Problem 9
Question
Suppose population is described by Equation \(17.31, P(t)=p_{0} M
/\left(p_{0}+\left(M-p_{0}\right) e^{-r t}\right)\) and \(0
Step-by-Step Solution
Verified Answer
The function and derivatives are derived using calculus rules.
1Step 1: Derive First Derivative
We need to find the derivative of the given function \( P(t) = \frac{p_{0} M}{p_{0} + (M - p_{0}) e^{-rt}} \). To do this, we apply the quotient rule, \( \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), where \( u = p_{0}M \) and \( v = p_{0} + (M - p_{0})e^{-rt} \).Taking the derivatives, \( u' = 0 \) (since \( u \) is a constant) and \( v' = -(M - p_{0})re^{-rt} \). Applying the quotient rule gives:\[P'(t) = \frac{0\cdot v - p_{0}M \cdot (-(M - p_{0})re^{-rt})}{(p_{0} + (M - p_{0})e^{-rt})^2} = \frac{p_{0}M (M - p_{0})re^{-rt}}{(p_{0} + (M - p_{0})e^{-rt})^2}\]
2Step 2: Derive Second Derivative
To find \( P''(t) \), we must differentiate \( P'(t) \). Here we once again apply the quotient rule to \( P'(t) = \frac{p_{0}M (M - p_{0})re^{-rt}}{(p_{0} + (M - p_{0})e^{-rt})^2} \).Let \( a = p_{0}M (M - p_{0})re^{-rt} \) and \( b = (p_{0} + (M - p_{0})e^{-rt})^2 \).Then, use \( a' = -r a \) since it's a product involving \( e^{-rt} \). \( b' = -2r(M - p_{0})re^{-rt}(p_{0} + (M - p_{0})e^{-rt}) \).Thus,\[P''(t) = \frac{-ra \cdot b - a \cdot (-2r(M - p_{0})re^{-rt}(p_{0} + (M - p_{0})e^{-rt}))}{b^2}\]
Key Concepts
Population DynamicsDerivativesQuotient RuleInflection Points
Population Dynamics
Population dynamics is a captivating area of study in biology and ecology. It involves understanding how populations of species change over time.
This includes examining factors like birth rates, death rates, immigration, and emigration.
In mathematical modeling of population dynamics, differential equations play a critical role. They allow scholars to predict how populations will evolve under various conditions.
For instance, the function provided in the equation describes a specific population model, where:
This includes examining factors like birth rates, death rates, immigration, and emigration.
In mathematical modeling of population dynamics, differential equations play a critical role. They allow scholars to predict how populations will evolve under various conditions.
For instance, the function provided in the equation describes a specific population model, where:
- \( P(t) \) represents the population size at time \( t \).
- \( p_0 \) is the initial population size.
- \( M \) stands for the carrying capacity, the maximum population that the environment can sustain.
Derivatives
Derivatives are fundamental concepts in calculus. They represent how a function changes as its input changes. More formally, a derivative is the slope of the tangent line to the curve at a point, providing the instantaneous rate of change.
In population dynamics, derivatives offer insights into how population sizes grow or decline over time.
To compute the derivative of a function, such as our population function \( P(t) \), we need to examine how values change as \( t \) changes. This involves applying various rules of differentiation, like the product, quotient, and chain rules, depending on the structure of the function.
Knowing \( P'(t) \), for example, helps us understand the rate of population growth or decline at any point in time.
In population dynamics, derivatives offer insights into how population sizes grow or decline over time.
To compute the derivative of a function, such as our population function \( P(t) \), we need to examine how values change as \( t \) changes. This involves applying various rules of differentiation, like the product, quotient, and chain rules, depending on the structure of the function.
Knowing \( P'(t) \), for example, helps us understand the rate of population growth or decline at any point in time.
Quotient Rule
The quotient rule is a specific technique in calculus for finding the derivative of a ratio of two functions. It applies when a function is expressed as a quotient, or division, of two other functions.
The general rule states that if you have a function \( \frac{u}{v} \), then the derivative, \( \frac{d}{dt} \left( \frac{u}{v} \right) \), is given by:
We perform operations on both the numerator and the denominator separately before combining these with the formula.
To practice the quotient rule means to identify your two functions, differentiate each as needed, and apply the rule accurately, keeping a keen eye on every step to ensure precision.
The general rule states that if you have a function \( \frac{u}{v} \), then the derivative, \( \frac{d}{dt} \left( \frac{u}{v} \right) \), is given by:
- \[ \frac{u'v - uv'}{v^2} \]
We perform operations on both the numerator and the denominator separately before combining these with the formula.
To practice the quotient rule means to identify your two functions, differentiate each as needed, and apply the rule accurately, keeping a keen eye on every step to ensure precision.
Inflection Points
An inflection point in calculus is where a curve changes its concavity, moving from being concave up to concave down or vice versa. This point is significant because it often corresponds to a maximum rate of change in the context of population dynamics.
For our population function, the inflection point can be found when the second derivative \( P''(t) \) equals zero. This reveals where the population grows at the fastest rate.
To compute it, we must set the second derivative to zero and solve for \( t \), which will yield \( t_{\text{steep}} \). At \( t_{\text{steep}} \), the population experiences dynamic change, indicating critical times in population management and strategies for species conservation.
For our population function, the inflection point can be found when the second derivative \( P''(t) \) equals zero. This reveals where the population grows at the fastest rate.
To compute it, we must set the second derivative to zero and solve for \( t \), which will yield \( t_{\text{steep}} \). At \( t_{\text{steep}} \), the population experiences dynamic change, indicating critical times in population management and strategies for species conservation.
Other exercises in this chapter
Problem 8
Let \(M=10\) and \(r=0.1\) and plot the graphs of $$p(t)=\frac{M p_{0}}{p_{0}+\left(M-p_{0}\right) e^{-r t}}$$ for \(0 \leq t \leq 80\) and a. \(p_{0}=1\) $$\te
View solution Problem 8
a. Show that Ricker's equation, $$p^{\prime}(t)=\alpha p e^{-p / \beta}-\gamma p$$ is equivalent to $$v^{\prime}(\tau)=v e^{-v}-\gamma_{0} v$$ with the substitu
View solution Problem 9
Identify the stable and nonstable solutions of $$u^{\prime}(t)=u(t)(1-u(t))$$
View solution Problem 10
Exercise 17.4.10 Suppose a marine fish population when not subject to harvest is reasonably modeled by $$u^{\prime}(t)=r \times u(t) \times(1-u(t))$$ with time
View solution