A predator-prey interaction is described by the Lotka-Volterra model x'= -0.3x + 0.04xy y' = 0.4y0.04xy. (a) Find the critical point in the first quadrant. (x, y) = =([ Use a numerical solver to sketch some population cycles. (Choose x(0) = 9 and y(0) = 6.) 14 14 12 x(t) 12 x(t) 10 10 8 ~~ 4 y(t) 5 100 20 40 60 O 0 80 100 20 40 60 80 12 x(t) 10 10 wwwwww 4y(t) 4-y(t) 20 40 80 60 100 00 20 100 00 40 60 80 (b) Estimate the period of the periodic solutions that are close to the critical point in part (a). (Choose x(0) = 9 and y(0) = 6. Use a CAS to estimate the period and round your answer to two decimal places.) 8 4 y(t) O ⁰
A predator-prey interaction is described by the Lotka-Volterra model x'= -0.3x + 0.04xy y' = 0.4y0.04xy. (a) Find the critical point in the first quadrant. (x, y) = =([ Use a numerical solver to sketch some population cycles. (Choose x(0) = 9 and y(0) = 6.) 14 14 12 x(t) 12 x(t) 10 10 8 ~~ 4 y(t) 5 100 20 40 60 O 0 80 100 20 40 60 80 12 x(t) 10 10 wwwwww 4y(t) 4-y(t) 20 40 80 60 100 00 20 100 00 40 60 80 (b) Estimate the period of the periodic solutions that are close to the critical point in part (a). (Choose x(0) = 9 and y(0) = 6. Use a CAS to estimate the period and round your answer to two decimal places.) 8 4 y(t) O ⁰
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![### Lotka-Volterra Predator-Prey Model
**Problem Statement:**
A predator-prey interaction is described by the Lotka-Volterra model:
\[ x' = 0.3x + 0.04xy \]
\[ y' = 0.4y - 0.04xy \]
**Tasks:**
(a) **Find the critical point in the first quadrant.**
\[ (x, y) = \left( \_\_\_ , \_\_\_ \right) \]
**Numerical Solution and Population Cycles:**
Use a numerical solver to sketch some population cycles. (Choose \( x(0) = 9 \) and \( y(0) = 6 \)).
**Graph Descriptions:**
- **Top Left Graph:**
- `x(t)` in blue exhibits oscillatory behavior peaking around 12 and dropping to 6.
- `y(t)` in red oscillates with peaks around 8 and troughs near 4.
- **Top Right Graph:**
- `x(t)` in blue oscillates with peaks around 10 and troughs near 8.
- `y(t)` in red exhibits smoother oscillations peaking near 6 and dipping to 4.
- **Bottom Left Graph:**
- `x(t)` in blue shows regular oscillations with peaks around 12 and troughs near 6.
- `y(t)` in red exhibits a more prominent oscillation peaking around 10 and troughs near 4.
- **Bottom Right Graph:**
- `x(t)` in blue exhibits irregular oscillatory behavior, dropping significantly at certain points.
- `y(t)` in red also shows irregular oscillations dropping to lower values.
(b) **Estimate the period of the periodic solutions that are close to the critical point in part (a).**
(Choose \( x(0) = 9 \) and \( y(0) = 6 \). Use a CAS to estimate the period and round your answer to two decimal places.)
\[ \text{Estimated Period:} \_\_\_\_ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77d2f4c8-4183-4052-877b-b5f84c5df55f%2F09c638fd-ab80-48bb-b235-fc960fc00eda%2Fjqezpie_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Lotka-Volterra Predator-Prey Model
**Problem Statement:**
A predator-prey interaction is described by the Lotka-Volterra model:
\[ x' = 0.3x + 0.04xy \]
\[ y' = 0.4y - 0.04xy \]
**Tasks:**
(a) **Find the critical point in the first quadrant.**
\[ (x, y) = \left( \_\_\_ , \_\_\_ \right) \]
**Numerical Solution and Population Cycles:**
Use a numerical solver to sketch some population cycles. (Choose \( x(0) = 9 \) and \( y(0) = 6 \)).
**Graph Descriptions:**
- **Top Left Graph:**
- `x(t)` in blue exhibits oscillatory behavior peaking around 12 and dropping to 6.
- `y(t)` in red oscillates with peaks around 8 and troughs near 4.
- **Top Right Graph:**
- `x(t)` in blue oscillates with peaks around 10 and troughs near 8.
- `y(t)` in red exhibits smoother oscillations peaking near 6 and dipping to 4.
- **Bottom Left Graph:**
- `x(t)` in blue shows regular oscillations with peaks around 12 and troughs near 6.
- `y(t)` in red exhibits a more prominent oscillation peaking around 10 and troughs near 4.
- **Bottom Right Graph:**
- `x(t)` in blue exhibits irregular oscillatory behavior, dropping significantly at certain points.
- `y(t)` in red also shows irregular oscillations dropping to lower values.
(b) **Estimate the period of the periodic solutions that are close to the critical point in part (a).**
(Choose \( x(0) = 9 \) and \( y(0) = 6 \). Use a CAS to estimate the period and round your answer to two decimal places.)
\[ \text{Estimated Period:} \_\_\_\_ \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman