Problem 24 - Calculus for Biology and Medicine

Question

Find the recursion for a population that triples in size every unit of time and that has 6 individuals at time \(0 .\)

Step-by-Step Solution

Answer

The recursion is \( P(0) = 6 \) and \( P(n+1) = 3 \times P(n) \).

Step 1: Understanding the Problem

We need to find a recursive formula for a population that triples in size every time unit. Initially, the population is 6 individuals at time 0.

Step 2: Define the Base Case

The base case refers to the initial condition provided. Here, the population at time 0 is given as 6. We define the population at time 0: \( P(0) = 6 \).

Step 3: Develop the Recursive Formula

A recursive formula expresses the population at time \( n+1 \) in terms of the population at time \( n \). Given that the population triples every unit of time, we have: \( P(n+1) = 3 \times P(n) \).

Step 4: Combine Base Case and Recursive Formula

Combine the base case and the recursive formula to give a complete recursion relation: \( P(0) = 6 \) and \( P(n+1) = 3 \times P(n) \) for \( n \geq 0 \). This fully describes the growth of the population over time.

Key Concepts

Understanding Recursion

When we talk about recursion, we're diving into a process where the outcome at one stage helps to determine the outcome at the next one. Imagine looking in a mirror that reflects another mirror; it goes on and on! That's similar to how recursion works in mathematical and computer science concepts. In population growth, recursion becomes useful because each new population size depends directly on the size of the population at the previous time period.

• Recursion often relies on initial conditions, also known as the base case, to kickstart the process.
• In the population model, knowing the number of individuals at time 0 helps us predict the future counts.

It’s like starting a journey with a map. We need the first point, and recursion tells us how to get to the next step based on where we currently are.

Recursive Formula in Population Growth

The recursive formula acts like a blueprint for determining future population sizes based on the current size. In our exercise, we learned that the population triples every unit of time. This is the key relationship used in the recursive formula.
To break it down:

• The given information is the initial population at time 0, which is 6 individuals.
• Each subsequent population size can be found using the formula: \( P(n+1) = 3 \times P(n) \). This implies that the next population size is three times the current size.

Think of it as baking a cake where each layer is three times the size of the previous one. You need the current layer (population at the current time) to calculate the next layer (population at the next time). This recursive formula is powerful because it can be applied repeatedly to predict long-term outcomes.

Exploring Exponential Growth

Exponential growth occurs when the growth rate of a quantity is proportional to its current size. This is a hallmark of processes where the larger the system gets, the faster it grows, like our population in the exercise.
In mathematical terms:

• If you start with a base population of 6 and it triples every period, what you have is exponential growth, which can be more generally represented by \( P(n) = P(0) \times r^n \), where \( r \) is the growth factor.
• In our case, \( r = 3 \), meaning each unit time the population size is 3 times its previous size.

Exponential growth is fascinating and slightly counterintuitive because it starts slow but can quickly explode into enormous numbers. Think of it like a snowball rolling down a hill; it's small at first, but with every turn, it gathers more snow, growing rapidly.