Mathematical Methods in Biology

1. Introduction To Ecological Modeling.

1.1 Mathematical Models.

1.2 Rates of Change.

1.3 Balance Laws.

1.4 Temperature in the Environment.

1.5 Dimensionless Variables.

1.6 Descriptive Statistics.

1.7 Regression and Curve Fitting.

1.8 Reference Notes.

2. Population Dynamics for Single Species.

2.1 Laws of Population Dynamics.

2.2 Continuous Time Models.

2.3 Qualitative Analysis of Population Models.

2.4 Dynamics of Predation.

2.5 Discrete Time Models.

2.6 Equilibria, Stability, and Chaos.

2.7 Reference Notes.

3. Structure and Interacting Populations.

3.1 Structure--Juveniles and Adults.

3.2 Structured Linear Models.

3.3 Nonlinear Interactions.

3.4 Appendix--Matrices.

3.5 Reference Notes.

4. Interactions in Continuous Time.

4.1 Interacting Populations.

4.2 Phase Plane Analysis.

4.3 Linear Systems.

4.4 Nonlinear Systems.

4.5 Bifurcation.

4.6 Reference Notes.

5. Concepts of Probability.

5.1 Introductory Examples and Definitions.

5.2 The Hardy-Weinberg Law.

5.3 Continuous Random Variables.

5.4 Discrete Random Variables.

5.5 Joint Probability Distributions.

5.6 Covariance and Correlation.

5.7 Reference Notes.

6. Statistical Inference.

6.1 Introduction.

6.2 Interval Analysis.

6.3 Estimating Proportions.

6.4 The Chi-Squared Test.

6.5 Hypothesis Testing.

6.6 Bootstrap Methods.

6.7 Reference Notes.

7. Stochastic Processes.

7.1 Introduction.

7.2 Randomizing Discrete Dynamics.

7.3 Random Walk.

7.4 Birth Processes.

7.5 Stochastic Differential Equations.

7.6 SDEs from Markov Models.

7.7 Solving SDEs.

7.8 The Fokker-Planck Equation.

7.9 Reference Notes.

A. Hints and Solutions to Exercises