Exploring Modeling with Data and Differential Equations Using R

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
List of Figures
Welcome
I. Models with Differential Equations
	1. Models of Rates with Data
		1.1. Rates of change in the world: a model is born
		1.2. Modeling in context: the spread of a disease
		1.3. Model solutions
		1.4. Which model is best?
		1.5. Start here
		1.6. Exercises
	2. Introduction to R
		2.1. R and RStudio
		2.2. First steps: getting acquainted with R
		2.3. Increasing functionality with packages
		2.4. Working with R: variables, data frames, and datasets
		2.5. Visualization with R
		2.6. Defining functions
		2.7. Concluding thoughts
		2.8. Exercises
	3. Modeling with Rates of Change
		3.1. Competing plant species and equilibrium solutions
		3.2. The Law of Mass Action
		3.3. Coupled differential equations: lynx and hares
		3.4. Functional responses
		3.5. Exercises
	4. Euler’s Method
		4.1. The flu and locally linear approximation
		4.2. A workflow for approximation
		4.3. Building an iterative method
		4.4. Euler’s method and beyond
		4.5. Exercises
	5. Phase Lines and Equilibrium Solutions
		5.1. Equilibrium solutions
		5.2. Phase lines for differential equations
		5.3. A stability test for equilibrium solutions
		5.4. Exercises
	6. Coupled Systems of Equations
		6.1. Flu with quarantine and equilibrium solutions
		6.2. Nullclines
		6.3. Phase planes
		6.4. Generating a phase plane in R
		6.5. Slope fields
		6.6. Exercises
	7. Exact Solutions to Differential Equations
		7.1. Verify a solution
		7.2. Separable differential equations
		7.3. Integrating factors
		7.4. Applying the verification method to coupled equations
		7.5. Exercises
II. Parameterizing Models with Data
	8. Linear Regression and Curve Fitting
		8.1. What is parameter estimation?
		8.2. Parameter estimation for global temperature data
		8.3. Moving beyond linear models for parameter estimation
		8.4. Parameter estimation with nonlinear models
		8.5. Towards model-data fusion
		8.6. Exercises
	9. Probability and Likelihood Functions
		9.1. Linear regression on a small dataset
		9.2. Continuous probability density functions
		9.3. Connecting probabilities to linear regression
		9.4. Visualizing likelihood surfaces
		9.5. Looking back and forward
		9.6. Exercises
	10. Cost Functions and Bayes’ Rule
		10.1. Cost functions and model-data residuals
		10.2. Further extensions to the cost function
		10.3. Conditional probabilities and Bayes’ rule
		10.4. Bayes’ rule in action
		10.5. Next steps
		10.6. Exercises
	11. Sampling Distributions and the Bootstrap Method
		11.1. Histograms and their visualization
		11.2. Statistical theory: sampling distributions
		11.3. Summary and next steps
		11.4. Exercises
	12. The Metropolis-Hastings Algorithm
		12.1. Estimating the growth of a dog
		12.2. Likelihood ratios for parameter estimation
		12.3. The Metropolis-Hastings algorithm for parameter estimation
		12.4. Exercises
	13. Markov Chain Monte Carlo Parameter Estimation
		13.1. The recipe for MCMC
		13.2. MCMC parameter estimation with an empirical model
		13.3. MCMC parameter estimation with a differential equation model
		13.4. Timing your code
		13.5. Further extensions to MCMC
		13.6. Exercises
	14. Information Criteria
		14.1. Model assessment guidelines
		14.2. Information criteria for assessing competing models
		14.3. A few cautionary notes
		14.4. Exercises
III. Stability Analysis for Differential Equations
	15. Systems of Linear Differential Equations
		15.1. Linear systems of differential equations and matrix notation
		15.2. Equilibrium solutions
		15.3. The phase plane
		15.4. Non-equilibrium solutions and their stability
		15.5. Exercises
	16. Systems of Nonlinear Differential Equations
		16.1. Introducing nonlinear systems of differential equations
		16.2. Zooming in on the phase plane
		16.3. Determining equilibrium solutions with nullclines
		16.4. Stability of an equilibrium solution
		16.5. Graphing nullclines in a phase plane
		16.6. Exercises
	17. Local Linearization and the Jacobian
		17.1. Competing populations
		17.2. Tangent plane approximations
		17.3. The Jacobian matrix
		17.4. Exercises
	18. What are Eigenvalues?
		18.1. Introduction
		18.2. Straight line solutions
		18.3. Computing eigenvalues and eigenvectors
		18.4. What do eigenvalues tell us?
		18.5. Concluding thoughts
		18.6. Exercises
	19. Qualitative Stability Analysis
		19.1. The characteristic polynomial (again)
		19.2. Stability with the trace and determinant
		19.3. A workflow for stability analysis
		19.4. Stability for higher-order systems of differential equations
		19.5. Exercises
	20. Bifurcation
		20.1. A series of equations
		20.2. Bifurcations with systems of equations
		20.3. Functions as equilibrium solutions: limit cycles
		20.4. Bifurcations as analysis tools
		20.5. Exercises
IV. Stochastic Differential Equations
	21. Stochastic Biological Systems
		21.1. Introducing stochastic effects
		21.2. A discrete dynamical system
		21.3. Environmental stochasticity
		21.4. Discrete systems of equations
		21.5. Exercises
	22. Simulating and Visualizing Randomness
		22.1. Ensemble averages
		22.2. Repeated iteration
		22.3. Exercises
	23. Random Walks
		23.1. Random walk on a number line
		23.2. Iteration and ensemble averages
		23.3. Random walk mathematics
		23.4. Continuous random walks and diffusion
		23.5. Exercises
	24. Diffusion and Brownian Motion
		24.1. Random walk redux
		24.2. Simulating Brownian motion
		24.3. Exercises
	25. Simulating Stochastic Differential Equations
		25.1. The stochastic logistic model
		25.2. The Euler-Maruyama method
		25.3. Adding stochasticity to parameters
		25.4. Systems of stochastic differential equations
		25.5. Concluding thoughts
		25.6. Exercises
	26. Statistics of a Stochastic Differential Equation
		26.1. Expected value of a stochastic process
		26.2. Birth-death processes
		26.3. Wrapping up
		26.4. Exercises
	27. Solutions to Stochastic Differential Equations
		27.1. Meet the Fokker-Planck equation
		27.2. Deterministically the end
		27.3. Exercises
References
Index