Problem 35 - Calculus for Biology and Medicine

Question

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=5 N_{t} \text { with } N_{0}=1 $$

Step-by-Step Solution

Answer

For \( t = 0, 1, 2, 3, 4, 5 \), the populations are 1, 5, 25, 125, 625, 3125. The formula is \( N_t = 5^t \).

Step 1: Define the Base Case

Given the initial value, we start our solution with \( N_0 = 1 \). This is the population size at \( t = 0 \).

Step 2: Calculate Population for t=1

Use the provided recursion formula \( N_{t+1} = 5N_t \). Plug \( t = 0 \) into the formula: \( N_1 = 5 \times 1 = 5 \).

Step 3: Calculate Population for t=2

Apply the recursion formula using \( N_1 = 5 \): \( N_2 = 5 \times 5 = 25 \).

Step 4: Calculate Population for t=3

Use the value from the previous step, \( N_2 = 25 \): \( N_3 = 5 \times 25 = 125 \).

Step 5: Calculate Population for t=4

Use \( N_3 = 125 \) to find \( N_4 \): \( N_4 = 5 \times 125 = 625 \).

Step 6: Calculate Population for t=5

Continue the pattern with \( N_4 = 625 \): \( N_5 = 5 \times 625 = 3125 \).

Step 7: Formulate the General Equation

Recognize the pattern \( N_t = 5^t \). This exponential relationship represents the population size after \( t \) iterations.

Key Concepts

Population Modeling

Population modeling is a mathematical method used to represent how populations change over time. It's like having a blueprint for understanding the growth or decline of a group---this could be people, animals, or even cells. In calculus, we often use models to predict future population sizes.

• Growth and Decay Models: Some populations grow larger (like bacteria), while others may shrink (endangered species). Modeling helps us visualize these changes.
• Discrete and Continuous Models: Calculator methods vary based on whether changes happen at specific intervals (discrete) or continuously (like a stream of water).
• Applications: Population models are useful in biology, environmental science, and economics, helping plan for future needs and resources.

By understanding population models, one can predict how a population might evolve with changes in conditions or resources.

Exponential Growth

Exponential growth refers to a process where the growth rate of a population is proportional to its current size. In simple terms, the bigger the population, the faster it grows. This type of growth is common in nature under ideal conditions.

• Characteristics: Exponential growth is not linear; it accelerates over time if resources are unlimited.
• Mathematical Representation: Mathematically, this is often represented by an equation like the one from our exercise, where the next population size is a multiple of the current one.
• Graphical Representation: When graphed, exponential growth shows a curve that starts slowly, but then steeply rises, resembling a J-shape.

Understanding exponential growth helps in recognizing the potential for rapid increases in populations, which is critical for planning and resource management.

Recursion Formula

A recursion formula provides the instructions for building a sequence of numbers step by step. In the context of our problem, the recursion formula is essential for calculating population sizes over time frames.

• Basic Principle: Recursion uses a starting value (base case) and a rule to get the next value, repeatedly applying this rule to move forward in the sequence.
• Example from Exercise: Here, we started with an initial population ( N_0 = 1 d), and used the rule N_{t+1} = 5N_t to find subsequent population sizes.
• Benefits: Recursion is efficient for breaking complex problems into smaller, manageable steps, making it easier to compute sequences.

Understanding recursion is crucial for solving problems that require iterative calculation of values, presenting a methodical approach to problem-solving.