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

• 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 = Cn • Find eigenvalues,

– General Solution: • Linear Combinations of all basic solutions

• Behavior of Solutions• Dominant eigenvalue

Linear Systems of Discrete Equations

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

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β <1+ γ < 2

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( • 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