Ecological Modeling

  1. Mathematical and Statistical Foundations
    1. Essential Calculus
      1. Derivatives and Rates of Change
        1. Instantaneous Change in Populations
          1. Slope Interpretation
            1. Chain Rule Applications
            2. Integrals and Accumulation
              1. Area Under Curves
                1. Cumulative Effects in Ecology
                  1. Fundamental Theorem of Calculus
                  2. Ordinary Differential Equations (ODEs)
                    1. First-Order ODEs
                      1. Second-Order ODEs
                        1. Systems of ODEs
                          1. Analytical Solutions
                            1. Numerical Solutions
                              1. Euler's Method
                                1. Runge-Kutta Methods
                                  1. Adaptive Step Size Methods
                                2. Partial Differential Equations (PDEs)
                                  1. Spatial and Temporal Dynamics
                                    1. Diffusion Equations
                                      1. Advection Equations
                                        1. Reaction-Diffusion Systems
                                          1. Numerical Methods for PDEs
                                            1. Finite Difference Methods
                                              1. Finite Element Methods
                                          2. Linear Algebra
                                            1. Vectors and Matrices
                                              1. Vector Operations
                                                1. Matrix Operations
                                                  1. Representation of Populations and Interactions
                                                  2. Eigenvalues and Eigenvectors
                                                    1. Calculation Methods
                                                      1. Stability and Long-Term Behavior
                                                        1. Applications in Structured Models
                                                          1. Dominant Eigenvalue Interpretation
                                                          2. Matrix Decomposition
                                                            1. LU Decomposition
                                                              1. Singular Value Decomposition
                                                              2. Stability Analysis of Linear Systems
                                                                1. Equilibrium Points
                                                                  1. Local Stability Criteria
                                                                    1. Jacobian Matrix Analysis
                                                                  2. Probability and Statistics
                                                                    1. Probability Theory Fundamentals
                                                                      1. Sample Spaces and Events
                                                                        1. Conditional Probability
                                                                          1. Independence
                                                                          2. Probability Distributions
                                                                            1. Discrete Distributions
                                                                              1. Binomial Distribution
                                                                                1. Poisson Distribution
                                                                                  1. Negative Binomial Distribution
                                                                                  2. Continuous Distributions
                                                                                    1. Normal Distribution
                                                                                      1. Exponential Distribution
                                                                                        1. Gamma Distribution
                                                                                          1. Beta Distribution
                                                                                        2. Statistical Inference
                                                                                          1. Parameter Estimation
                                                                                            1. Method of Moments
                                                                                              1. Maximum Likelihood Estimation
                                                                                                1. Least Squares Estimation
                                                                                                2. Hypothesis Testing
                                                                                                  1. Type I and Type II Errors
                                                                                                    1. Power Analysis
                                                                                                      1. Multiple Testing Corrections
                                                                                                      2. Confidence Intervals
                                                                                                      3. Regression Analysis
                                                                                                        1. Simple Linear Regression
                                                                                                          1. Multiple Linear Regression
                                                                                                            1. Nonlinear Regression
                                                                                                              1. Model Diagnostics
                                                                                                                1. Residual Analysis
                                                                                                                2. Bayesian Statistics
                                                                                                                  1. Bayes' Theorem
                                                                                                                    1. Prior Distributions
                                                                                                                      1. Posterior Distributions
                                                                                                                        1. Markov Chain Monte Carlo (MCMC)
                                                                                                                          1. Gibbs Sampling
                                                                                                                            1. Metropolis-Hastings Algorithm
                                                                                                                            2. Time Series Analysis
                                                                                                                              1. Autocorrelation
                                                                                                                                1. Stationarity
                                                                                                                                  1. ARIMA Models
                                                                                                                                    1. State-Space Models
                                                                                                                                  2. Difference Equations for Discrete Time Models
                                                                                                                                    1. First-Order Difference Equations
                                                                                                                                      1. Higher-Order Difference Equations
                                                                                                                                        1. Systems of Difference Equations
                                                                                                                                          1. Stability Analysis in Discrete Systems
                                                                                                                                            1. Bifurcation Theory
                                                                                                                                              1. Fixed Points
                                                                                                                                                1. Periodic Orbits
                                                                                                                                                  1. Chaos

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                                                                                                                                              1. Introduction to Ecological Modeling

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                                                                                                                                              3. A Typology of Ecological Models