1. TEST 1 REVIEW

2. 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 Behavior of solutions - determined by the magnitude of ‘a’ –Increasing, decreasing, oscillating

3. Tumor Growth, Segmental Growth, Red Blood Cell Production, Blood CO 2 –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

4. Solutions –Characteristic equation Look for solution of the form x n = C n Find eigenvalues, –General Solution: Linear Combinations of all basic solutions Behavior of Solutions Dominant eigenvalue Linear Systems of Discrete Equations

5. Linear Discrete Essentials You should be able to: –Characterize and know the properties of the equations –Solve Linear equations –Describe the behavior of solutions

6. 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’(x e )| < 1 for stability Don’t worry about –2 point cycles –Chaos Look at 2.1, 2.2, 2.5, 3.1

7. 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

8. Nonlinear Discrete Essentials You should be able to: –Find steady states –Determine their stability –Describe the behavior of solutions –Interpret model behavior

9. Bifurcation Review Bifurcations –What are they? Bifurcation diagrams What are they, why are they useful? Generate them Read them and interpret them

10. 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!

11. 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 )

12. 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

13. Model Building Given a description of a biological problem, be able to derive a mathematical model