Problem 34 - 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 \(t\). $$ N_{t+1}=3 N_{t} \text { with } N_{0}=7 $$

Step-by-Step Solution

Answer

The population sizes are 7, 21, 63, 189, 567, 1701, and the function is \( N_t = 7 \times 3^t \).

Step 1: Identify Initial Population

The initial population size given is \( N_0 = 7 \). This provides the starting point for calculating the population sizes for subsequent time steps.

Step 2: Calculate Population for t=1

Using the recursion formula \( N_{t+1} = 3N_t \), calculate the population for \( t=1 \). Since \( N_0 = 7 \), we have \( N_1 = 3 \times 7 = 21 \).

Step 3: Calculate Population for t=2

Continue using the recursion formula: \( N_2 = 3N_1 = 3 \times 21 = 63 \).

Step 4: Calculate Population for t=3

Calculate \( N_3 \) using \( N_2 \): \( N_3 = 3N_2 = 3 \times 63 = 189 \).

Step 5: Calculate Population for t=4

For \( t=4 \), calculate \( N_4 = 3N_3 = 3 \times 189 = 567 \).

Step 6: Calculate Population for t=5

Finally, calculate \( N_5 = 3N_4 = 3 \times 567 = 1701 \).

Step 7: Write the General Formula for Nt

To express \( N_t \) as a function of \( t \), observe the pattern. With each step, \( N_t = 3^t \times 7 \). Thus, the general formula is \( N_t = 7 \times 3^t \).

Key Concepts

Recursion Formula in Population Dynamics

In population dynamics, a **recursion formula** is a mathematical tool used to predict future population sizes based on a current population size. It involves defining a relationship, typically of the form:

• \( N_{t+1} = f(N_t) \)

Here, \( f(N_t) \) is a function of the current population size \( N_t \) that determines the next population size, \( N_{t+1} \).
In the given exercise, the recursion formula specifies that each generation of population is three times the size of the previous:

• \( N_{t+1} = 3N_t \)

The beauty of recursion lies in its simplicity. We keep applying the same formula to calculate subsequent values. This ongoing sequence allows us to model growth easily step-by-step.
Recursion formulas are particularly helpful for understanding discrete systems in biology, economics, and computer science. They enable predictions of how systems evolve over time based on initial conditions and iterated processes.

Understanding Exponential Growth

Exponential growth describes a process where the quantity grows by a constant factor over equal increments of time. It's like snowballing—starting slow, then accelerating rapidly as time progresses. In mathematics, it's expressed by the formula:

• \( N_t = N_0 imes r^t \)

Where \( N_0 \) is the initial amount, \( r \) is the growth factor, and \( t \) represents the number of periods elapsed.
In our example, the population grows according to the rule \( N_{t+1} = 3N_t \). The factor of 3 indicates each population's size triples compared to its predecessor, making it classic exponential growth.
This type of increase is quite common in natural populations under ideal, resource-abundant conditions. So long as conditions remain constant—unlimited food, space, and no predators—the population can grow exponentially without bounds.
In reality, populations will often hit limits on resources, and the growth will slow down, showing the practical importance of modeling assumptions in predicting real-world dynamics.

Significance of Initial Population

The **initial population** \( N_0 \) represents the starting count from which growth begins. It serves as the baseline for all future calculations.
In our exercise, \( N_0 = 7 \) signifies the starting point. This is crucial because the initial population size directly affects each subsequent population size; the entire growth pattern and predictions are fundamentally tied to this initial number.

• Every calculated future population: \( N_1, N_2, N_3, \) etc., is derived from \( N_0 \).
• The general formula \( N_t = 7 imes 3^t \), continues to factor in \( N_0 \) alongside growth rate.

Understanding the role of the initial population helps clarify why the accuracy of initial measurements is so vital in dynamic models. In essence, it anchors the entire exponential growth pattern.
Misestimating the initial population can lead to significantly wrong forecasts. Hence, accurately determining \( N_0 \) is often the first critical step in population modeling.