Problem 31 - Calculus for Biology and Medicine

Question

In Problems 31-42, 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 \(t\). $$ N_{t+1}=2 N_{t} \text { with } N_{0}=3 $$

Step-by-Step Solution

Answer

The population sizes are 3, 6, 12, 24, 48, 96, and \(N_t = 3 \times 2^t\).

Step 1: Identify Initial Population

The problem provides the initial population size. We have \(N_0 = 3\). This is the starting point from which we will calculate subsequent population sizes.

Step 2: Apply the Recursion Formula

The recursion formula is \(N_{t+1} = 2N_t\). This means that each successive population size is twice the previous population size.

Step 3: Calculate Population for t=1

Using the initial population: \(N_1 = 2N_0 = 2 \times 3 = 6\).

Step 4: Calculate Population for t=2

Using the population size for \(t=1\): \(N_2 = 2N_1 = 2 \times 6 = 12\).

Step 5: Calculate Population for t=3

Using the population size for \(t=2\): \(N_3 = 2N_2 = 2 \times 12 = 24\).

Step 6: Calculate Population for t=4

Using the population size for \(t=3\): \(N_4 = 2N_3 = 2 \times 24 = 48\).

Step 7: Calculate Population for t=5

Using the population size for \(t=4\): \(N_5 = 2N_4 = 2 \times 48 = 96\).

Step 8: Generalize the Recursion to a Function

Given that each population size is twice the previous one starting from \(N_0 = 3\), we can write \(N_t = 3 imes 2^t\) as the general formula.

Key Concepts

Recursion Formula

A recursion formula is an equation that describes how to calculate the next term in a sequence from the previous term. In the context of population growth, it allows us to determine future population sizes based on a known starting point. In the exercise given, the recursion formula is:
\[N_{t+1} = 2N_t\]This formula indicates that each generation's population is simply double the size of the previous one. This doubling process happens step by step, or iteratively, allowing us to find population size at any given 'step' or time, "t". By using recursion, rather than calculating individually each time, we save time and make the process systematic. An understanding of recursion is fundamental in exploring dynamic systems that evolve in repeated cycles.

Exponential Growth

Exponential growth refers to an increase that becomes more rapid in proportion to the growing total number or size. The recursion formula \(N_{t+1} = 2N_t\) is a classic example of exponential growth because the population grows by a multiplicative factor rather than an additive amount.
When using exponential growth in calculations, the growth can be represented as:

• Doubling time: For every unit of time, the population doubles.
• Rate: Constant multiplying factor, which in this case is 2.

In general, populations experiencing exponential growth will follow a formula like \(N_t = N_0 \times (growth\_factor)^t\), where 'growth_factor' is the rate of change. With exponential growth, small initial differences can lead to large differences over time, highlighting why many natural processes follow this model.

Initial Population

The concept of the initial population, denoted as \(N_0\), is the starting point for any recursion or exponential growth calculation. A clear understanding of the initial population size is crucial because it impacts all subsequent calculations.
In the context of our original exercise, the initial population at time \(t=0\) is given as 3, i.e., \(N_0 = 3\). This value serves as the baseline to apply the recursion formula.

• Starting Point: Establishes the baseline from which growth is calculated.
• Importance: Any error in the initial population will amplify in successive steps.
• Common Misunderstanding: Ensure clarity about \(N_0\) as starting populations often need external verification.

Understanding this starting point ensures that each subsequent stage of growth is correctly interpreted, making \(N_0\) a vital component of mathematical modeling in population studies.