Suppose that the population x, of bacteria satisfies the discrete time dynamical system X1 where r is a parameter, and 0
Suppose that the population x, of bacteria satisfies the discrete time dynamical system X1 where r is a parameter, and 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
![Suppose that the population \( x_t \) of bacteria satisfies the discrete time dynamical system:
\[
x_{t+1} = \frac{x_t}{r + (x_t)^2}
\]
where \( r \) is a parameter, and \( 0 < r < 1 \).
1. If \( x_t = 0 \), what is \( x_{t+1} \)?
- [Fill in the blank]
2. Is \( x^* = 0 \) an equilibrium?
- [Fill in the blank]
3. If \( x_t = \sqrt{1-r} \), what is \( x_{t+1} \)?
- [Fill in the blank]
4. Is \( x^* = \sqrt{1-r} \) an equilibrium?
- [Fill in the blank]
5. Write the updating function rule as \( f(x) = \frac{x}{r + x^2} \).
6. Compute the derivative: \( f'(x) = \)
- [Fill in the blank]
7. What condition on \( r \) guarantees that the equilibrium \( x^* = \sqrt{1-r} \) is stable?
- [Fill in the blank]
\[ < r < \]
---
This exercise involves analyzing the stability of equilibria in a discrete-time dynamical system that models a population of bacteria. The model examines how the population changes over time based on the parameter \( r \). You are asked to find conditions for equilibrium and its stability by evaluating specific population values and deriving functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58cffe43-d701-4c21-9740-08fe8d98ee79%2F3eb574af-32a7-4c86-bf89-0845c2dfe749%2Fhzw6kv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that the population \( x_t \) of bacteria satisfies the discrete time dynamical system:
\[
x_{t+1} = \frac{x_t}{r + (x_t)^2}
\]
where \( r \) is a parameter, and \( 0 < r < 1 \).
1. If \( x_t = 0 \), what is \( x_{t+1} \)?
- [Fill in the blank]
2. Is \( x^* = 0 \) an equilibrium?
- [Fill in the blank]
3. If \( x_t = \sqrt{1-r} \), what is \( x_{t+1} \)?
- [Fill in the blank]
4. Is \( x^* = \sqrt{1-r} \) an equilibrium?
- [Fill in the blank]
5. Write the updating function rule as \( f(x) = \frac{x}{r + x^2} \).
6. Compute the derivative: \( f'(x) = \)
- [Fill in the blank]
7. What condition on \( r \) guarantees that the equilibrium \( x^* = \sqrt{1-r} \) is stable?
- [Fill in the blank]
\[ < r < \]
---
This exercise involves analyzing the stability of equilibria in a discrete-time dynamical system that models a population of bacteria. The model examines how the population changes over time based on the parameter \( r \). You are asked to find conditions for equilibrium and its stability by evaluating specific population values and deriving functions.
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