Problem 45 - Calculus for Biology and Medicine

Question

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ 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 \(N_{0}\) $$ R=3, N_{0}=1 $$

Step-by-Step Solution

Answer

Plot points (1, 3) and (3, 9) on a line with slope 3 in the \(N_t - N_{t+1}\) graph.

Step 1: Understand the Relationship

The given relationship is \( N_{t+1} = R N_{t} \). This equation suggests that the population at time \( t+1 \) is equal to the population at time \( t \) multiplied by the growth rate \( R \). In this case, \( R = 3 \).

Step 2: Calculate \( N_1 \, \& \, N_2 \)

Start with \( N_0 = 1 \).- Calculate \( N_1 = RN_0 = 3 \times 1 = 3 \).- Use \( N_1 \) to find \( N_2 = RN_1 = 3 \times 3 = 9 \).Thus, \( N_1 = 3 \) and \( N_2 = 9 \).

Step 3: Determine Points for Plotting

Use the calculated values \( N_0 = 1 \), \( N_1 = 3 \), and \( N_2 = 9 \) to determine the points.- At \( t = 0 \), the point is \((N_0, N_1) = (1, 3)\).- At \( t = 1 \), the point is \((N_1, N_2) = (3, 9)\).- At \( t = 2 \), while \( N_{t+1} \) isn't needed, one more iteration provides \( N_2 \).

Step 4: Define the Line on the Graph

In the \( N_t - N_{t+1} \) plane, plot the line of the equation \( N_{t+1} = 3N_t \). This line passes through the origin \((0,0)\) with a slope of \( R = 3 \), rising 3 units in \( N_{t+1} \) for every unit increase in \( N_t \).

Step 5: Locate and Mark the Points

Plot and mark the points obtained:- \((1, 3)\): From \( N_0 = 1 \) leading to \( N_1 = 3 \).- \((3, 9)\): From \( N_1 = 3 \) leading to \( N_2 = 9 \).

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept within population dynamics and it can be described using the equation \( N_{t+1} = R N_t \). This equation shows how a population can expand rapidly over time when each individual, on average, contributes more than one offspring. The growth factor \( R \) plays a crucial role here.
In this scenario, we have \( R = 3 \), which means that the population triples at each time step. Starting with an initial population, denoted as \( N_0 \), the sequence progresses by multiplying the current population by \( R \) to obtain the next value: \( N_1, N_2, ... \).
For example:

• If \( N_0 = 1 \), then \( N_1 = R \times N_0 = 3 \times 1 = 3 \).
• Next, \( N_2 = R \times N_1 = 3 \times 3 = 9 \).

The chain reaction this set off results in a rapid escalation in numbers, manifesting the exponential nature of growth controlled by a constant multiplication factor.

Linear Graph Analysis

In linear graph analysis, visualization becomes a key tool to understand relationships defined by mathematical models. The graph of the equation \( N_{t+1} = 3N_t \) in the \( N_t - N_{t+1} \) plane provides a clear depiction of population changes over successive time steps.
Plotting the initial given values helps in mapping these changes. For this specific case:

• The point \( (N_0, N_1) = (1, 3) \) shows how the population moves from its initial state \( N_0 \) to its next state \( N_1 \).
• Similarly, the point \( (N_1, N_2) = (3, 9) \) indicates the transition from \( N_1 \) to \( N_2 \), further illustrating the pattern.

On this graph, the line defined by the equation is characterized by a slope, or steepness, of \( R = 3 \). This tells us that for every increase of one unit in \( N_t \), \( N_{t+1} \) increases by three units. This linear relation helps identify the nature of growth across different time intervals.

Mathematical Modeling

Mathematical modeling leverages equations to simulate real-world phenomena like population growth, aiming to predict future behavior. Here, the model used is \( N_{t+1} = R N_t \), where each term in the sequence represents the population size at a subsequent time point.
By implementing these calculations, predictions about the system's future state are made, offering insights into trends and potential outcomes. With \( R = 3 \), our model predicts what happens if each generation of a population had three times as many members as the previous.

• The importance lies in its ability to provide assumptions which can be adjusted to match real observable scenarios, making the model adaptable.
• In practical applications, such models help in decision-making processes related to resource allocations, understanding how populations can change under different conditions and constraints.

This approach of building mathematical models is fundamental in theoretical and applied sciences, assisting in grasping the complexities of numerous systems.