Problem 64 - Calculus for Biology and Medicine

Question

Suppose the size of an insect population, \(N(t)\), grows with time \(t\), according to the function $$N(t)=M t e^{-m t}$$ where \(M\) and \(m\) are coefficients. (a) Show that the model can be rewritten as: $$\ln \left(\frac{N}{t}\right)=\ln M-m t$$ (b) Explain how the coefficients \(m\) and \(M\) can be estimated from a plot of \(\ln (N / t)\) against \(t\). $$\begin{array}{lccccc} \hline \boldsymbol{t} & 0.1 & 0.3 & 0.5 & 0.8 & 1 \\ \boldsymbol{N} & 6.11 & 1.64 & 1.00 & 0.196 & 0.0633 \\ \hline \end{array}$$ (c) Use a least squares error method to fit \(M\) and \(m\) from the following experimental data.

Step-by-Step Solution

Answer

Rewrite equation using logs, plot to find slope and intercept, use least squares for coefficients.

Step 1: Rewrite the Provided Equation

We start with the given function for the insect population:\[ N(t) = M t e^{-m t} \]The task is to show that this can be expressed in the form:\[ \ln \left(\frac{N}{t}\right) = \ln M - m t \]We begin by manipulating the expression:\[\frac{N}{t} = M e^{-m t}\]Taking the natural logarithm of both sides gives:\[\ln \left( \frac{N}{t} \right) = \ln(M e^{-m t})\]By using logarithmic properties, this becomes:\[\ln \left( \frac{N}{t} \right) = \ln M + \ln e^{-m t} = \ln M - mt \] Thus, we have shown the required form.

Step 2: Explanation of Plot Analysis

The given equation \( \ln \left( \frac{N}{t} \right) = \ln M - mt \) suggests that \( \ln \left( \frac{N}{t} \right) \) is a linear function of \( t \) with slope \( -m \) and y-intercept \( \ln M \). By plotting \( \ln \left( \frac{N}{t} \right) \) against \( t \), the slope of the resulting line will provide the value of \( -m \), and the y-intercept will be \( \ln M \). Using these points, \( M \) can be found by taking the exponent of the y-intercept.

Step 3: Transform the Data

Firstly, we calculate \( \ln \left( \frac{N}{t} \right) \) for each \( t \) using the given data. 1. \( t = 0.1 \), \( N = 6.11 \): \( \ln \left( \frac{6.11}{0.1} \right) = \ln(61.1) \) 2. \( t = 0.3 \), \( N = 1.64 \): \( \ln \left( \frac{1.64}{0.3} \right) = \ln(5.47) \) 3. \( t = 0.5 \), \( N = 1.00 \): \( \ln \left( \frac{1.00}{0.5} \right) = \ln(2) \) 4. \( t = 0.8 \), \( N = 0.196 \): \( \ln \left( \frac{0.196}{0.8} \right) = \ln(0.245) \) 5. \( t = 1.0 \), \( N = 0.0633 \): \( \ln \left( \frac{0.0633}{1.0} \right) = \ln(0.0633) \)

Step 4: Apply Least Squares Method

For the least squares approach, let \( y = \ln \left( \frac{N}{t} \right) \). We set up equations to find the line of best fit:We solve for the coefficients of the linear relation \( y = a + bt \), where \( a = \ln M \) and \( b = -m \).Calculate:1. The sum of \( t \), the sum of \( y \), the sum of \( t\cdot y \), the sum of \( t^2 \), and the number of points, \( n \).2. Solve the normal equations: \[ b = \frac{n \sum (t\cdot y) - \sum t \sum y}{n\sum t^2 - (\sum t)^2} \] \[ a = \frac{\sum y - b \sum t}{n} \]Substitute calculated sums into these equations to find \( a \) (\( \ln M \)) and \( b \) (\( -m \)). Finally, solve for \( M \) by exponentiating \( a \).

Key Concepts

Insect Population Modeling

Insect population modeling is an important part of understanding ecological systems and can help scientists predict future population sizes under different conditions. The model provided in this exercise uses a mathematical function to represent how an insect population, denoted as \( N(t) \), changes over time \( t \). This model is particularly suited for populations that initially grow rapidly and then stabilize, mimicking the behavior often observed in nature.
The model utilized here is \( N(t) = M t e^{-m t} \), where \( M \) and \( m \) are coefficients that describe specific aspects of the population growth. The coefficient \( M \) is related to the initial population size and its potential growth rate, while \( m \) determines how quickly the growth rate decreases over time. By understanding these two coefficients, researchers can gain insights into the factors affecting population dynamics, such as resource availability and environmental pressures.
This specific model is particularly interesting because it incorporates both exponential growth and exponential decay, providing a flexible framework to simulate realistic population behaviors. This dual nature allows the model to represent populations that experience a sharp increase before leveling off or declining, capturing the intricacies of real-world scenarios.

Exponential Growth

Exponential growth refers to a process where quantities increase at a rate proportional to their current size. This type of growth is crucial in biological studies, including population dynamics. In the context of this exercise, the function \( N(t) = M t e^{-m t} \) initially experiences exponential growth. The term \( M t \) represents linear growth scaling with time, while the exponential decay part \( e^{-mt} \) eventually slows this growth down.
In the biological context, exponential growth occurs when resources are plentiful, and there are minimal constraints on the population, such as food supply or habitat space. Initially, the population grows rapidly as there are more individuals to reproduce. However, as the population increases, constraints like competition and predation start to play a significant role, slowing down the growth rate.
For insect populations, it is common to observe rapid population increases that taper off once environmental limits are reached. The model's exponential decay term reflects these limits as it reduces the influence of rapid growth over time. Understanding both aspects of growth and decay in the model is crucial for predicting future population sizes and making informed decisions in wildlife management and conservation.

Least Squares Method

The least squares method is a statistical technique used to find the best-fitting curve through a set of points by minimizing the sum of the squares of the vertical deviations from each point to the curve. In this exercise, it is used to estimate the coefficients \( M \) and \( m \) of the insect population model based on empirical data.
The technique involves plotting the transformed data points, \( \ln \left( \frac{N}{t} \right) \) against \( t \), allowing the relationship to be expressed linearly as \( y = a + bt \), where \( a = \ln M \) and \( b = -m \). By calculating the slope \( b \) and intercept \( a \) from the line of best fit, we can derive the values of \( m \) and \( M \).
To achieve this, you calculate several sums: the sum of the time values, the sum of the transformed growth values, the sum of the product of time and growth values, and the sum of the squares of the time values. These are then substituted into the equations for the slope and intercept to determine \( m \) and \( M \).
Applying the least squares method is critical as it provides a systematic way to derive the most accurate coefficients for the model. This ensures any predictions made using the model are based on the best possible representation of the underlying data, giving researchers confidence in their analyses and conclusions.