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.

Table of Contents

I. Models with Differential Equations

1. Models of Rates with Data

2. Introduction to R

3. Modeling with Rates of Change

4. Euler's Method

5. Phase Lines and Equilibrium Solutions

6. Coupled Systems of Equations

7. Exact Solutions to Differential Equations

II. Parameterizing Models with Data

8. Linear Regression and CuNe Fitting

9. Probability and Likelihood Functions

10. Cost Functions and Bayes' Rule

11. Sampling Distributions and the Bootstrap Method

12. The Metropolis-Hastings Algorithm

13. Markov Chain Monte Carlo Parameter Estimation

14. Information Criteria

Ill. Stability Analysis for Different ial Equations

15. Systems of Linear Differential Equations

16. Systems of Nonlinear Different ial Equations

17. Local Linearization and the Jacobian

18. What are Eigenvalues?

19. Qualitat ive Stability Analysis

20. Bifurcation

IV. Stochastic Differential Equations

21. Stochastic Biological Systems

22. Simulating and Visualizing Randomness

23. Random Walks

24. Diffusion and Brownian Motion

25. Simulating Stochast ic Different ial Equations

26. Stat ist ics of a Stochast ic Differential Equation

27. Solutions to Stochastic Different ial Equations

About the Author

John Zobitz is a Professor of Mathematics and Data Science at Augsburg University in Minneapolis, Minnesota. His scholarship in environmental data science includes ecosystem models parameterized with datasets from environmental observation networks. He is a member of the Mathematical Association of America (MAA) and previous president of the North Central Section of the MAA. He has served on the editorial board of MAA Notes. He was a recipient of the Fulbright-Saastamoinen Foundation Grant in Health and Environmental Sciences at the University of Eastern Finland in Kuopio, Finland. In addition, he is an affiliated member of the Ecological Forecasting Network and regularly taught at Fluxcourse, an annual summer course for measurements and modeling of ecosystem biogeochemical fluxes.