Problem 31 - Calculus for Biology and Medicine

Question

Suppose that the per capita growth rate of a population is \(3 \% ;\) that is, if \(N(t)\) denotes the population size at time \(t\), then $$\frac{1}{N} \frac{d N}{d t}=0.03$$ Suppose also that the population size at time \(t=4\) is equal to 100\. Use a linear approximation to compute the population size at time \(t=4.1\).

Step-by-Step Solution

Answer

The population size at time \( t=4.1 \) is approximately 100.3.

Step 1: Understand the Given Information

We are given that the per capita growth rate of a population is 3%, expressed as \( \frac{1}{N} \frac{dN}{dt} = 0.03 \). We also know that the population size at \( t = 4 \), \( N(4) \), is 100.

Step 2: Relate Growth Rate to Derivative

The equation \( \frac{1}{N} \frac{dN}{dt} = 0.03 \) implies that \( \frac{dN}{dt} = 0.03N \). This differential equation describes the rate of change of the population size, which can be used for approximation.

Step 3: Apply Linear Approximation Formula

For a linear approximation, the population at time \( t=4.1 \) can be approximated using \( N(t + \Delta t) \approx N(t) + \frac{dN}{dt}(t) \cdot \Delta t \). We have \( N(4) = 100 \) and \( \Delta t = 0.1 \).

Step 4: Calculate the Derivative

Calculate \( \frac{dN}{dt} \) at \( t=4 \): \( \frac{dN}{dt} = 0.03 \cdot 100 = 3 \).

Step 5: Compute Population Size at \( t=4.1 \)

Using the linear approximation formula: \( N(4.1) \approx 100 + 3 \cdot 0.1 = 100 + 0.3 = 100.3 \).

Key Concepts

Per Capita Growth Rate

The per capita growth rate is an essential concept in population dynamics. It measures how much each individual in a population contributes to the overall growth rate. Simply put, it's the percentage increase or decrease of a population per individual. When we talk about a 3% per capita growth rate, we mean every member of the population is theoretically boosting the population by 3% individually. In mathematical terms, for a population size represented by \(N(t)\), the per capita growth rate is expressed as \(\frac{1}{N} \frac{dN}{dt} = 0.03\). This equation tells us directly how much the population is expected to change at that specific rate.

• A positive per capita growth rate indicates a growing population.
• A negative rate suggests the population is shrinking.

Understanding this rate is crucial as it provides insights into how sustainable or dynamic a population is, given its current size and growth tendencies.

Differential Equation

Differential equations are mathematical tools used to model situations where change is a factor. In our exercise, the differential equation \( \frac{dN}{dt} = 0.03N \) describes how the population \(N\) changes over time. This particular differential equation falls under the category of first-order linear differential equations. Here, \( \frac{dN}{dt} \) is the rate of change of the population size over time, and it is directly proportional to the current population size.

• This property is typical of exponential growth or decay models.
• The solution to such an equation often involves finding a function \(N(t)\) that describes the population at any time \(t\).

Differential equations like this are powerful because they don't just give a snapshot; they provide a way to understand the whole dynamic process of the population's growth or decline.

Population Size

Population size in this context refers to the total number of individuals within a given population at a specific time. In our exercise, at time \(t = 4\), the population size \(N(4)\) is given as 100. Understanding the population size is critical because it serves as the baseline for calculating growth or decline. It helps us apply linear approximations and other mathematical methods to predict future population states.

• The beginning population size affects how much it can grow or shrink over time.
• A larger initial population size typically results in more significant changes when growth rates are positive.

By using a linear approximation, we can make reasonable predictions about population size at a slightly later time without solving complex differential equations entirely. For instance, predicting the population size at \(t = 4.1\) relies on the current size, the growth rate, and the small time change (\(\Delta t\)). Each of these factors plays a role in understanding population dynamics and predicting future trends.