Problem 38 - 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 \(t\). $$ N_{t+1}=\frac{3}{2} N_{t} \text { with } N_{0}=32 $$

Step-by-Step Solution

Answer

The population sizes are: 32, 48, 72, 108, 162, 243. The function is \(N_t = 32 \times \left(\frac{3}{2}\right)^t\).

Step 1: Understand the Given Problem

We have a recursive equation \(N_{t+1} = \frac{3}{2}N_t\) with the initial value \(N_0 = 32\). Our job is to find the population sizes at \(t = 0, 1, 2, \ldots, 5\) and also express \(N_t\) as a function of \(t\).

Step 2: Calculate Population Size at t = 0

We are given \(N_0 = 32\), so the population size at \(t = 0\) is already known. This serves as the starting point for the recursion.

Step 3: Calculate Population Size at t = 1

Apply the recursive formula: \(N_1 = \frac{3}{2}N_0\). Substitute \(N_0 = 32\): \(N_1 = \frac{3}{2} \times 32 = 48\).

Step 4: Calculate Population Size at t = 2

Using the recursion, compute \(N_2 = \frac{3}{2}N_1\). From previous step, \(N_1 = 48\), so \(N_2 = \frac{3}{2} \times 48 = 72\).

Step 5: Calculate Population Size at t = 3

Again, use the recursion: \(N_3 = \frac{3}{2}N_2\). With \(N_2 = 72\), it follows that \(N_3 = \frac{3}{2} \times 72 = 108\).

Step 6: Calculate Population Size at t = 4

For \(t = 4\), \(N_4 = \frac{3}{2}N_3\). Substitute \(N_3 = 108\): \(N_4 = \frac{3}{2} \times 108 = 162\).

Step 7: Calculate Population Size at t = 5

Use the same process. \(N_5 = \frac{3}{2}N_4\), with \(N_4 = 162\), giving \(N_5 = \frac{3}{2} \times 162 = 243\).

Step 8: Expressing N_t as a Function of t

Recognize the recursive relation as a geometric progression. The formula for \(N_t\) is: \(N_t = 32 \times \left(\frac{3}{2}\right)^t\). This formula describes the population at any time \(t\).

Key Concepts

Recursive Equations

A recursive equation establishes a relationship where the next term in a sequence or progression is derived from the previous terms. By using recursive formulas, we can describe complex processes in a more systematic manner. In the given exercise, the recursive equation is \(N_{t+1} = \frac{3}{2}N_t\). This means that to find the population size at the next time step, we multiply the current population size by \(\frac{3}{2}\). This relationship continues from a known starting point, \(N_0 = 32\), throughout the specified time steps. Recursive relationships are powerful especially in population modeling, as they allow for the step-by-step computation of population values over time, leading to a better understanding of growth dynamics.

Geometric Progression

Geometric progression is a sequence where each term is derived by multiplying the previous term by a constant factor, known as the common ratio. Recognizing geometric progression is essential in both mathematics and real-world applications, like population models. In our exercise, the formula \(N_t = 32 \times \left(\frac{3}{2}\right)^t\) illustrates a geometric sequence. Here, \(32\) is the initial term, and \(\frac{3}{2}\) is the common ratio. This sequence grows exponentially, meaning each term rapidly increases compared to the last. The convenience of geometric progression is its concise formulation, allowing us to express the population size at any time step \(t\) without having to calculate every preceding term individually.

Population Dynamics

Population dynamics explore the variations in population sizes and how they change over time due to births, deaths, immigration, and emigration. In discrete population models, like the one presented in the exercise, populations are examined at specific intervals or time steps. Here, the recursive formula \(N_{t+1} = \frac{3}{2}N_t\) represents a scenario where the population increases by 50% each time step. This exercise reveals exponential growth which is characteristic of many natural populations when resources are not limited. Such models are crucial for understanding ecological balances, predicting future population sizes, and making informed decisions in conservation and urban planning based on how populations might grow or decline.