# 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

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