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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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Single species discrete equations
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
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 equations1
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
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
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


Nonlinear discrete equations1
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


Nonlinear discrete essentials
Nonlinear Discrete Essentials

  • You should be able to:

    • Find steady states

    • Determine their stability

    • Describe the behavior of solutions

    • Interpret model behavior


Bifurcation review
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
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!


Continuous models review1
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)


Continuous model essentials
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


Model building
Model Building

  • Given a description of a biological problem, be able to derive a mathematical model


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