TEST 1 REVIEW

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# TEST 1 REVIEW - PowerPoint PPT Presentation

TEST 1 REVIEW. Single Species Discrete Equations. Chapter 1 in Text, Lecture 1 and 2 Notes Homogeneous (Bacteria growth), Inhomogeneous (Breathing model) x n +1 = ax n + b . Finding solutions Homogeneous: x n = C a n General Solution = Homogeneous Solution + Particular Solution

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Presentation Transcript
Single Species Discrete Equations
• Chapter 1 in Text, Lecture 1 and 2 Notes
• Homogeneous (Bacteria growth), Inhomogeneous (Breathing model)
• xn+1 =axn+ b.
• Finding solutions
• Homogeneous: xn = Can
• General Solution = Homogeneous Solution + Particular Solution
• Behavior of solutions - determined by the magnitude of ‘a’
• Increasing, decreasing, oscillating
Linear Systems of Discrete Equations
• Tumor Growth, Segmental Growth, Red Blood Cell Production, Blood CO2
• Sections1.3, 1.6, 1.8, 1.9 in Text, Lecture 3 Notes
• Order
• Number of previous generations needed to determine a future generation
• Any system of two or more linear, first order discrete equations can be written as a single higher order equation
Linear Systems of Discrete Equations
• Solutions
• Characteristic equation
• Look for solution of the form xn = Cln
• Find eigenvalues, l
• General Solution:
• Linear Combinations of all basic solutions
• Behavior of Solutions
• Dominant eigenvalue
Linear Discrete Essentials
• You should be able to:
• Characterize and know the properties of the equations
• Solve Linear equations
• Describe the behavior of solutions
Nonlinear Discrete Equations
• Single Species (Discrete Logistic)
• Chapter 2 in Text, Lecture 4-5 Notes
• Stability - analytically/graphically
• Cobweb Diagrams
• |f’(xe)| < 1 for stability
• 2 point cycles
• Chaos
• Look at 2.1, 2.2, 2.5, 3.1
Nonlinear Discrete Equations
• Nonlinear Systems: Host-Parasitoid Interactions
• Chapters 2.7, 2.8, 3.2-3.4
• Stability
• For other examples see section 3.5 and homework #3
Nonlinear Discrete Essentials
• You should be able to:
• Determine their stability
• Describe the behavior of solutions
• Interpret model behavior
Bifurcation Review
• Bifurcations
• What are they?
• Bifurcation diagrams
• What are they, why are they useful?
• Generate them
• Read them and interpret them
Continuous Models Review
• Single Species
• Logistic Equation and Spruce Budworm
• Lectures 7 and 8
• Nondimensionalization
• (Lecture 7 Notes, Section 4.5 in Text)
• Be able to do it, express why its useful and interpret the scales (eg HW #5)
• Steady states (Lecture 7 Notes )
• Graphically and Analytically
• Stability (Lecture Notes 7)
• Graphically and Analytically
Continuous Models Review
• Systems of ODEs: The Chemostat,
• Lectures 9 and 10, Lab 5, Chapters: 4.2 - 6.2
• Nondimensionalization
• Lecture 9 and 10 Notes, Sections 4.6 and 5.5 in Text
• Analytically
• Graphically (5.5)
• Intersection of Nullclines
• Stability (Lecture Notes 9 and 10, 4.7, 4.9)
• Analytically
• RE(
• Graphically
• Phase portraits (Chapter 5, Lectures 9 and 10)
Continuous Model Essentials
• Nondimensionalize