# The Jacobian Matrix at X0 Part 4

Miran Anwar, mth 235 ss21 1: Hw18-5.3-5.4-SDE-2x2NS-CS. Due: 04/26/2021 at 11:00pm EDT.

Similar problem in LN, § 5.3, Example 5.3.6, 5.3.7. 1. (10 points)

Find the critical points (also called equilibrium solutions) of the predator-prey system

x′ =−3 x + 2 x y y′ = 7 y−8 x y

Critical Points:

Note: A point is an ordered pair (x,y), and your answer must be a comma separated list of points.

Similar problem in LN, § 5.3, Example 5.3.6, 5.3.7. 2. (10 points)

Find the critical points (also called equilibrium solutions) of the competing species system

x′ = 3 x−x2 −2 x y y′ = 2 y−y2 −x y

Equilibrium Points:

Note: A point is an ordered pair (x,y), and your answer must be a comma separated list of points.

See in LN, § 5.3, See Examples 5.3.8-5.3.10. 3. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system

x′ =−9 x + x3

y′ =−2 y

(a) The critical points of the system above have the form

x0 = [

0 0

] , x1 =

[ x1 0

] , x2 =

[ x2 0

] .

where x1 > 0 > x2. Find these components.

x1 = x2 =

Part 2: The Jacobian Matrix 1

Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 See in LN, § 5.3, See Examples 5.3.8-5.3.10. 4. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system

x′ =−16 x + x3

y′ = 3 y

(a) The critical points of the system above have the form

x0 = [

0 0

] , x1 =

[ x1 0

] , x2 =

[ x2 0

] .

where x1 > 0 > x2. Find these components.

x1 = x2 =

Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 See in LN, § 5.3, See Examples 5.3.8-5.3.10. 5. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system

x′ = 4 y−y3

y′ =−9 x−y2

(a) The critical points of the system above have the form

x0 = [

0 0

] , x1 =

[ x1 y1

] , x2 =

[ x2 y2

] .

where y1 > 0 > y2. Find these components.

x1 = y1 =

x2 = y2 =

Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2

2

See in LN, § 5.4, See Examples 5.4.1, 5.4.2. 6. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system

x′ = x (1− x− y)

y′ = y (1

2 −

1 4

y− 3 4

x )

(a) The critical points of the system above have the form

x0 = [

0 0

] , x1 =

[ 0 y1

] , x2 =

[ x2 0

] , x3 =

[ x3 y3

] .

where y1 > 0 > y2. Find these components.

y1 = x2 =

x3 = y3 =

Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 Part 6: The Jacobian Matrix at X3 See in LN, § 5.4, See Examples 5.4.1, 5.4.2. 7. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system

x′ = x (1− x− y)

y′ = y (3

4 − y−

1 2

x )

(a) The critical points of the system above have the form

x0 = [

0 0

] , x1 =

[ 0 y1

] , x2 =

[ x2 0

] , x3 =

[ x3 y3

] .

where y1 > 0 > y2. Find these components.

y1 = x2 =

x3 = y3 =

Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1

3

Part 5: The Jacobian Matrix at X2 Part 6: The Jacobian Matrix at X3

See in LN, § 5.4, See Examples 5.4.1, 5.4.2. 8. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system

x′ = x ( 3 2 − x−

1 2

y)

y′ = y (

2− y− 3 4

x )

(a) The critical points of the system above have the form

x0 = [

0 0

] , x1 =

[ 0 y1

] , x2 =

[ x2 0

] , x3 =

[ x3 y3

] .

where y1 > 0 > y2. Find these components.

y1 = x2 =

x3 = y3 =

Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 Part 6: The Jacobian Matrix at X3

See in LN, § 5.4, See Examples 5.4.1, 5.4.2. 9. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system

x′ = x ( 3 2 − x−

1 2

y)

y′ = y (

2− 1 2

y− 3 2

x )

(a) The critical points of the system above have the form

x0 = [

0 0

] , x1 =

[ 0 y1

] , x2 =

[ x2 0

] , x3 =

[ x3 y3

] .

where y1 > 0 > y2. Find these components.

y1 = x2 =

x3 = y3 = 4

Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 Part 6: The Jacobian Matrix at X3

Generated by c©WeBWorK, http://webwork.maa.org, Mathematical Association of America

5

## Needs help with similar assignment?

We are available 24x7 to deliver the best services and assignment ready within 3-4 hours? Order a custom-written, plagiarism-free paper

## Math homework help

Do not let math assignments discourage you from trying your best. One of our math experts can help you with math homework online. Our math tutors can help you with any level of algebra, calculus, or geometry. Ask your question to get the support you need 24/7.Order Over WhatsApp Place an Order Online