Problem 46 - 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 \(N_{0}\) $$ R=4, N_{0}=2 $$

Step-by-Step Solution

Answer

Plot points (2, 8) and (8, 32) on the plane.

Step 1: Understand the Equation

The given recursive relation is \(N_{t+1} = R \times N_{t}\). This means for each subsequent time \(t\), the value \(N_{t+1}\) is obtained by multiplying the current value \(N_{t}\) by the constant \(R\).

Step 2: Substitute Initial Values in the Equation

Start by calculating \(N_{1}\) using \(N_{0} = 2\) and \(R = 4\). Substitute 2 into the relation: \[ N_{1} = R \times N_{0} = 4 \times 2 = 8. \]

Step 3: Calculate Subsequent Values

Now calculate \(N_{2}\) using \(N_{1} = 8\): \[ N_{2} = R \times N_{1} = 4 \times 8 = 32. \]

Step 4: Plot Points in the Plane

List the calculated points based on the values of \(N_{t}\) and \(N_{t+1}\):- For \(t=0\), the point is \((N_{0}, N_{1}) = (2, 8)\).- For \(t=1\), the point is \((N_{1}, N_{2}) = (8, 32)\).Plot these points on the \(N_{t}\)-\(N_{t+1}\) plane.

Key Concepts

Recursive Sequences: Understanding the Basics

A recursive sequence is a way of defining a series of numbers where each term is derived from one or more previous terms using a fixed rule. In the given exercise, the sequence follows the formula \( N_{t+1} = R \times N_{t} \). This indicates each term in the sequence is generated by multiplying the previous term \( N_{t} \) by a constant \( R \).
To put it simply, you start from an initial term, which in this exercise is given as \( N_{0} = 2 \). You'll then apply the recursive rule to find succeeding terms.
For example:

• At \( t=0 \), \( N_{0} = 2 \).
• At \( t=1 \), we calculate \( N_{1} = 4 \times N_{0} = 8 \).
• At \( t=2 \), \( N_{2} = 4 \times N_{1} = 32 \).

Recursive sequences are powerful tools in mathematics because they allow us to express complex iterative processes simply and are often used in computer algorithms and problem-solving contexts.

Graphing Functions: Visualizing Mathematical Relationships

Graphing is a crucial skill that helps visualize how different mathematical concepts are related. In the context of this exercise, graphing \( N_{t+1} = R \times N_{t} \) allows us to see how the sequence progresses.
The graph is plotted in the \( N_{t} - N_{t+1} \) plane. Here's how to go about it:

• Plot each calculated point as a pair \((N_{t}, N_{t+1})\).
• For our specific example, start with the points: \((2, 8)\) for \( t=0 \) and \((8, 32)\) for \( t=1 \).
• The line connecting these points shows the rate of change determined by the value \( R \) which is \( 4 \).

By connecting the plotted points, you get a visual representation of the pattern dictated by the recursion rule. The steeper the line, the larger the multiplier, showing how quickly values grow or shrink. Graphs make it easier to predict future values and understand the behavior of sequences at a glance.

Biological Applications of Calculus: Population Dynamics

Calculus is more than just an abstract mathematical concept—it's immensely useful in real-world applications, including biology. One key application is modeling population dynamics.
In population studies, recursive sequences similar to the equation \( N_{t+1} = R \times N_{t} \) are used to model how a population of organisms grows over time given a constant rate of growth \( R \).
This kind of model is instrumental in:

• Predicting the future size of a population under constant growth conditions.
• Assessing the impact of environmental changes or resources on growth rates.
• Developing conservation or resource management strategies.

Calculus, combined with recursive sequences, helps biologists understand and predict changes in populations, contributing to strategic decisions. This exercise displays a simplified model of such real-world applications, demonstrating how mathematical principles form the backbone in understanding and solving practical problems.