Transition files

These files are designed to instruct the popMod module of MDiG on how to build a transition matrix in order to apply stage- or age-based population models in the spatial context that MDiG provides. The popMod module is designed to implement two type of matrices; either static (all values within the matrix are fixed and do not change) or stochastic (values within the matrix are subject to random variation). The transition file uses the approach of specifying each individual transition value which are then combined to form the transition matrix.

Table of Contents Population Stages File Structure Parameters Expressions Transition Value Definition Stochasticity Environmental Stochasticity Demographic Stochasticity

Population Stages

Each stage of the population is represented by a raster which contains the number of individuals at that stage in each cell. These stages and their order are determined by their placement within the <output></output> tags in the main DispersalML file. Within each cell, all of the different stages are queried to form a one dimensional population matrix which contains the number of individuals in each stage within that cell

<math>\begin{bmatrix}

12 \\ 1 \\ 3 \end{bmatrix}</math>

The cell-level population matrix is then mulitplied by one or more transition matrices to obtain the population distribution for the following timestep. Transition matrices can be uniquely generated for each application; therefore each cell has the potential for a unique trajectory.

File Structure

Like the model file, the transition files are written in xml and parsed by the software to extract the relevant and appropriate information. Each transition file represents the application of a single matrix to the population within a single cell. The file structure contains two main sections; the first defines the data source for parameters, and the second applies the parameters in an expression which is evaluated to calculate the individual transition values.

Parameters

Each parameter definition is performed using the following syntax:

    <ParameterValue>
        <parameterName>Fec</parametername>
         <source>static</source>
        <index>1</index>
        <distribution>None</distribution>
        <values>
            <d>82.04</d>
            <d></d>
            <d></d>
        </values>
    </parametervalue>    

Each of the arguments is as follows:

- parameterName: Defines a string used to identify the parameter in the expression (see below)

- source: Determines how the parameter is derived; can take the values 'static', 'random', or 'CODA'

- index: Specifies the value that this is associated with from the <indexmap></indexmap>. Can also be specified as the path to an index file associated with a CODA file from a Bayesian estimation using OpenBUGS.

- distribution: Name of the distribution if the value is derived from a standard distribution (see below), otherwise, 'None'

- values: Associated parameter values. Either a static number, the arguments for a number generated from a standard distribution, or the pathname to a chain from a Bayesian fitting performed in OpenBUGS

Expressions

An expression must be specified for each position in the transition matrix. These are defined as mathematical functions which may or may not incorporate parameters defined as above. Expressions are specified using the following syntax:

    &lt&#59;expression&gt&#59; 
        &lt&#59;position&gt&#59;0&lt&#59;/position&gt&#59;
        &lt&#59;formula&gt&#59;0.2&lt&#59;/formula&gt&#59;
    &lt&#59;/expression&gt&#59;

The <position></position> tag specifies which transition value is defined by the corresponding <formula></formula>. Position is determined by numbering each position in the matrix left to right and top to bottom, starting at zero. For example, the positions in a 3x3 matrix are as follows:

<math>\begin{bmatrix}

0 & 1 & 2 \\ 3 & 4 & 5 \\ 6 & 7 & 8 \end{bmatrix}</math>

The <formula></formula> tags contain the right side of a mathematical equation that determines the value of the transition value. The formula consists of a number of values

Transition Value Definition

Stochasticity

PopMod has the ability to incorporate either demographic or environmental stochasticity within the population modeling.

Environmental Stochasticity

Environmental stochasticity refers to stochasticity in the transition values themselves. This variation inherently represents fluctuation in environmental conditions. This may be specified explicitly (ie. referencing a map of growing-degree days) or implicitly by defining a sampling space for the parameter. Parameter sample space can be determined using a standard distribution...

Random Distributions

Sampling from a random distribution uses the "numpy.random" routine, included in the numpy module for python. A list of the available distributions for the routine that can be specified can be found here. For example, specifying a parameter as a random draw from a normal (Gaussian) distribution with mean=0 and standard dev=10 can be achieved by:

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Demographic Stochasticity