# The Jacobian Matrix

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 systemx′ =−3 x + 2 x y y′ = 7 y−8 x yCritical PointsNote: 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 systemx′ = 3 x−x2 −2 x y y′ = 2 y−y2 −x yEquilibrium 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 systemx′ =−9 x + x3y′ =−2 y(a) The critical points of the system above have the formx0 = [0 0] , x1 =[ x1 0] , x2 =[ x2 0] .where x1 > 0 > x2. Find these components.x1 = x2 =Part 2: The Jacobian Matrix 1Part 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 systemx′ =−16 x + x3y′ = 3 y(a) The critical points of the system above have the formx0 = [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 systex′ = 4 y−y3y′ =−9 x−y2

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