Exploring Modeling with Data and Differential Equations Using R provides a unique introduction to differential equations with applications to the biological and other natural sciences. Additionally, model parameterization and simulation of stochastic differential equations are explored, providing additional tools for model analysis and evaluation. This unified framework sits "at the intersection" of different mathematical subject areas, data science, statistics, and the natural sciences. The text throughout emphasizes data science workflows using the R statistical software program and the tidyverse constellation of packages. Only knowledge of calculus is needed; the text’s integrated framework is a stepping stone for further advanced study in mathematics or as a comprehensive introduction to modeling for quantitative natural scientists.
The text will introduce you to:
- modeling with systems of differential equations and developing analytical, computational, and visual solution techniques.
- the R programming language, the tidyverse syntax, and developing data science workflows.
- qualitative techniques to analyze a system of differential equations.
- data assimilation techniques (simple linear regression, likelihood or cost functions, and Markov Chain, Monte Carlo Parameter Estimation) to parameterize models from data.
- simulating and evaluating outputs for stochastic differential equation models.
An associated R package provides a framework for computation and visualization of results. It can be found here: https://cran.r-project.org/web/packages/demodelr/index.html.
Author(s): John Zobitz
Publisher: CRC Press/Chapman & Hall
Year: 2022
Language: English
Pages: 377
City: Boca Raton
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
