Invitation to Mathematical Biology

David G Costa is a mathematician interested in Partial Differential Equations (PDEs), Ordinary Differential Equations (ODEs) and the Calculus of Variations. In particular, he is interested in the use of so-called variational and topological techniques to study qualitatively and visualization of phenomena in PDEs and ODEs. Such phenomena are present in various areas of sciences, including physics, biology, and chemistry, among others. He teaches a variety of courses at the undergraduate level (including Calculus, Linear Algebra, ODEs, PDEs, and Introductory Real Analysis), and graduate level (including PDEs, and Real Analysis), as well as a course in Biomathematics jointly offered by the Department of Mathematical Sciences and School of Life Sciences.Paul J Schulte is a plant physiologist interested in biophysical approaches to studying internal processes in plants.  These commonly involve applications of mathematical approaches asrealized through computational solutions.  Plants are dependent on water for survival and their ability to acquire water from the soil and transport it throughout the plant is determined in part by the hydraulic properties of the plant's tissues.  Most of his work considers transport processes such as water flow in the xylem tissues or sugar flow in the phloem tissues.  He teaches a variety of courses such as Plant Physiology, Plant Anatomy, Introduction to Biological Modeling, and Biomathematics jointly offered in the School of Life Sciences and Department of Mathematical Sciences.

• Preface.- 1 Introduction.- 2 Exponential Growth and Decay.- 2.1 Exponential Growth.- 2.2 Exponential Decay.- 2.3 Summary.- 2.4 Exercises.- 2.5 References- 3 Discrete Time Models.- 3.1 Solutions of the discrete logistic.- 3.2 Enhancements to the Discrete Logistic Function.- 3.3 Summary.- 3.4 Exercises.- 3.5 References- 4 Fixed Points, Stability, and Cobwebbing.- 4.1 Fixed Points and Cobwebbing.- 4.2 Linear Stability Analysis.- 4.3 Summary.- 4.4 Exercises.- 4.5 References- 5 Population Genetics Models.- 5.1 Two Phenotypes Case.- 5.2 Three Phenotypes Case.- 5.3 Summary.- 5.4 Exercises.- 5.5 References- 6 Chaotic Systems.- 6.1 Robert May’s Model.- 6.2 Solving the Model.- 6.3 Model Fixed Points.- 6.4 Summary.- 6.5 Exercises.- 6.6 References- 7 Continuous Time Models.- 7.1 The Continuous Logistic Equation.- 7.2 Equilibrium States and their Stability.- 7.3 Continuous Logistic Equation with Harvesting.- 7.4 Summary.- 7.5 Exercises.- 7.6 References-.- 8 Organism-Organism Interaction Models.-8.1 Interaction Models Introduction.- 8.2 Competition.- 8.3 Predator-Prey.- 8.4 Mutualism.- 8.5 Summary.- 8.6 Exercises.- 8.7 References- 9 Host-Parasitoid Models.- 9.1 Beddington Model.- 9.2 Some Solutions of the Beddington Model.- 9.3 MATLAB Solution for the Host-Parasitoid Model.- 9.4 Python Solution for the Host-Parasitoid Model.- 9.5 Summary.- 9.6 Exercises.- 9.7 References- 10 Competition Models with Logistic Term.- 10.1Addition of Logistic Term to Competition Models.- 10.2 Predator-Prey-Prey Three Species Model.- 10.3Predator-Prey-Prey Model Solutions.- 10.4 Summary.- 10.5Exercises.- 10.6References- 11 Infectious Disease Models.- 11.1 Basic Compartment Modeling Approaches.- 11.2SI Model.- 11.3SI model with Growth in S.- 11.4 Applications using Mathematica.- 11.5 Applications using MATLAB.- 11.6 Summary.- 11.7 Exercises.- 11.8 References- 12 Organism Environment Interactions.- 12.1 Introduction to Energy Budgets.- 12.2 Radiation.- 12.3 Convection.- 12.4 Transpiration.- 12.5 Total Energy Budget.- 12.6 Solving the Budget: Newton’s Method for Root Finding.- 12.7 Experimenting with the Leaf Energy Budget.- 12.8 Summary.- 12.9 Exercises.- 12.10 References- 13 Appendix 1: Brief Review of Differential Equations in Calculus- 14 Appendix 2: Numerical Solutions of ODEs- 15 Appendix 3: Tutorial on Mathematica- 16 Appendix 4: Tutorial on MATLAB- 17 Appendix 5: Tutorial on Python Programming- Index

“This very brief volume is designed for biology undergraduates with only a basic calculus background. … it is very self contained, with the appendixes providing the basics of standard numerical packages.” (Louis J. Gross, The Quarterly Review of Biology, Vol. 100 (1), March, 2025) “I think the authors have succeeded in their goal of providing an ‘invitation’ to mathematical biology. Not only can this book serve as a stepping stone to more comprehensive mathematical biology books, it also offers the reader a glimpse into a variety of topics that they can pursue in deeper detail later on if they are interested.” (Anita T. Layton, SIAM Review, Vol. 66 (3), August, 2024)