Models in Biology

What is a model?
    Reality and Validity

Types of Models
    Deterministic vs. Stochastic
    Descriptive vs. Mechanistic
    Dynamic vs. Static
    Analytical vs. Numerical

Computers and Modelling
    Using Computers
    Modelling Objectives

Elements of Modelling
    Problem Identification
    Implementation of Simple Version
    Debugging of Full Version
    Verification of Model
    Sensitivity Analysis

Population Growth

What are populations
    What Controls Populations?

A Model of Continuous Population Growth
    Model development

Modelling Continuous Population Growth Rate
    Instantaneous rate of increase ‘r’
    Doubling times
    Model Assumptions

Modelling Discrete Population Growth
    Finite rate of increase "lambda’
    Relationship between r and lambda

 Logistic Growth

Examining the assumption of constant b and d
    The effect of N (population size)
    K = the carrying capacity

The Logistic Growth Model
    Assumptions of the logistic model

Variations in the Logistic Growth Model
    Time lags (tau)
    Asymptotes and oscillations

The Discrete Model of Logistic Growth and tau
    More cycles and chaos

Optimal Yield: A practical example
    The Peruvian Anchovy Fishery

 Calculus

Differential Calculus
    Functions
    Limits
    Four Step Rule
    Derivatives
    Regular
    Transcendental
    Second Derivatives
    Min and Max

Integral Calculus
    Indefinite Integrals
    Definite Integrals
    Common Applications
    Areas
     Forecasting

Linear Algebra

Background on Applications

Matrices and Matrix Operations
    Matrix and Vector
    Matrix elements
    Matrix diagonal

Matrix Mathematics
    Addition
    Subtraction
    Multiplication
    Division (needs a diversion)

    Transpose
    Identity Matrix
    Symmetry
    Determinants (Cramer’s Rule)
    Adjugate Matrix
    Inverse

Eigenvalues and Eigenvectors (Solutions)
    Characterisitc Equation |A-lambdaI| = 0
    Eigenvalues (roots)

Linear Algebra Applications

Matrix Addition
    Population Surveys

Matrix Subtraction
    Weight Change Under Different Diets

 Matrix Multiplication (three of the following **)
    Fecundity of Moose **
    Dominance Hierarchy of Competitive Interactions
    Disease Transmission
    Population Dynamics **
    Statistical Analyses **

Determinants of Matrix
    Solution to Linear Equations

Eigenvalues of Matrices
    Stable Age Distribution Population Models
    Principal Components Analysis
 
Alternative Growth Models

Growth Models
    Exponential Growth
    Exponential Decay
    Logistic Growth

Analytical Solution
    General Form
    Growth of Organisms
    Allometric Growth Model
    Richard’s General Formula
    Monomolecular Growth Model
    Von Bertalanffy Growth Model
    Logistic Growth Model
    Gompertz Growth Model

 Chaos

Developing Models
    Step One: Identify the Problem
    Step Two:  Graphical and Mathematical Model
    Step Three: Equilibrium Conditions
    Step Four: The Nature of the Equilibrium Conditions

Ricker Stock Recruitment Model
    Defining the Problem
    Generating the Model
    Equilibrium Stock Size N*
    Numerical Approach to Equilibrium
    Difference Eqn. and Brute Force Technique
    Analytical Approach to Equilibrium
     Differential Eqn. and calculus

The Nature of Chaos
    Period Doubling
    Strange Attractors
    Self Similarity
    Fractals and Pattern Formation

Curve Fitting

Objectives
Deterministic vs. Stochastic Models

Estimation of variables
    Linear Regression
    Residuals
    Fitting the Model

Polynomials Models
    Non-linear Models

Transformation

Non-linear Least Squares Regression
    Model Distributions
    Estimating the Variance

General Linear Models
    Model
    Measures of Fit
    Outliers, Leverage, and Colinearity
 
Models and Random Elements

Random Variables
    The Normal Distribution

Stochasticity
 
Environmental Stochasticity
    Random variation in r
    Random variation in K
 
Demographic Stochasticity
    Chance of extinction

Random Sampling
    Types of random distributions
    Random Normal
    White noise
    Binomial Distribution
    Negative Exponential
    Continuous Distribution
    Lognormal Distribution

