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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

- Homogeneous (Bacteria growth), Inhomogeneous (Breathing model)

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 Equations

- Solutions
- Characteristic equation
- Look for solution of the form xn = Cln
- Find eigenvalues, l

- General Solution:
- Linear Combinations of all basic solutions

- Characteristic equation
- Behavior of Solutions
- Dominant eigenvalue

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

- 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 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

- You should be able to:
- Find steady states
- Determine their stability
- Describe the behavior of solutions
- Interpret model behavior

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

- 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!

- Logistic Equation and Spruce Budworm

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

- Lecture 9 and 10 Notes, Sections 4.6 and 5.5 in Text
- Stability (Lecture Notes 9 and 10, 4.7, 4.9)
- Analytically
- RE(

- Graphically
- Phase portraits (Chapter 5, Lectures 9 and 10)

- Analytically

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

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

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