Modeling in Biology

3.

Mathematical Foundations for Modeling

3.1.

Calculus for Dynamic Systems

3.1.1.

Derivatives and Rates of Change

3.1.1.1.

Instantaneous Change

3.1.1.2.

Biological Growth Rates

3.1.1.3.

3.1.2.

Integrals and Accumulation

3.1.2.1.

Area Under Curves

3.1.2.2.

Cumulative Quantities

3.1.2.3.

Conservation Laws

3.1.3.

Multivariable Calculus

3.1.3.1.

Partial Derivatives

3.1.3.2.

Gradient and Divergence

3.2.

Differential Equations

3.2.1.

Ordinary Differential Equations

3.2.1.1.

First-Order ODEs

3.2.1.2.

Higher-Order ODEs

3.2.1.3.

Systems of ODEs

3.2.1.4.

Analytical Solutions

3.2.1.5.

Numerical Solutions

3.2.2.

Partial Differential Equations

3.2.2.1.

Diffusion Equations

3.2.2.2.

Reaction-Diffusion Equations

3.2.2.3.

3.2.2.4.

Boundary Conditions

3.3.

Difference Equations

3.3.1.

Discrete Time Systems

3.3.1.1.

Recurrence Relations

3.3.1.2.

Population Models in Discrete Time

3.3.1.3.

Stability Analysis

3.4.

3.4.1.

Vectors and Matrices

3.4.1.1.

3.4.1.2.

Transition Matrices

3.4.1.3.

Matrix Operations

3.4.2.

Eigenvalues and Eigenvectors

3.4.2.1.

Stability Analysis

3.4.2.2.

Long-Term Behavior

3.4.2.3.

Principal Component Analysis

3.5.

Probability and Statistics

3.5.1.

Probability Distributions

3.5.1.1.

Discrete Distributions

3.5.1.2.

Continuous Distributions

3.5.1.3.

Joint Distributions

3.5.2.

Statistical Inference

3.5.2.1.

Parameter Estimation

3.5.2.2.

Hypothesis Testing

3.5.2.3.

Confidence Intervals

3.5.3.

Bayesian Methods

3.5.3.1.

Bayesian Inference

3.5.3.2.

Prior and Posterior Distributions

3.5.3.3.

Markov Chain Monte Carlo

3.6.

Optimization Theory

3.6.1.

Constrained Optimization

3.6.2.

Unconstrained Optimization

3.6.3.

Global vs Local Optima

3.6.4.

Evolutionary Algorithms

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4. Modeling Paradigms and Techniques