Problem 33 - 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 \(t\). $$ N_{t+1}=3 N_{t} \text { with } N_{0}=2 $$

Step-by-Step Solution

Answer

The population sizes are: 2, 6, 18, 54, 162, 486. The function is \( N_t = 2 \cdot 3^t \).

Step 1: Understanding the Problem

We have a recursive equation: \( N_{t+1} = 3N_t \) with an initial population size \( N_0 = 2 \). We need to find the population sizes from \( t = 0 \) to \( t = 5 \) and then express \( N_t \) as a function of \( t \).

Step 2: Calculate Population for Each Time Point

Using the recursive formula, we calculate:- At \( t = 0 \), \( N_0 = 2 \).- At \( t = 1 \), \( N_1 = 3N_0 = 3 \times 2 = 6 \).- At \( t = 2 \), \( N_2 = 3N_1 = 3 \times 6 = 18 \).- At \( t = 3 \), \( N_3 = 3N_2 = 3 \times 18 = 54 \).- At \( t = 4 \), \( N_4 = 3N_3 = 3 \times 54 = 162 \).- At \( t = 5 \), \( N_5 = 3N_4 = 3 \times 162 = 486 \).

Step 3: Write the General Formula for \(N_t\)

The recursive relation shows that each \( N_{t+1} \) is 3 times \( N_t \). This suggests a geometric progression. The general form for a geometric series is \( N_t = a \cdot r^t \), where \( a \) is the initial term and \( r \) is the common ratio.Here, \( a = 2 \) and \( r = 3 \). Therefore, \( N_t = 2 \times 3^t \).

Key Concepts

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence can model phenomena that grow or shrink at a constant percentage rate. For example, if you have an initial number, say 2, and a common ratio of 3, the sequence will be:

• 1st term: 2
• 2nd term: 2 x 3 = 6
• 3rd term: 6 x 3 = 18
• 4th term: 18 x 3 = 54
• 5th term: 54 x 3 = 162

Notice how each term is 3 times the previous one. In mathematical terms, for our example, the sequence follows the formula \( N_t = 2 \times 3^t \). Here, 2 is the initial term \( a \), and 3 is the common ratio \( r \). When dealing with sequences like this, understanding the starting value and how each term relates to the previous one is crucial. This pattern can describe many real-world processes, like growth populations, investments, and more.

Recursive Formula

A recursive formula defines each term of a sequence using the preceding terms. It's a step-by-step process where the next term is derived from the current term. For example, with our given problem, the recursive formula is \( N_{t+1} = 3N_t \). This tells us that each new term equals three times the previous term.

• Start with an initial value \( N_0 = 2 \).
• To find \( N_1 \), we compute \( N_1 = 3 \times N_0 = 6 \).

This recursive approach gives a direct way to build sequences progressively. It's particularly useful in computing scenarios where calculating each value independently is essential, like programming or creating prediction models. By repeatedly applying the operation defined by the recursive formula, you can explore how things evolve over time.

Population Dynamics

Population dynamics is a study of how populations change over time. This field can involve modeling using mathematical formulas, like recursive equations or geometric progressions, to predict future sizes or behaviors of populations, whether they are animal, human, or even data clusters.In our example, the recursive equation \( N_{t+1} = 3N_t \) models a population multiplying by a factor of 3 every cycle. The initial population \( N_0 \) starts at 2, and over each time step, this technique helps us understand the pattern of change efficiently:

• At \( t = 0 \), we start with 2 individuals.
• At \( t = 1 \), there are 6.
• At \( t = 2 \), it grows to 18, and so on.

Population dynamics can be affected by many factors like resources, space, and competition. Understanding these equations can give insight into predictable patterns or irregular changes in real-world systems.