Mathematical Modeling in Biology

Shandelle M. Henson is Professor of Mathematics and Professor of Ecology at Andrews University, Michigan, USA. She uses dynamical systems theory and bifurcation theory to study nonlinear population dynamics and the effects of climate change on marine organisms. Shandelle earned a Ph.D. in mathematics at the University of Tennessee, Knoxville and did several years of postdoctoral work at the University of Arizona. She serves as editor-in-chief of the journal Natural Resource Modeling and is a coauthor of the book Chaos in Ecology: Experimental Nonlinear Dynamics (Academic Press 2003) which documented chaotic dynamics in laboratory populations of insects. James L. Hayward is Professor Emeritus of Biology at Andrews University, Michigan, USA. He earned a Ph.D. in zoology from Washington State University, and he has studied the behaviors and population dynamics of marine birds and mammals in the Pacific Northwest of North America since he began graduate studies in 1972. In addition, Jim has studied the behavior of marine iguanas and flightless cormorants on the remote and uninhabited island of Fernandina, the westernmost island in the Galapagos. He also is a recognized expert in the fossilization of eggshell in birds and dinosaurs. Shandelle and Jim, a wife-husband research team, are widely published in both the technical and popular literature, and both have won awards for their teaching. They have applied mathematical methods to the behavioral dynamics of seabirds at Protection Island National Wildlife Refuge in the Strait of Juan de Fuca, Washington, USA since 2002. They reside in Niles, Michigan, USA and they have a daughter, son-in-law, and four grandsons. They enjoy hiking, geology, art, music, and literature.

I. Introduction to Modeling 1. Mathematical Modeling. 1.1. What You Should Know About This Chapter. 1.2 The modeling cycle. 1.3. Biology. 1.4. Mathematics. 1.5. Statistics. 1.6. Epistemology: How We Know. 1.7. Exercises. 2. Avian Bone Growth: A Case Study. 2.1. What you should know about this chapter. 2.2. Scientific Problem. 2.3. Translation into Mathematics. 2.4. Model Parametrization. 2.5. Model Selection. 2.6. Model Validation. 2.7. Exercises. II. Discrete-Time Models. 3. Discrete-Time Maps. 3.1. What you should know about this chapter. 3.2. Compartmental Model. 3.3. Linear Maps. 3.4. Nonlinear Maps. 3.5. Linearization. 3.6. The Ricker Nonlinearity. 3.7. Exercises. 4. Chaos: Simple Rules Can Generate Complex Results. 4.1. What you should know about this chapter. 4.2. Ricker Model Revisited. 4.3. New Paradigms Associated from Chaos. 4.4. May’s Hypothesis. 4.5. Exercises. 5. Higher Dimensional Discrete-time Models. 5.1. What you should know about this chapter. 5.2. Intraspecific Interactions. 5.3. Interspecific Interactions. 5.4. Example of an Age-structured Single-Species Model. 5.5. Example of a Two-Species Model. 5.6. n-dimensional Linear Difference Equations. 5.7. Solving Linear Systems of Difference Equations. 5.8. Nonlinear Systems. 5.9. Exercises. 6. Flour Beetle Dynamics: A Case Study. 6.1. What you should know about this chapter. 6.2. Flour Beetles. 6.3. Data. 6.4. Assumptions. 6.5. Alternative Deterministic Models. 6.6. Stochastic Models. 6.7. Model Parametrization. 6.8. Model Selection. 6.9. Model Validation. 6.10. The "Hunt for Chaos" Experiments. 6.11. Exercises. III. Continuous-time Models 7. Introduction to Differential Equations. 7.1. What you should know about this chapter. 7.2. Compartmental Models. 7.3. Exercises. 8. Scalar Differential Equations. 8.1. What you should know about this chapter. 8.2. Linear Equations. 8.3. Nonlinear Equations. 8.4. Bifurcations. 8.5. Exercises. 9. Systems of Differential Equations. 9.1. What you should know about this chapter. 9.2. Linear Systems of ODEs and Phase Plane Analysis. 9.3. Nonlinear Systems of ODEs. 9.4. Limit Cycles, Cycle Chains, and Bifurcations. 9.5. Lotka-Volterra Models and Nullcline Analysis. 9.6. Exercises. 10. Seabird Behavior: A Case Study. 10.1. What you should know about this chapter. 10.2. The Scientific Problem. 10.3. Historical Data. 10.4. General Model. 10.5. Alternative Models. 10.6. Model Parametrization. 10.7. Model Selection. 10.8. Model Validation. 10.9. Test of a priori predictions. 10.10. Steady-state model. 10.11. Discussion. Exercises. IV. Regression Models. 11. Regression Models. 11.1. What You Should Know About This Chapter. 11.2. Linear Regression. 11.3. Logistic Regression. 11.4. Generalized Linear Models (GLMs). 11.5. Interaction Terms. 11.6. Exercises. 12. Climate Change and Seabird Cannibalism: A Case Study. 12.1. What You Should Know About This Chapter. 12.2. The Scientific Problem. 12.3. Data. 12.4. Logistic Regression Analysis. 12.5. Model Validation. 12.6. Outcomes. 12.7. Climate Change, Cannibalism, and Reproductive Synchrony. 12.8. Exercises. V. Appendix. A. Linear Algebra Basics. B. MATLAB: The Basics. C. Connecting Models to Data: A Brief Summary with Sample Codes.