Problem 9 - Calculus for Biology and Medicine

Question

give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. . \(N_{0}=2\); population quadruples every 30 minutes; one unit of time is 15 minutes

Step-by-Step Solution

Answer

The formula for the population at time \(t\) is \(N(t) = 2 \cdot 2^t\).

Step 1: Understanding the Initial Condition

The problem states that the initial population, denoted as \(N_0\), is 2. This is the population at time \(t=0\). This forms the basis for our population growth calculation.

Step 2: Identifying the Growth Factor

The population quadruples, which means it becomes four times larger over a set period. That period is 30 minutes. This implies that every 30 minutes the population is multiplied by 4.

Step 3: Determining the Unit of Time

The problem states one unit of time is 15 minutes. Therefore, when we say the population quadruples every 30 minutes, it relates to two units of 15 minutes (since 30 minutes = 2 times 15 minutes). Therefore, every 2 time units, the population quadruples.

Step 4: Finding the Growth Rate per Time Unit

Since the population quadruples or is multiplied by 4 every 2 time units, we need to find the growth rate per single time unit. Let the growth factor per time unit be denoted by \(r\). It must satisfy the equation \(r^2 = 4\). Solving for \(r\), we have \(r = \sqrt{4} = 2\). Thus, the population doubles every 15 minutes, or every unit of time.

Step 5: Formulating the Population Expression

Given that we start with a population of 2 and it doubles every time unit, the population at time \(t\) can be expressed using the formula: \[ N(t) = N_0 \cdot r^t = 2 \cdot 2^t \] This accounts for the initial population and the doubling every 15 minutes.

Key Concepts

Understanding Population Dynamics

Population dynamics is a fascinating field that explores how populations change over time. In this context, we are looking at how a population grows under certain conditions. Specifically, it examines the changes in size and structure of populations due to birth rates, death rates, and migration.
Population dynamics are governed by various factors:

• Initial Population: This is the starting point or initial condition. In our problem, this is denoted as \(N_0\), which is 2 at time \(t=0\).
• Growth Pattern: The rate at which the population increases or decreases. Here, the population quadruples every 30 minutes. Understanding how quickly a population grows helps in determining future sizes.

These factors are crucial for predicting future population sizes under various scenarios, making them essential in ecology, economics, and resource management.

Growth Rate Calculation

Calculating the growth rate is a vital step in understanding population changes. The growth rate tells us by how much the population changes over a set period of time.
In our exercise, we have a known condition that the population quadruples every 30 minutes. The time unit given is 15 minutes, which means each 30-minute period encompasses two time units.
To determine the growth rate per time unit, you:

• Understand that quadrupling every 30 minutes implies multiplication by 4 for that duration.
• Because 30 minutes equals two time units, you must find the growth per single time unit. Let this growth rate be \(r\).
• Solve the equation \(r^2 = 4\) to find the single-unit growth, giving \(r = \sqrt{4} = 2\).

This calculation shows that the population doubles every 15 minutes, or every unit of time.

Introducing the Discrete Time Model

The discrete time model is useful for modeling populations that change at specific intervals. In the exercise, we use a time unit of 15 minutes.
In this model:

• The population size at any given time \(t\) is calculated using specific formula based methods, which take into account the consistent growth rate at each time interval.
• We use the formula \(N(t) = N_0 \cdot r^t\) where \(N_0\) is the initial population (2 in this case), and \(r\) is the growth factor per 15-minute interval (which we found to be 2).
• This model assumes that changes only happen at the end of each interval, making it simple yet powerful for predicting future population sizes based on past growth.

The discrete nature of this model aligns well with populations that do not change continuously but rather in stages or increments at regular intervals.