Problem 1
Question
Find \(r\) and \(M\) so that the solution \(p(t)=\frac{M p_{0}}{p_{0}+\left(M-p_{0}\right) e^{-r t}} \quad\) of the logistic equation \(\quad p^{\prime}=r \times p \times\left(1-\frac{p}{M}\right)\) approximates the data $$\begin{array}{|r|r|r|} \hline {\mathrm{pH} 6.25} \\\\\hline \text { Time } & \text { Index } & \text { Population } \\ \text { (min) } & t & \text { Density } \\\\\hline 0 & 0 & 0.022 \\ 16 & 1 & 0.036 \\\32 & 2 & 0.060 \\\48 & 3 & 0.101 \\ 64 & 4 & 0.169 \\\80 & 5 & 0.266 \\\96 & 6 & 0.360 \\\112 & 7 & 0.510 \\\128 & 8 & 0.704 \\\144 & 9 & 0.827 \\\160 &10& 0.928 \\\\\hline\end{array}$$
Step-by-Step Solution
Verified Answer
Estimate \( r \) and \( M \) using non-linear regression with the given data.
1Step 1: Understand the Logistic Model
The given logistic model is a mathematical model used for predicting how a population grows over time. The solution to the logistic equation is given by \( p(t) = \frac{M p_0}{p_0 + (M - p_0) e^{-r t}} \), where \( M \) is the carrying capacity, \( r \) is the growth rate, and \( p_0 \) is the initial population density.
2Step 2: Identify Initial Parameters
From the data provided, notice that at \( t = 0 \), the population density \( p(0) = 0.022 \). So, the initial condition \( p_0 = 0.022 \). Now, work with other data points to estimate \( r \) and \( M \).
3Step 3: Use Known Points to Set Equations
Using one data point, such as \( t = 1 \) where \( p(1) = 0.036 \), substitute into the logistic formula: \( 0.036 = \frac{M \times 0.022}{0.022 + (M - 0.022) e^{-r \times 1}} \). This can be used to create an equation involving \( M \) and \( r \).
4Step 4: Solve the System of Equations
Utilize additional data points such as \( t = 8 \), \( p(8) = 0.704 \) and set up another equation \( 0.704 = \frac{M \times 0.022}{0.022 + (M - 0.022) e^{-r \times 8}} \). Solve this system of equations for \( M \) and \( r \). This typically involves substituting one equation into the other or using numerical methods.
5Step 5: Approximate Using Non-Linear Regression
Given the complexity in directly solving the non-linear equations, use a non-linear regression method or software tool to fit the experimental data to the logistic equation by adjusting \( r \) and \( M \) to minimize error.
Key Concepts
Population DynamicsCarrying CapacityGrowth Rate
Population Dynamics
Population dynamics is a field of biology that explores how populations of organisms change over time and space. In the context of the logistic growth model, it specifically examines the growth patterns and density variations of a population. The population's size at any given time is influenced by several factors, including birth rates, death rates, immigration, and emigration.
In mathematical models like the logistic growth model, these dynamics are captured using differential equations. The logistic model is particularly useful because it describes how a population will grow rapidly at first when resources are abundant and then slow down as resources become limited. This creates an S-shaped curve, representing exponential growth that levels off near the carrying capacity.
These dynamics are crucial for understanding real-world scenarios, such as predicting wildlife populations, managing conservation efforts, and analyzing human population growth.
In mathematical models like the logistic growth model, these dynamics are captured using differential equations. The logistic model is particularly useful because it describes how a population will grow rapidly at first when resources are abundant and then slow down as resources become limited. This creates an S-shaped curve, representing exponential growth that levels off near the carrying capacity.
These dynamics are crucial for understanding real-world scenarios, such as predicting wildlife populations, managing conservation efforts, and analyzing human population growth.
Carrying Capacity
Carrying capacity, denoted as \( M \) in the logistic growth equation, represents the maximum population size that an environment can sustain indefinitely. This is an essential concept when studying population dynamics because it dictates the upper limit of population density due to available resources like food, space, and other ecological factors.
In the logistic model used in the exercise, the carrying capacity is a parameter that one must often estimate from real data. As a population approaches its carrying capacity, the growth rate decreases, demonstrating the influence of limited resources.
In the logistic model used in the exercise, the carrying capacity is a parameter that one must often estimate from real data. As a population approaches its carrying capacity, the growth rate decreases, demonstrating the influence of limited resources.
- Carrying capacity is affected by environmental changes and human impacts like deforestation, pollution, and urban development.
- In ecological studies, understanding carrying capacity helps predict how ecosystems respond to various pressures.
Growth Rate
The growth rate, symbolized by \( r \) in the logistic model, is a measure of how quickly a population increases in size. It is a fundamental parameter in population dynamics, standing for the rate of reproduction minus the death rate per capita. High growth rates indicate fast-expanding populations, often found in environments with abundant resources.
In the logistic growth model, the growth rate is initially exponential when the population is small and resources are plentiful. As the population density increases, growth slows down due to environmental resistance based on the carrying capacity.
In the logistic growth model, the growth rate is initially exponential when the population is small and resources are plentiful. As the population density increases, growth slows down due to environmental resistance based on the carrying capacity.
- Calculating the growth rate involves analyzing changes in population size over time against the model's predictions.
- Growth rates can be adjusted by factors such as birth controls, natural disasters, and changes in resource availability.
Other exercises in this chapter
Problem 1
Draw the direction field for \(y^{\prime}(t)=\sqrt{y(t)}\) and decide whether the equilibrium solution \(y(t)=0\) is stable.
View solution Problem 1
Show that each solution satisfies the initial condition and the differential equation. $$\begin{aligned} &\text { Solution } \quad \text { Initial Condition } \
View solution Problem 2
Write an equation that describes the temperature of an egg after it is uncovered (the adult bird leaves the nest to feed). Assume that the rate of change of the
View solution