Appendix A. A description of the REPEATER software.
We developed the REPEATER program to implement the optimization principles discussed in the main article. The REPEATER program automates the execution of multiple runs of the landscape and population models of RAMAS Landscape; this process previously required continual user interaction. The main article provides a context for the REPEATER program and outlines its functionality. The REPEATER executable, source code and user manual are available on-line. Here we discuss some technical issues relating to the software implementation.
In the main article we assume that the random component of MP associated with landscape stochasticity (XL) is a normal random variable:
where μL is the mean and σL2 is the variance of the distribution, and that the random component of MP associated with population stochasticity (XP) is a normal random variable with zero mean and constant variance σP2 independent of the particular landscape realization on which the simulation takes place:
Thus, the random variable representing MP is also a normal random variable:
For a given DLMP (dynamic landscape metapopulation) model, the REPEATER program seeks to minimize the variance of EMP (expected minimum population size) estimates. It does this by estimating an optimal value of nP, the number of population repetitions per landscape according to the following equation:
where σL2 is the variance associated with landscape stochastictiy, σP2 is the variance associated with population stochasticity, tL is the running time of the landscape module, and tP is the running time of the population module (see main article for derivation).
In terms of software implementation, the terms on the right hand side of (A.1) are generally unknown at the beginning of the simulation. From a conceptual point of view, tL, the time required to make an observation on XL, actually comprises two components: tL1, the time required for a run of the landscape model; and tL2, the overhead associated with a run of the metapopulation model. Also, tP, the time required to make an observation on XP, is a fraction of the time required for a run of the metapopulation model. However, for a given value of nP, REPEATER can measure directly only the time taken for a run of the landscape model (tL1) and the total time taken for a run of the metapopulation model (tG). Assuming a linear relationship between nP and tG (this assumption is upheld in practice for RAMAS GIS), we can write:
Because tL2 and tP cannot be measured directly, REPEATER estimates them from a regression of tG on nP in accordance with (A.2). To facilitate this, REPEATER uses an initial user-specified value of nP for the first landscape simulation, and doubles this for the second landscape simulation; estimates of the optimal value of nP are used thereafter. Under normal operating conditions, where model running times are stable, the parameters estimated from the regression converge to the true running times, regardless of the initial value of nP.
To estimate σL2 and σP2, the REPEATER program reads in the output from the metapopulation model and collates the data on minimum population size for each landscape and population repetition. The program estimates the total underlying variance between independent observations (i.e., observations on separate landscapes):
where is the observed variance in mean MP across landscapes, si2 is the observed variance of observations on the i th landscape (see Appendix B for derivation).
The program then uses an unbiased estimator of within-landscape variance to provide an estimate of σP:
where nPi is the number of population repetitions on the ith landscape and is the total number of observations across all landscapes (see Appendix B for derivation).
The underlying between-landscape variance is then estimated by subtracting (A.4) from (A.3):
If (A.5) gives a negative value, the REPEATER program simply uses = 0. This can occur by chance, and corresponds to a situation where the landscape means are under-dispersed, i.e., the variance between landscape means is less than would be expected based on the observed within-landscape variances.
Using these relations, REPEATER calculates progressively refined estimates of the parameters on the right hand side of (A.1), and hence of the optimal nP value, as more simulations are performed and more data become available. The estimates of optimal nP are used on subsequent landscape realizations.
Modellers should be aware that under certain conditions, REPEATER may not converge to an optimal solution over the course of a simulation. This can happen if model running times are variable (perhaps due to CPU time being shared with other processes), or if the assumptions of normality and homogeneity of variance are violated. Convergence should be assessed by examining plots of the optimal nP estimate and standard error of the EMP estimate over the course of the simulation.