TEST 1 REVIEW
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
• 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
• 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
  1   2
• 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!
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(l
• Graphically
– Phase portraits (Chapter 5, Lectures 9 and 10)
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