4.2.1
Defining expressions
One of the most powerful features in PopMod is the flexibility with which the transition matrix can be specified. Each transition value within the transition matrix is produced by evaluating a user-defined expression. This expression is evaluated for each iteration of the transition matrix (defined as individual cell x timestep combinations), so that any stochastic elements of the matrix will be re-drawn for each cell in each timestep of the simulation.
In the simplest application, each expression in the transition matrix can evaluate to a single value, resulting in a static matrix that is exactly the same in every cell across the raster layer at every timestep. This static population model applied across a homogenous landscape will result in identical growth trajectories in each cell. In this case, any di↵eren- tiation between results in individual cells will be completely dependent on either starting values or immigration/emigration processes that result from dispersal of individuals. More complex applications can incorporate variation in the transition matrices between di↵erent cells. One way to incorporate that variation is to include environmental stochas- ticity into the expressions, implemented as variation in the parameter values (Caswell 2001). In PopMod this stochasticity is defined by specifying a distribution for each stochastic parameter, which serves to define the sample space from which the param- eter values are drawn. Values are then drawn from this distribution at random, and are then substituted into the expression when the model is applied. Currently, this sam- ple space can be defined using any of the 35 standard distributions currently utilised by Python’s numpy extension, providing a wide range of possibilities (Appendix A).
Figure 4.2: Diagram describing how PopMod integrates matrix population modelling into the the MDiG framework. Beginning at the top of the diagram, three di↵erent rasters represent three di↵erent lifestages (S, J, and A) at time t. At this point other modules available through MDiG can be applied; in this example, the dispersal module is applied to only individuals in the ‘seed’ stage, as established plants (J and A stages) are sessile. The lifestage vectors are assembled by querying the di↵erent lifestage rasters at a particular cell location (c) to construct the population vector for that location at time t, represented by the 3x1 matrix in the lower right. A transition matrix is then generated using a range of potential data sources. The population vector is then multiplied by the transition matrix to project the subsequent population vector at time t+1. This process is reiterated for all cell locations and then rewritten to a new set of rasters, which record the count of individuals in each stage at each location across the entire landscape. Variable definitions are the same as in Figure 1, with the addition of the location specificity (c).
In addition to using standard distributions to define the sample space, PopMod has the ability to utilise the posterior distributions produced by hierarchical Bayesian modelling obtained from BUGS software (Lunn et al. 2000). These posterior distributions are used to represent an estimated variable, and are produced in the form of a vector of estimates. The benefit of using a hierarchical Bayesian approach for parameter estimation is that a properly formulated model will produce a result where the confidence interval is not influenced by increasing sample size, as is the case with many traditional frequentist approaches. Instead, the distribution better reflects the uncertainty in the parameter value (Clark 2003), providing a more realistic sample space for simulation behaviour. PopMod
is currently set up to process posterior estimates in the form of the files produced by the BUGS software. The output from other MCMC sampling software can be incorporated, but it must be modified to match the BUGS output format. In addition, multiple factor levels in the posterior distribution can be referenced through the the indexing of parameter levels (see below). PopMod draws values for parameters in the same expression from the same position in the chains in order to account for any covariance between simultaneously estimated parameters.
Parameter values can also be sourced from additional rasters, allowing them to reflect spatial variability across the landscape. Any data raster can be used to provide the pa- rameter values as long as it has the same extent and resolution as the population rasters. This approach allows the user to incorporate any spatially explicit environmental data, such as moisture availability, light levels, nutrient availability, or any other relevant spatial data. In addition to environmental variables, this approach can be used to query previous population rasters, which are also stored and accessed as a raster format. This feature allows the number of individuals in the previous time step to be used as a parameter in the expression, providing an ideal avenue for implementing density-dependent regulatory processes. As each lifestage is assigned to a unique parameter in the expressions, their in- fluence can even be weighted according to the lifestage they currently occupy. This would allow, for example, adults to have a greater influence on density-dependent processes than seedlings (or vice-versa).
Lastly, di↵erences in the expressions for parameter values can be implemented through an indexing feature. This feature works by classifying the landscape into a finite number of classes. Each cell of the index raster is assigned an index integer, analogous to a factor level. This index is then used to determine which of multiple possible parameter definitions is used to source the parameter value to be implemented into the expression. The indexed parameters can be as similar or di↵erent to each other as desired; it may be di↵erent static values, represent an alternative parameterisation of a distribution for a stochastic parameter, or even contain a mix of stochastic and static parameter sources, which are dependant on the physical location. This indexing function provides an excellent option for integrating coarser delineations of the landscape such as habitats or biomes into the analysis.
4.2.2
Periodic matrix modelling
PopMod includes the functionality to implement multiple transition matrices during a single time step in a periodic approach to matrix modelling. This allows the transitions of a single timestep to be partitioned across a number of transition matrices (Figure 4.3). There are three direct benefits to this approach; first, applying matrix modelling to project plant populations contains a logistical dilemma; incorporating a seed stage inherently in- troduces a dormant year when the matrix is applied in annual increments (Caswell 2001).
Utilising the periodic approach allows one matrix to specify the recruitment of seed from the previous year, while a second can specify the production of seed. This allows seed that are produced in one year the opportunity to recruit to a plant stage the following year, as opposed to serving a timestep in the dormant seed stage. Secondly, while a de- mographic model could incorporate fecundity as a single step, separating seed production and recruitment provides an opportunity to enact a dispersal process on individuals in the seed stage. This is an important step as germination probabilities are more likely to be influenced by conditions the seed experiences after dispersal, as opposed to conditions at its origin. Incorporating the di↵erential response to the seeds eventual location in a single function would require accounting for all possible destinations, creating a function which is both unwieldy and ine↵ective. Lastly, separating the seed production and recruitment processes provides an opportunity to incorporate one or more seedbank stages, which can be important and influential aspect of plant success at the population level (Anderson et al. 1996; Lonsdale et al. 1988).
2 4SJtt At 3 5 ⇥ 2
4PPSS,aSJ,a PPJS,aJJ,a PPAS,aAJ,a
PSA,a PJA,a PAA,a
3 5 ⇥
2
4PPSS,bSJ,b PPJS,bJJ,b PPAS,bAJ,b
PSA,b PJA,b PAA,b
3 5 ⇥
2
4PPSS,cSJ,c PPJS,cJJ,c PPAS,cAJ,c
PSA,c PJA,c PAA,c
3 5 = 2 4SJt+1t+1 At+1 3 5
Figure 4.3: Example of a periodic matrix application where instead of a single transition matrix, multiple transition matrices are utilised in the calculation of the population vector at t+1.