Problem 67 - Calculus for Biology and Medicine

Question

A growing population contains \(N(t)\) individuals \(N(t)\) at time \(t\) is modeled by the equation $$ N(t)=N_{0} e^{r t} $$ where \(N_{0}\) denotes the population size at time \(0 .\) The constant \(r\) is called the intrinsic rate of growth. (a) Plot \(N(t)\) as a function of \(t\) if \(N_{0}=100\) and \(r=2\). Compare your graph against the graph of \(N(t)\) when \(N_{0}=100\) and \(r=3\). Which population grows faster? (b) You are given the following data for the size of the population. $$ \begin{array}{cc} \hline \boldsymbol{t} & \boldsymbol{N}(\boldsymbol{t}) \\ \hline 0 & 100 \\ 2 & 300 \\ \hline \end{array} $$ (i) Calculate the parameters \(N_{0}\) and \(r\) to make the mathematical model fit the data. (ii) When will the population size first reach 1000 individuals? (iii) When will the population size first reach 10,000 individuals?

Step-by-Step Solution

Answer

The population with \(r=3\) grows faster. For \(r=\frac{\ln(3)}{2}\), the population reaches 1000 at \(t \approx \frac{2 \ln(10)}{\ln(3)}\) and 10,000 at \(t \approx \frac{2 \ln(100)}{\ln(3)}\).

Step 1: Understand the population equation

The given formula for the population size over time is \( N(t) = N_{0} e^{rt} \). \(N_{0}\) is the initial population size, and \(r\) is the growth rate. This is an exponential growth model.

Step 2: Graph the population growth for different rates

For part (a), we need to graph \(N(t) = 100 e^{2t}\) and \(N(t) = 100 e^{3t}\). Since both functions have the same initial population \(N_{0} = 100\), but different growth rates, \(r = 3\) will result in a population that grows faster than \(r = 2\). Thus, we can predict that the plot for \(r = 3\) will be steeper.

Step 3: Solve for parameters using given data

For part (b), given that \(N(0) = 100\), it directly gives \(N_{0} = 100\). With \(N(2) = 300\), we use the equation \(300 = 100 e^{2r}\) to solve for \(r\): \(e^{2r} = 3\). Taking the natural logarithm on both sides, we have \(2r = \ln(3)\), thus \(r = \frac{\ln(3)}{2}\).

Step 4: Calculate when population reaches 1000

Using the parameters, to find when \(N(t) = 1000\), substitute into the population formula: \(1000 = 100 e^{\frac{\ln(3)}{2}t}\) so \(e^{\frac{\ln(3)}{2}t} = 10\). Taking the natural logarithm, \(\frac{\ln(3)}{2}t = \ln(10)\). Solving for \(t\), we get \(t = \frac{2 \ln(10)}{\ln(3)}\).

Step 5: Calculate when population reaches 10,000

For \(N(t) = 10,000\), use the equation \(10,000 = 100 e^{\frac{\ln(3)}{2}t}\) so \(e^{\frac{\ln(3)}{2}t} = 100\). Take the natural logarithm to get \(\frac{\ln(3)}{2}t = \ln(100)\). Solving for \(t\), yields \(t = \frac{2 \ln(100)}{\ln(3)}\).

Key Concepts

Population Dynamics

Population dynamics is the study of how and why populations change over time. It is a crucial area of biology that helps us understand the factors affecting the survival and reproduction of different species. At the heart of population dynamics is the concept of exponential growth, described by the equation \( N(t) = N_0 e^{rt} \). Here, \( N(t) \) represents the population size at time \( t \), \( N_0 \) is the initial size of the population, and \( r \) is the intrinsic growth rate.

This model assumes that the population has unlimited resources, which is rarely the case in nature. Still, it provides a fundamental understanding of how populations could grow under optimal conditions.

• The model tells us that populations with higher growth rates \( r \) will expand more quickly.
• Changes in population size can be used to predict when a population will reach a certain size.

The model is applicable in various fields like ecology, conservation, and even human population studies, where understanding growth patterns can guide resource management and policy-making.

Growth Rate

The growth rate, denoted as \( r \) in the exponential growth model, is a measure of how quickly the population increases over time. In the equation \( N(t) = N_0 e^{rt} \), \( r \) determines the steepness of the growth curve.

When \( r \) is greater, the population grows faster, leading to a steeper curve in the graph of \( N(t) \) over \( t \). Conversely, if \( r \) is smaller, the population grows more slowly. The growth rate is intrinsic, meaning it is a natural characteristic of the population under specific conditions.

• It can be influenced by factors like birth rates, death rates, immigration, and emigration.
• Understanding \( r \) is essential for predicting population trends and making decisions about conservation and resource allocation.

By analyzing the current population data and comparing it to predicted values, scientists and policymakers can discern if an increase in the growth rate would be sustainable or if efforts should be made to control the growth.

Natural Logarithm

The natural logarithm, represented by \( \ln \), is a powerful mathematical function used to transform equations and solve problems involving exponential growth. In the context of population dynamics, the natural logarithm helps solve for unknowns like time \( t \) and growth rate \( r \). For instance, given \( e^{2r} = 3 \), taking the natural logarithm of both sides allows us to isolate \( r \):

\[ 2r = \ln(3) \]

From here, we can find \( r \) by dividing both sides by 2:

\[ r = \frac{\ln(3)}{2} \]

The natural logarithm is a key feature of the exponential growth model because it reverses the exponential function \( e^{rt} \).

• This makes it easier to work with equations involving growth rates and predict when a population will reach a specific size.
• Understanding how to use natural logarithms is critical for analyzing population growth data accurately.

It is an essential tool in mathematics, especially when working with continuous compounding, like in financial calculations and population studies.