test 1 review
Download
Skip this Video
Download Presentation
TEST 1 REVIEW

Loading in 2 Seconds...

play fullscreen
1 / 13

TEST 1 REVIEW - PowerPoint PPT Presentation


  • 76 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'TEST 1 REVIEW' - noelle


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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
ad