t\).

Let's break down the graphing steps for this exercise:

• Determine the initial points: For \(N_0 = 2\), calculate \(N_1 = 4\), \(N_2 = 8\), and \(N_3 = 16\).
• Plot the points: Place the coordinates \((2, 4)\), \((4, 8)\), and \((8, 16)\) onto the graph.
• Draw the connecting line: Since the function is linear, the points will align in a straight line. This line illustrates the consistent growth rate.

Graphing these points allows one to see that the pattern of growth is consistent and makes it easier to perceive how the population doubles at every step for this particular rate.

Using Linear Equations in Population Models

Linear equations, such as \(N_{t+1} = R \cdot N_t\), form the backbone of understanding consistent growth patterns in population models. These equations express relationships where change is proportional and predictable.

A linear relationship suggests that if you increase the length of time, the change in population remains proportional. This is why the line drawn through the points during graphing is straight—it symbolically represents the consistency and predictability of population growth, given a constant rate.

• The equation \(N_{t+1} = 2 \cdot N_t\) embodies a simple linear model. It's straightforward but effective in scenarios where the growth rate is steady.
• Using linear equations, even more complex models can also incorporate varying rates for more sophisticated forecasts.
• They provide a starting point for more complex studies involving alternating growth patterns.

Linear equations in population models serve as a clear way to anticipate changes and assess how populations can evolve based on different factors. Their simplicity is key in educating and providing insight into more intricate real-world dynamics.