Problem 40 - Calculus for Biology and Medicine

Question

In Problems , find the population sizes for \(t=0,1,2, \ldots, 5\) for each recursion. $$ N_{t+1}=7 N_{t} \text { with } N_{0}=4 $$

Step-by-Step Solution

Answer

The population sizes are: \(N_0 = 4\), \(N_1 = 28\), \(N_2 = 196\), \(N_3 = 1372\), \(N_4 = 9604\), \(N_5 = 67228\).

Step 1: Understand the Recursion Formula

The recursion formula provided is \(N_{t+1} = 7N_t\). This means that each successive population size is obtained by multiplying the previous population size by 7.

Step 2: Initial Population

It's given that at time \(t=0\), the population size \(N_0\) is 4. This is our starting point.

Step 3: Compute Population for t=1

To find \(N_1\), use the formula: \(N_1 = 7N_0 = 7 \times 4 = 28\).

Step 4: Compute Population for t=2

For \(t=2\), calculate \(N_2 = 7N_1 = 7 \times 28 = 196\).

Step 5: Compute Population for t=3

Apply the recursion to find \(N_3 = 7N_2 = 7 \times 196 = 1372\).

Step 6: Compute Population for t=4

For \(t=4\), use \(N_4 = 7N_3 = 7 \times 1372 = 9604\).

Step 7: Compute Population for t=5

Finally, calculate \(N_5 = 7N_4 = 7 \times 9604 = 67228\).

Key Concepts

Population Modeling

Population modeling refers to the mathematical study of how populations change over time. It utilizes equations or formulas to predict the future size of a population based on current conditions or trends. The recursion formula used in our exercise is one such model. It describes the population size at any time based on the size at the previous time.

Population models often involve factors like birth rates, death rates, immigration, and emigration. These models help make predictions about population growth or decline. In the recital example provided, we are dealing with a geometric growth model, illustrated by the equation \( N_{t+1} = 7N_t \). Here, the population multiplies by a fixed number, 7, every time unit.

This type of model is generally used for populations that grow rapidly, such as bacteria, which under ideal conditions, can multiply exponentially. Understanding such models helps in fields like ecology, economics, and any domain where population size is a critical metric.

Step-by-Step Solution

Solving our population modeling recurrence involves a step-by-step evaluation. Here's a breakdown to understand how this helps ascertain populations at different times.

• **Initial Formula and Population**: We start with a given recursive formula \( N_{t+1} = 7N_t \) and initial population \( N_0 = 4 \).
• **Sequential Calculation**: Using the formula, we sequentially find the population sizes at times \( t=1, 2, \ldots, 5 \) by multiplying each previous population by 7. This involves recalculating at each time step which helps illustrate how rapid and exponential growth is quantified.
• **Computation Verification**: By calculating step-by-step for each \( t \), you can verify each result. For example, at \( t=1 \): \( N_1 = 7 \times 4 = 28 \). Continuing this process up to \( t=5 \), confirms understanding of the growth mechanism.

Calling this a step-by-step solution reveals the method of growing a population through recurrent multiplication, a key concept in models of geometric growth.

Initial Condition

An initial condition is the starting point of a problem, particularly in recursive formulas or differential equations. It provides the necessary information to solve the model. In our example, the initial condition is given as \( N_0 = 4 \), meaning the population size starts at 4 when \( t = 0 \).

Initial conditions are crucial because they set the stage for everything that follows. They guide the computation of all future terms in the sequence or model. In population studies, this can represent an observed starting population in a controlled environment or a natural setting.

Moreover, the importance of the initial condition cannot be overstated; alter the initial condition, and you'll fundamentally change the entire trajectory of the model. Imagine if the initial population were different; the resultant trajectory in a recursive formula would exponentially accelerate or decelerate growth outcomes. Hence, always pay close attention to these "starting points" when tackling population modeling problems.