1 State Space

1.1 Introduction to State Variables and State Space

1.2 Defining Vectors: Working With n-Dimensional Space

2 Operating with Vectors

2.1 State Space: Rules for Operating with Vectors

2.2 Addition and Scalar Multiplication of 2-Dimensional Vectors

3 Trajectories and Time Series

3.1 Vectors and Trajectories - a little history

3.2 Change in State Space: from time series to trajectories and back again

3.3 Functions

4 Models and Change Vectors

4.1 Introduction to Models and Change Equations: The Bathtub

4.2 Models and Change Equations in 2 Dimensions: Shark Meets Tuna

4.3 From Change Equations to Change Vectors

4.4 State spaces and change vectors in 2D

4.5 Trajectories in 2D

4.6 Logistic Growth

4.7 Models and Vectorfields: a nonlinear example in 1D

4.8 Modeling Chemical Reactions

4.9 Epidemiology

4.10 Immune Dynamics

5 Euler's Method

5.1 Introducing Euler's Method

5.2 Algebra of Euler's Method in 1D

5.3 Algebra of Euler's Method in 2D

5.4 Katherine Johnson and Euler's Method

5.5 Euler's Method to Create a Model of Ventricular Fibrillation

6 Equilibrium Points

6.1 Equilibrium Points in 1D

6.2 Stable and Unstable Equilibrim Points

7 Linear Stability Analysis

7.1 Linear Stability Analysis 1

7.2 Linear Stability Analysis 2

7.3 Review of EPs in 1D

8 Higher Dimensions

8.1 Linear EPS in 2D and higher

8.2 Nonlinear EPs - the Deer Moose model

8.3 Assessing the Stability of Equilibrium Points in 2D using nullclines and extreme test points: the Deer-Moose model

8.4 Stable and Unstable EPs in 1 and 2D

9 The Lac operon: a biological switch

9.1 Basins and the definition of a “switch”

9.2 The lac operon

9.3 The EP structure of the lac operon model

10 Bifurcation

10.1 Equilibrium Points in the Lac Operon model

10.2 The Bifurcation Diagram of Lac Operon

10.3 The Outbreak of the Spruce Budworm

11 Additional Bifurcations

11.1 Pitchfork Bifurcation

11.2 Transcritical Bifurcation

12 The Central Dogma

12.1 The Central Dogma

12.2 Central Dogma vs. Multiple feedbacks on gene expression

12.3 Bifurcation as qualitative change

13 Stable Oscillations in Science (and Music)

13.1 Oscillation in the Sciences

13.2 Rayleigh’s Clarinet

13.3 Attractors and the Tacoma Narrows Bridge

14 Oscillations in Physiology

14.1 The Hypothalamo-Pituitary-Gonadal Axis

14.2 Control of CO2 by Respiration, Muscle Tremor - the Mackey-Glass model

14.3 Insulin Glucose Oscillations - Time Delays

14.4 Oscillatory gene expression

15 Biological Oscillations

15.1 The Holling Tanner model

15.2 Bifurcations and forms of motion

16 Chaos 1

16.1 A new kind of behavior: fluid turbulence and cardiac fibrillation

16.2 A mathematical model of a 3-species food chain

16.3 Chaos in the Food Chain model

16.4 Chaos and its Properties

16.5 Extreme Sensitivity and the Butterfly Effect

17 Chaos 2

17.1 Discrete Time Dynamical Systems

17.2 Discrete Time Logistic Function

17.3 Sensitive Dependence in the Logistic Model

17.4 The Cause of Chaos in the Logistic Model

17.5 Routes to Chaos in the Logistic Model

17.6 A Therapy for Chaos in the Logistic Model

18 Chaos 3

18.1 The Flickering Candle

18.2 Diagnosing Chaos in Nature

19 Chaos 4

19.1 A Route to Chaos in the Logistic Model

19.2 Route to Chaos in the Food Chain Model

19.3 Review of Chaos in Nature

19.4 The Dripping Faucet

19.5 Streching and Folding

20 Chaos and Cardiology

20.1 Cardiac Chaos

20.2 Periodically Stimulated Cardiac Cells

20.3 Ventricular Fibrillation

21 Applications of Linear Algebra

21.1 Applications of Linear Algebra 1 - WWW2 Strategic Bombing

21.2 Applications of Linear Algebra 2 - The Google PageRank Algorithm

21.3 Applications of Linear Algebra 3 - Google PageRank and Foodwebs

22 Interactive Simulations

22.1 A Model of HIV Immunology within an Individual Person

22.2 Grass-Sheep-Wolves

22.3 Shark-Tuna

22.4 Lac Operon

22.5 Epidemic modeling

A mathematical model of a 3-species food chain

Summary

Developing a model of a 3-species food chain, where \(Z\) eats \(Y\) and \(Y\) eats \(X\).