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Magazine Chapter 3: Problem 37
Consider a solution to the logistic equation with initial population \(y_{0}\)
where \(0
Short Answer
Expert verified
The solution is concave up for \(0<y<\frac{r}{2a}\) and concave down for \(\frac{r}{2a}<y<\frac{r}{a}\).
Step by step solution
01
Write the Logistic Equation
The logistic equation is given by: \[ \frac{dy}{dt} = ry(1 - \frac{ay}{r}) \]
02
Differentiate the Right Side
Differentiate the right side of the logistic equation with respect to the population, y: \[ \frac{d}{dy} \bigg[ ry - \frac{a y^2}{r} \bigg] = r - 2 \frac{a y}{r} \]
03
Set the Result Equal to Zero
Set the differentiated equation to zero to find critical points: \[ r - 2 \frac{a y}{r} = 0 \]Solving for y gives: \[ y = \frac{r}{2a} \]
04
Test Intervals for Concavity
Check the sign of the second derivative within the intervals \( 0<y<\frac{r}{2a} \) and \( \frac{r}{2a}<y<\frac{r}{a} \). First differentiate \( \frac{dy}{dt} \) again: \[ \frac{d^2 y}{dt^2} = \frac{d}{dy} \bigg[ r - 2 \frac{a y}{r} \bigg] (\frac{dy}{dt}) = -\frac{2a}{r} (\frac{dy}{dt}) \]For \( 0<y<\frac{r}{2a} \), the second derivative is positive, so the curve is concave up. For \( \frac{r}{2a}<y<\frac{r}{a} \), the second derivative is negative, so the curve is concave down.
05
Describe Behavior of \( \frac{dy}{dt} \)
\( \frac{dy}{dt} \) is positive when \( 0\frac{r}{2a} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differential Equations
Differential equations are mathematical expressions that describe how quantities change. These equations involve derivatives, which represent rates of change. In the context of population dynamics, we often deal with differential equations to describe how a population evolves over time. In the logistic equation, we have \(\frac{dy}{dt} = ry(1 - \frac{ay}{r})\). Here, \(\frac{dy}{dt}\) represents the rate of change of the population size \(y\), with respect to time \(t\). The terms \(r\) and \(a\) are parameters that determine the growth rate and the carrying capacity of the environment, respectively.
Population Dynamics
Population dynamics explore how populations change over time under various influences like birth rates, death rates, and migration. The logistic equation is a classical model used in population dynamics. It captures the idea that populations cannot grow indefinitely. The logistic equation: \(\frac{dy}{dt} = ry(1 - \frac{ay}{r})\) incorporates a growth rate \(r\) and a carrying capacity \(\frac{r}{a}\). Initially, when the population \(y\) is small, the term \(1 - \frac{ay}{r}\) is close to 1, leading to exponential growth. As \(y\) approaches the carrying capacity, this term becomes smaller, slowing the growth and ultimately leading to a stable population size.
Second Derivative Test
The second derivative test is a mathematical method used to determine the concavity of a function. It helps us understand the behavior of the population in our logistic model. In the provided solution, we differentiated the logistic equation to find \( \frac{d}{dy} \bigg[ ry - \frac{a y^2}{r} \bigg]= r - 2 \frac{a y}{r}\). Setting this to zero gives us the critical point \( y = \frac{r}{2a} \). To check concavity, we look at the second derivative: \( \frac{d^2 y}{dt^2} = \frac{d}{dy} \bigg[ r - 2 \frac{a y}{r} \bigg] (\frac{dy}{dt}) = -\frac{2a}{r} (\frac{dy}{dt})\). This helps us analyze whether the population is increasing or decreasing at different rates.
Concavity
Concavity tells us whether a curve is 'bending' upwards or downwards. If the second derivative of a function is positive, the curve is concave up, resembling a cup \(\cup\). If the second derivative is negative, the curve is concave down, resembling a cap \(\cap\). For the logistic equation, we determined that:
\(\text{For } 0
\(\text{For } \frac{r}{2a}
\(\text{For } 0
\(\text{For } \frac{r}{2a}
Critical Points
Critical points are where the first derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection. In our logistic equation, we set the first derivative equal to zero: \( r - 2 \frac{a y}{r} = 0 \), solving for \(y\) gives us the critical point \( y = \frac{r}{2a} \). This critical point helps identify where the growth rate starts to change. It separates the concave up and concave down regions, indicating a shift in the growth behavior of the population.
