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An Introduction to Mathematical Modelling by Glenn Marion


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An Introduction to Mathematical Modelling written by Glenn Marion, Bioinformatics and Statistics Scotland. Given 2008 by Glenn Marion and Daniel Lawson

An Introduction to Mathematical Modelling written by Glenn Marion cover the following topics.

  • 1. Introduction
    1.1 What is mathematical modelling?
    1.2 What objectives can modelling achieve?
    1.3 Classifications of models
    1.4 Stages of modelling

  • 2. Building models
    2.1 Getting started
    2.2 Systems analysis
    2.2.1 Making assumptions
    2.2.2 Flow diagrams
    2.3 Choosing mathematical equations
    2.3.1 Equations from the literature
    2.3.2 Analogies from physics
    2.3.3 Data exploration
    2.4 Solving equations
    2.4.1 Analytically
    2.4.2 Numerically

  • 3. Studying models
    3.1 Dimensionless form
    3.2 Asymptotic behaviour
    3.3 Sensitivity analysis
    3.4 Modelling model output

  • 4. Testing models
    4.1 Testing the assumptions
    4.2 Model structure
    4.3 Prediction of previously unused data
    4.3.1 Reasons for prediction errors
    4.4 Estimating model parameters
    4.5 Comparing two models for the same system

  • 5. Using models
    5.1 Predictions with estimates of precision
    5.2 Decision support

  • 6. Discussion
    6.1 Description of a model
    6.2 Deciding when to model and when to stop

  • A Modelling energy requirements for cattle growth

  • B Comparing models for cattle growth

  • List of Figures
    1 A schematic description of a spatial model
    2 A flow diagram of an Energy Model for Cattle Growth
    3 Diffusion of a population in which no births or deaths occur
    4 The relationship between logistic growth a population data
    5 Numerical estimation of the cosine function
    6 Scaling of two logistic equations, dy/dt = ry(a - y) to dimensionless form
    7 Graph of dy/dt against y for the logistic curve given by dy/dt = ry(a - y)
    8 Plots of dy/dt against y for modified logistic equations
    9 Phase plane diagram for the predator-prey system: dx dt = x(1 - y) (prey) & dy dt = -y(1 - x) (predator), showing the states passed through between times t1 (state A) and time t2 (state B)
    10 Graph of yi against i for the chaotic difference equation yi+1 = 4yi(1 - yi)
    11 Left: The behaviour of the deterministic Lotka-Volterra predator-prey system. Right: The same model with stochastic birth and death events. The deterministic model predicts well defined cycles, but these are not stable to even tiny amounts of noise. The stochastic model predicts extinction of at least one type for large populations. If regular cycles are observed in reality, this means that some mechanism is missing from the model, even though the predictions may very well match reality.
    12 Comparison of two models via precision of parameter estimates.
    13 AIC use in a simple linear regression model. Left: The predictions of the model for 1,2,3 and 4 parameters, along with the real data (open circles) generated from a 4 parameter model with noise. Right: the AIC values for each number of parameters. The most parsimonious model is the 2 parameter model, as it has the lowest AIC.
    14 Distribution functions F(x) = Probability(outcome¡x) comparing two scenarios A and B.

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