persistence. Obviously measures of patch contribution should therefore support accurate predictions about the importance of a patch; however, in order to have maximum impact, they should also be relatively simple to compute from available data.
4.2
Characterising patch value
In this section, we provide a brief overview of existing work to characterise the value of a patch in a metapopulation framework. To date, most of the theory has been conducted using a deterministic framework in which long-term persistence is either certain or the system ultimately tends to extinction. A common approach is to establish a persistence threshold that separates these two regimes; metapopulation size can then be evaluated in the case where persistence is guaranteed1. Hanski and Ovaskainen (2000) introduce a measure of patch value Vithat captures the contribution of a patch to metapopulation capacity λM, at the
threshold between persistence and extinction. Metapopulation capacity itself measures the capacity of a fragmented landscape to support long-term species persistence.
In contrast, in many standard stochastic population models, extinction is the ultimate fate of the metapopulation even if it persists for a long time. Instead of a persistence threshold, we can consider whether the metapopulation enters a quasi-stationary regime, the characteristics of that regime, and the mean time to extinction from quasi-stationarity. The quasi-stationary distribution is defined as the probability distribution of system states (i.e. particular patterns of patch occupancy) conditioned on non-extinction, once the effect of initial conditions has been lost (see Section 2.4.2 for a formal definition). Given a parameterised metapopulation model, provided the system is not too large, the full state transition matrix can be constructed and methods for finding the quasi-stationary distribution and mean time to extinction are well understood (Darroch and Seneta, 1967; Nasell, 1996, 1999, 2001a; Artalejo, 2012). Although the quasi-stationary distribution can be thought of as serving a similar role for stochastic systems as the long-run (or steady-state) solution for deterministic systems, its characteristics differ in a number of respects. As a result, it is generally unclear whether conclusions about persistence that are drawn from deterministic models apply to stochastic models, and the stochastic situation needs to be investigated separately.
The effect of particular patches on determining extinction times and the distribution of states in the quasi-stationary regime has attracted little systematic attention. Additional definitions of patch value in a stochastic model are provided in Ovaskainen and Hanski (2003b), with 1Assuming that the deterministic system is described by a set of differential equations, a persistent system is often defined as one with a stable nontrivial equilibrium solution, in which case metapopulation size is is given by a single number. A nontrivial long-run solution may take more complex forms such as cyclical patterns; however, these issues are beyond the scope of this chapter as the deterministic version of the model used here does have a stable equilibrium state.
4.2. Characterising patch value 57
patch value considered in relation to contributions to colonisation events wi, to metapopula-
tion size ui, and to time to extinction ti. However, there has been little exploration of how
one might evaluate these measures in the absence of a fully parameterised metapopulation model, the construction of which is technically involved and requires considerable data. Fur- thermore, computing the full state transition matrix is also only possible for relatively small systems (for a metapopulation in which each patch has only two states and n patches, the full state transition matrix has dimension 2n× 2n), and memory limitations mean that this is infeasible for systems with more than around 20 patches. As a result, alternative approaches for estimating patch values would be highly valuable.
One option is to test the possibility of using deterministic patch value measures such as Vito
predict the contribution of patches to relevant stochastic outcomes. The extent to which Vi
is a useful measure for this purpose has not been addressed. Other possible measures can be found in an applied context. Beyer et al. (2012) test a range of algorithms for vaccine allo- cation in a metapopulation model (with individuality), implicitly using patch value measures that combine the ‘risk’ of a patch becoming infected and of it generating new infections. A further alternative is to investigate the potential of using measures based on the quasi- stationary distribution. Day and Possingham (1995) note that for a discrete-time metapop- ulation model (with 8 patches both of equal and different sizes), the ranking of patches according to their probability occupancy in the quasi-stationary distribution was identical to that of the effect of their removal on 100-year extinction probability. Despite their comment that ‘this suggests that, in some circumstances, the quasi-stationary probability of occupancy is a good measure of the importance of a patch for metapopulation persistence’ (Day and Possingham, 1995, p345), the relationship appears not to have been explored further.
Our main aim in this chapter is to compare predictors of the effect of patch removal on the mean time to metapopulation extinction Tm and metapopulation size in the quasi-stationary
regime Sπ. These predictors are derived from both deterministic patch value measures and measures based on the quasi-stationary distribution. We evaluate their accuracy as predictors of the ‘true’ value of a patch for these outcomes, the size of the effect on measurable system outcomes, as well as the ease with which they can be computed. In the next section, we provide more information on the metapopulation model employed, the overall experimental procedures and implementation, the persistence outcome measures Tm and Sπ, and the pre-
dictors tested. We then present our results and discuss the accuracy of the predictors and the relative ease with which they can be calculated.
4.3. Materials and methods 58