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 (*X _{L}*) is a normal random variable:

where *μ _{L}* is the mean and

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 *n _{P}*, the number of population repetitions per landscape according to the following equation:

where *σ _{L}*

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, *t _{L}*, the time required to make an observation on

(A.2) |

Because *t _{L}*

To estimate *σ _{L}*

(A.3) |

where is the observed variance in mean MP across landscapes, *s _{i}*

The program then uses an unbiased estimator of within-landscape variance to provide an estimate of *σ _{P}*:

(A.4) |

where *n _{Pi}* is the number of population repetitions on the

The underlying between-landscape variance is then estimated by subtracting (A.4) from (A.3):

(A.5) |

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 *n _{P}* value, as more simulations are performed and more data become available. The estimates of optimal

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 *n _{P}* estimate and standard error of the EMP estimate over the course of the simulation.