(c) Now suppose that K= 8, but the value of r is not given (it is assumed that r> 0.) Show that the reproduction rate g(N) is a decreasing function of N for all N> 0. The function g must be a decreasing function because is always a number for all r>0.
(c) Now suppose that K= 8, but the value of r is not given (it is assumed that r> 0.) Show that the reproduction rate g(N) is a decreasing function of N for all N> 0. The function g must be a decreasing function because is always a number for all r>0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
4-2:
Drop down boxes on second half of question:
-first box: g'(N) or g(N)
-second box: negative or positive
Transcribed Image Text:**Question:**
Select the correct choice below and, if necessary, fill in the answer box within your choice.
**A.** The function \( g \) is decreasing on \(\_\_\_\) and is not increasing on any interval.
(Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.)
**B.** The function \( g \) is increasing on \(\_\_\_\) and is not decreasing on any interval.
(Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.)
**C.** The function \( g \) is increasing on \(\_\_\_\) and is decreasing on \(\_\_\_\).
(Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answers.)
---
**(c)** Now suppose that \( K = 8 \), but the value of \( r \) is not given (it is assumed that \( r > 0 \).) Show that the reproduction rate \( g(N) \) is a decreasing function of \( N \) for all \( N > 0 \).
The function \( g \) must be a decreasing function because \(\_\_\_\) is always a \(\_\_\_\) number for all \( r > 0 \).
![**Logistic Model of Population Growth**
For a population growing according to the logistic model, a per capita reproductive rate can be calculated, which is defined to be equal to the equation below:
\[ g(N) = \frac{f(N)}{N} = r \left( 1 - \frac{N}{K} \right), \, N \geq 0 \]
**Explanation of Variables:**
- \( g(N) \): Per capita reproductive rate.
- \( f(N) \): Total reproductive rate of the population.
- \( N \): Current population size.
- \( r \): Intrinsic growth rate (maximum growth rate of the population).
- \( K \): Carrying capacity (maximum population size that the environment can sustain indefinitely).
**Conceptual Overview:**
This equation describes how the growth rate decreases as the population approaches the carrying capacity. The term \( \left( 1 - \frac{N}{K} \right) \) reflects the limitation on growth imposed by the carrying capacity. When \( N \) is much smaller than \( K \), the population grows nearly exponentially. As \( N \) approaches \( K \), the growth rate reduces to zero, stabilizing the population size.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dce3dae-6456-4f3a-a2bd-6bde155fd2f1%2F34954a1d-37dc-49d4-87b8-1f0d535ee194%2Fwn29jfm_processed.png&w=3840&q=75)
Transcribed Image Text:**Logistic Model of Population Growth**
For a population growing according to the logistic model, a per capita reproductive rate can be calculated, which is defined to be equal to the equation below:
\[ g(N) = \frac{f(N)}{N} = r \left( 1 - \frac{N}{K} \right), \, N \geq 0 \]
**Explanation of Variables:**
- \( g(N) \): Per capita reproductive rate.
- \( f(N) \): Total reproductive rate of the population.
- \( N \): Current population size.
- \( r \): Intrinsic growth rate (maximum growth rate of the population).
- \( K \): Carrying capacity (maximum population size that the environment can sustain indefinitely).
**Conceptual Overview:**
This equation describes how the growth rate decreases as the population approaches the carrying capacity. The term \( \left( 1 - \frac{N}{K} \right) \) reflects the limitation on growth imposed by the carrying capacity. When \( N \) is much smaller than \( K \), the population grows nearly exponentially. As \( N \) approaches \( K \), the growth rate reduces to zero, stabilizing the population size.
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Answer:
First box:
Second box: Negative
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