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Understanding Single Species Discrete Equations: A Comprehensive Review

This review covers single species discrete equations, emphasizing both homogeneous models, such as bacterial growth, and inhomogeneous models, such as breathing dynamics. Key concepts include finding solutions, the behavior of solutions based on characteristic equations, and analyzing stability through graphs and diagrams. We delve into nonlinear discrete equations, steady states, stability analysis, and bifurcation theory. This guide offers a foundational understanding of these models, enhances problem-solving skills, and prepares you for modeling biological phenomena analytically and graphically.

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Understanding Single Species Discrete Equations: A Comprehensive 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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