Problem 34
Question
Suppose that the size of a population at time \(t\) is \(N(t)\) and its growth rate is given by the logistic growth function $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right), \quad t \geq 0 $$ where \(r\) and \(K\) are positive constants. The per capita growth rate is defined by $$ g(N)=\frac{1}{N} \frac{d N}{d t} $$ (a) Show that $$ g(N)=r\left(1-\frac{N}{K}\right) $$ (b) Graph \(g(N)\) as a function of \(N\) for \(N \geq 0\) when \(r=2\) and \(K=100\), and find the population size for which the per capita growth rate is maximal.
Step-by-Step Solution
Verified Answer
The per capita growth rate is maximal at \( N = 0 \).
1Step 1: Start with the Equation for Per Capita Growth Rate
The per capita growth rate \( g(N) \) is defined as \( g(N) = \frac{1}{N} \frac{dN}{dt} \). We need to replace \( \frac{dN}{dt} \) with its expression from the logistic growth rate equation, \( \frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) \).
2Step 2: Substitute and Simplify the Expression
Substitute \( \frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) \) into the per capita growth rate equation: \[ g(N) = \frac{1}{N} \cdot rN\left(1 - \frac{N}{K}\right) \]. Simplifying this, we find that terms of \( N \) cancel, leaving us with \( g(N) = r\left(1 - \frac{N}{K}\right) \). This confirms the expression.
3Step 3: Define Parameters for Graphing
To graph \( g(N) = r\left(1 - \frac{N}{K}\right) \), substitute the given values: \( r = 2 \) and \( K = 100 \). The function becomes \( g(N) = 2\left(1 - \frac{N}{100}\right) \).
4Step 4: Graph the Function
Plot the graph of \( g(N) = 2\left(1 - \frac{N}{100}\right) \) for \( N \geq 0 \). Observe that the graph is a linear function beginning from \( g(0) = 2 \) and decreasing linearly as \( N \) increases towards \( K \).
5Step 5: Determine Maximum Per Capita Growth Rate
The per capita growth rate is maximal at \( N = 0 \) since \( g(N) = 2\left(1 - \frac{N}{100}\right) \) decreases linearly from its maximum at \( N = 0 \), where \( g(N) = 2 \).
Key Concepts
Population DynamicsPer Capita Growth RateGraphing Functions
Population Dynamics
Population dynamics is the study of how populations change over time. Populations are groups of individuals from the same species, and their dynamics can be affected by various factors such as birth rates, death rates, immigration, and emigration. In the case of the logistic growth model, the focus is on the changing size of a population over time. This model provides a more realistic depiction of population growth than the simple exponential model.
In the logistic growth model, the size of the population at any time \(t\) is represented by \(N(t)\), and its change over time is determined by the equation:
\[ \frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) \]
Here, \(r\) represents the intrinsic growth rate, which is the maximum potential growth rate of the population when not limited by resources. \(K\) is the carrying capacity, indicating the maximum population size that the environment can sustain indefinitely, taking into account available resources. As the population approaches \(K\), growth slows down, capturing the natural limits of resource availability.
In the logistic growth model, the size of the population at any time \(t\) is represented by \(N(t)\), and its change over time is determined by the equation:
\[ \frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) \]
Here, \(r\) represents the intrinsic growth rate, which is the maximum potential growth rate of the population when not limited by resources. \(K\) is the carrying capacity, indicating the maximum population size that the environment can sustain indefinitely, taking into account available resources. As the population approaches \(K\), growth slows down, capturing the natural limits of resource availability.
Per Capita Growth Rate
The per capita growth rate is a crucial concept in population dynamics. It measures the average increase in population per individual and provides insights into how quickly a population can grow relative to its size. It's given by the expression:
\[ g(N) = \frac{1}{N} \frac{dN}{dt} \]
When dealing with the logistic growth model, this formula can be further simplified by substituting in the expression for \(\frac{dN}{dt}\). We see that:
\[ g(N) = \frac{1}{N} \cdot rN\left(1 - \frac{N}{K}\right) = r\left(1 - \frac{N}{K}\right) \]
This calculation shows that the per capita growth rate declines linearly as the population size \(N\) increases. At the start, when the population is small, the growth rate is near its maximum possible value, \(r\). As the population approaches the carrying capacity \(K\), the growth rate slows because resources become limited and environmental resistance increases.
\[ g(N) = \frac{1}{N} \frac{dN}{dt} \]
When dealing with the logistic growth model, this formula can be further simplified by substituting in the expression for \(\frac{dN}{dt}\). We see that:
\[ g(N) = \frac{1}{N} \cdot rN\left(1 - \frac{N}{K}\right) = r\left(1 - \frac{N}{K}\right) \]
This calculation shows that the per capita growth rate declines linearly as the population size \(N\) increases. At the start, when the population is small, the growth rate is near its maximum possible value, \(r\). As the population approaches the carrying capacity \(K\), the growth rate slows because resources become limited and environmental resistance increases.
Graphing Functions
Graphing functions is an essential skill in understanding population dynamics and analyzing how different factors influence growth over time. In the logistic growth context, the function we graph represents the per capita growth rate over various population sizes. With the given parameters where \(r = 2\) and \(K = 100\), the function becomes:
\[ g(N) = 2\left(1 - \frac{N}{100}\right) \]
This linear function starts at \(g(0) = 2\), indicating the maximum growth when the population size is zero. As we graph this for \(N \geq 0\), we observe a straight line declining to zero at \(N = 100\).
The graph is primarily a visual representation of how the growth rate changes with population size. The maximum per capita growth rate occurs when the population is smallest (starting point), and it gradually decreases to zero as the population size reaches the carrying capacity. Understanding the graph allows us to predict how quickly a population can grow in different scenarios, providing a clearer picture of its trajectory under given conditions.
\[ g(N) = 2\left(1 - \frac{N}{100}\right) \]
This linear function starts at \(g(0) = 2\), indicating the maximum growth when the population size is zero. As we graph this for \(N \geq 0\), we observe a straight line declining to zero at \(N = 100\).
The graph is primarily a visual representation of how the growth rate changes with population size. The maximum per capita growth rate occurs when the population is smallest (starting point), and it gradually decreases to zero as the population size reaches the carrying capacity. Understanding the graph allows us to predict how quickly a population can grow in different scenarios, providing a clearer picture of its trajectory under given conditions.
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