Problem 27 - Calculus for Biology and Medicine

Question

Find the recursion for a population that triples in size every unit of time and that has 10 individuals at time \(0 .\)

Step-by-Step Solution

Answer

The recurrence relation is \( P(n+1) = 3P(n) \) with \( P(0) = 10 \).

Step 1: Identify the Initial Condition

We are given that the population at time 0 is 10. Let this be represented by \( P(0) = 10 \).

Step 2: Determine the Growth Pattern

The problem states that the population triples every unit of time. This means if the population at time \( n \) is \( P(n) \), then after one time unit, the population is 3 times \( P(n) \).

Step 3: Establish the Recurrence Relation

From the growth pattern, we can write the recurrence relation as \( P(n+1) = 3P(n) \). This indicates that each successive population size is three times the previous size.

Step 4: Formulate the General Solution

With the recurrence relation \( P(n+1) = 3P(n) \) and initial condition \( P(0) = 10 \), we can find the population at any time \( n \) using the formula: \( P(n) = 10 \cdot 3^n \).

Key Concepts

Initial Condition

In the context of population growth, the initial condition plays a crucial role as it establishes the starting point for any analysis. It's like the starting line in a race, providing us with a known beginning from which all future calculations will spring.
For our exercise, the initial condition is given as the population count at time zero. This is represented by the equation: \( P(0) = 10 \). This means that at the starting point, we have 10 individuals in the population.
This foundation allows us to map out the changes as time progresses and helps set the framework for understanding how the population will grow over time.

Recurrence Relation

The recurrence relation is a mathematical model that describes how the population evolves over discrete time intervals. It's like a set of instructions that tell us how to calculate the future based on the present.
In our exercise, the population triples every time unit. If we represent this with \( P(n) \), where \( n \) is the time elapsed, then the recurrence relation is expressed as \( P(n+1) = 3P(n) \). This indicates that every new time point's population is three times the size of the previous point.
Recurrence relations are a powerful way to model processes that evolve step-by-step, making them particularly useful in situations like modeling population growth.

Growth Pattern

Understanding the growth pattern in population studies involves identifying how the number of individuals increases or decreases over time.
For this exercise, the growth pattern is exponential, characterized by the population tripling each unit of time. This is a common pattern in populations with abundant resources and no limiting factors, indicating a rapid increase.
In our formulated solution, the equation \( P(n) = 10 \cdot 3^n \) embodies this growth pattern, showing how population size \( P(n) \) is derived from the initial condition and the recurrent multiplication factor over time. Observing this pattern allows scientists and researchers to predict future population sizes and make informed decisions.