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TEST 1 REVIEW

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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TEST 1 REVIEW

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  1. TEST 1 REVIEW

  2. 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

  3. 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

  4. 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

  5. Linear Discrete Essentials • You should be able to: • Characterize and know the properties of the equations • Solve Linear equations • Describe the behavior of solutions

  6. Nonlinear Discrete Equations • Single Species (Discrete Logistic) • Chapter 2 in Text, Lecture 4-5 Notes • Steady states - analytically/graphically • Stability - analytically/graphically • Cobweb Diagrams • |f’(xe)| < 1 for stability • Don’t worry about • 2 point cycles • Chaos • Look at 2.1, 2.2, 2.5, 3.1

  7. Nonlinear Discrete Equations • Nonlinear Systems: Host-Parasitoid Interactions • Chapters 2.7, 2.8, 3.2-3.4 • Steady states • Stability • For other examples see section 3.5 and homework #3

  8. Nonlinear Discrete Essentials • You should be able to: • Find steady states • Determine their stability • Describe the behavior of solutions • Interpret model behavior

  9. Bifurcation Review • Bifurcations • What are they? • Bifurcation diagrams • What are they, why are they useful? • Generate them • Read them and interpret them

  10. 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 • Don’t worry about hysteresis!

  11. Continuous Models Review • Systems of ODEs: The Chemostat, • Lectures 9 and 10, Lab 5, Chapters: 4.2 - 6.2 • Nondimensionalization • Steady states • 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)

  12. Continuous Model Essentials • Nondimensionalize • Find Steady States • Name/Interpret them • Determine their existence conditions • Determine and Characterize Stability • Draw Phase Portraits • Provide mathematical conclusions regarding model behavior • Interpret the results in terms of the biological problem

  13. Model Building • Given a description of a biological problem, be able to derive a mathematical model

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