Spatial Analysis

General Considerations
    Uniform Distribution
    Hyperdispersed Distribution
    Random Distribution
    Poisson Model
    Clustered Distribution
    Negative Binomial Model

Problems with Clustering
    Grid Size
    False Negatives
    False Positives

Lloyd’s mean crowding and patchiness
Morisita’s Index of Dispersion
Nearest Neighbour Techniques

 Time Series Analysis

Some Definitions

What are we looking for?
    Systematic Patterns
    Seasonal Effects
    Cycles and Quasi-cycles
    Trends

Residual Variation
    Random Variation
    Stationary Time Series
    White Noise

Techniques for Decomposing Time Series
    Smoothing
    Moving Average
    Curve Fitting
    Differencing
    Autocorrelation
    Lags
    Measures of Accuracy
    Partial Autocorrelation, Autoregression etc.
 
Spectral Analysis

Purpose
Terminology
    Ensembles
    Stationary
    Transients
    Ergotic

Review of Period Functions
    Fourier Series
    Periodic Functions
    Fourier Series
    Polar Form
    Complex Form

Fourier Transforms
    Spectral Techniques
    Nyquist Frequency
    Filters
    Power Spectrum

Examples
 
Biological Interactions I

Types of Interactions

Competition
    Intra-Specific Competition
    Inter-Specific Competition
    Population Growth
        & Intra-Specific Competition
        & Intra + Inter-Specific Competition
 
Lotka-Volterra Models
    Competition Coefficients
    Equilibrium Conditions
    State Space
    Outcomes
 
Biological Interactions II

Equilibrium Conditions
    State Space

Outcomes of Competitive Interactions
    Species 1 Wins
    Species 2 Wins
    Species Coexistence
    Competitive Exclusion

Mathematical Formulation of Outcomes
    K1/K2, alpha and 1/beta
 
Biological Interactions III

Conditions for Species Coexistence
    Strong Competitors
    Weak Competitors

Assumptions of the Competition Model

Intraguild Predation
    Guilds

Biological Interactions
    Encounter Rates (N1N2)
    Conversion Constants (gamma, delta)
 
Predator - Prey Models I

Predatory-Prey Models
    Encounters (PV)
    Conversion Rates (alpha, beta)
    Functional Response (alphaV)
    Numerical Response (betaV)

Equilibrium Conditions
    Prey: P* = r /alpha
    Predators V* = q /beta
 
State Space
    Time Series

 Predator-prey Cycles
Assumptions of the Model
 
Predator-Prey Interactions II

Lotka-Volterra Model
    Equilibrium Conditions

Modifications to the Lotka-Volterra Model

    (1) Adding a Carrying Capacity to V
        Equilibrium Conditions

    (2) Predator Functional Responses
        Type I Functional Response

        Type II Functional Response
            tfeeding = t search + thandling
            Feeding Rate (n/t)
            Michaelis-Menten Kinetics (k, D)
            Equilibrium Conditions

 Type III Functional Response
    Logistic Type Response
    Equilibrium Conditions

 Predator-Prey Interactions III

Modifications to the Prey Isoclines

Allee Effect
    The hump-shaped isocline
        P isocline to left of V isocline
        P isocline intersects V isocline
        P isocline to right of V isocline

    Paradox of Enrichment
    The U-shaped prey isocline

Modifications to the Predator Isoclines
    Three Outcomes

The Effects of Changing P and V Isocline
    Clockwise Rotation
    Counterclockwise Rotation

Some Examples From Nature

 Mathematics and Ecology

What is Ecology

Ten Equations that Changed Biology (Jungck, 1997)

Hypothesis Testing
    Parametric Statistics
    Clustering and Ordination

Pattern Formation and Description
    Allometric Growth
    Mapping
    Advection-Diffusion Equations

Biological Interactions
    Competitive Interactions
    Predator-Prey Systems

Ecology and the Future

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Last modified on January 6, 2002.