Problem 23 - Calculus for Biology and Medicine

Question

. Find the recursion for a population that doubles in size every unit of time and that has 11 individuals at time \(0 .\)

Step-by-Step Solution

Answer

The recursion is \( P_0 = 11 \) and \( P_{t+1} = 2P_t \).

Step 1: Understand the Problem

We need to set up a recursive formula for a population that doubles every unit of time. We're given that the initial population at time \( t=0 \) is 11 individuals.

Step 2: Identify the Base Case

The base case is given by the initial condition of the problem. At time \( t=0 \), the population \( P_0 \) is 11. So, \( P_0 = 11 \).

Step 3: Develop the Recursive Formula

Since the population doubles every unit of time, the population at time \( t+1 \) is twice the population at time \( t \). This can be expressed as \( P_{t+1} = 2P_t \).

Step 4: Combine Formulas to Write the Recurrence Relation

Combine the base case and the recursive formula: The recurrence relation for the population is \( P_0 = 11 \) and \( P_{t+1} = 2P_t \) for \( t \geq 0 \).

Key Concepts

Population Growth

Population growth refers to the change in the size of a population over time. It is a key concept in understanding how species, including humans, expand their numbers. In our example, we are looking at a population that doubles every unit of time. This signifies exponential growth as each time period results in a multiplication of the existing number of individuals. Such growth can be observed in natural environments, typically under conditions where resources are plentiful and there are no significant constraints or predators. In mathematical terms, exponential growth models capture this rapid increase, helping us predict population sizes at future time steps.

Recursive Formula

A recursive formula is a tool used to generate a sequence of numbers where each term is defined in relation to the preceding term(s). It begins with one or more initial terms, and each subsequent term is derived using a given rule or formula. In our example, because the population doubles each time unit, we set up a recursive formula. This formula allows us to find the population at any time without computing explicitly each preceding time step:

• The initial population at time zero is given as 11.
• The formula, which doubles the population, is used to find the population at each subsequent time step.

Through the recursive formula, we can efficiently calculate the population at any desired time without repetitive calculations.

Base Case

The base case in a recursive process is the starting point or initial condition from which all further calculations begin. It provides the essential foundation for the sequence generated by the recursive formula. In our exercise, the base case describes the population at time zero:

• The population at time zero, denoted as \( P_0 \), is 11 individuals.
• This stable point is pivotal as it enables the recursive formula to be applied effectively to determine future populations.

Without specifying the base case, the recursive method could not commence, as there would be no benchmark for calculating subsequent values in the sequence.

Recurrence Relation

A recurrence relation is an equation that recursively defines a sequence. It specifies how each term in the sequence depends on its preceding term(s). In our context, the recurrence relation links the population at each time step with its predecessor. Here's how:

• The initial term, or base case, is \( P_0 = 11 \).
• The relation \( P_{t+1} = 2P_t \) indicates that each term is twice the previous term. This reflects the pattern of doubling population every time unit.

Recurrence relations are fundamental in discrete mathematics for describing sequences and are particularly useful for representing processes that unfold over time without requiring explicit formulations for every step.