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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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• 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

• 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

• 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

• You should be able to:

• Characterize and know the properties of the equations

• Solve Linear equations

• Describe the behavior of solutions

• 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 Systems: Host-Parasitoid Interactions

• Chapters 2.7, 2.8, 3.2-3.4

• Stability

• For other examples see section 3.5 and homework #3

• You should be able to:

• Determine their stability

• Describe the behavior of solutions

• Interpret model behavior

• Bifurcations

• What are they?

• Bifurcation diagrams

• What are they, why are they useful?

• Generate them

• Read them and interpret them

• 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

• 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)

• Nondimensionalize

• 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

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