the largest proportional effect on metapopulation size in the QSD when the mean propor- tion of patches occupied in the original landscape was just under half. Shading indicates the change in the proportion of patches occupied in the QSD as a proportion of those occupied in the original landscape, showing a similar trend to that of the effect on metapopulation size. This finding implies that we are most likely to observe the effect of patch removal on landscapes that are initially just under 50% occupied.
Equal Exponential LogNormal
0.0 0.2 0.4 0.6
0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
Mean proportion of patches occupied in QSD
∇ S π 1 2 3 4 5 Sπ
Figure 4.19: The effect of patch removal on mean metapopulation size for different initial mean occupancy levels in the QSD (n = 10). Each line represents the removal of patch zero for one landscape.
4.5
Discussion
In this chapter we sought to compare predictors of the effect of patch removal on the mean time to metapopulation extinction from quasi-stationarity Tm and metapopulation size in the
quasi-stationary regime Sπ. The findings show that the best predictor of percentage change ∇Tm is (pπi)η with η ≈ 2, and that the best predictor of ∇Sπ is our combined measure Ui.
These findings strike us as surprising in their simplicity and accuracy. In effect, they tell us that if we are willing to assume only that our real world system is in the quasi-stationary regime and that the system is well-modelled by a presence-absence SRLM with functional forms for colonisation and extinction similar to those found to give good accuracy here, then the effect of patch removal on both∇Tmand∇Sπcan be predicted with good accuracy from
only information on the probability of patch occupancy.
Presence-absence data that can be used to compute occupancy probability is collected rou- tinely by ecologists and epidemiologists. Where the above assumptions regarding quasi- stationarity and the applicability of an SRLM model with appropriate functional forms are
4.5. Discussion 81
met, this suggests that presence-absence data can be used almost directly to choose between possible interventions, without the need for complex model fitting.
In the case where the primary object of study is a model, the findings also have practical implications. Firstly, they reduce the number of necessary calculations required for estimat- ing∇Tmand∇Sπ. In order to find the true value of these outcome measures, computations
need to be conducted for both the original landscape and for the landscape with each patch be removed, leading to n + 1 computations (where n is the number of patches). If one is willing to approximate patch value by the square of the probability occupancy in the QSD, a single QSD computation can be carried out. While this may not make much difference if parameter values are known fairly accurately (and the number of computations is thus n+1 as above), if parameter values are estimated and one wants to conduct a sensitivity analysis, this may be a significant advantage, allowing much wider exploration of the parameter space. In addition, computing the full QSD for large systems raises other computational issues because of the number of possible states. What this finding suggests is that it may be sufficient to simulate according to an algorithm that tracks a reduced version of the QSD from which pπi can be accurately reconstructed, potentially reducing the required storage very considerably. A number of possible extensions to this work would be valuable for future investigation. The most obvious extension is to conduct robustness testing for the measure of metapopulation size, along similar lines to that conducted for Tm above. Additional suggestions include
testing the robustness of these findings to alternative patch groupings, systems with a larger number of patches, and whether the QSD assumption is appropriate for metapopulations that are below the critical threshold for persistence. In more detail, it would be valuable to test to what extent the pπi heuristics described here are robust to the way in which patches are grouped. A practical problem in applying patch models with real world data is that it is often difficult to know exactly where to situate patch boundaries, such that it would be useful to know whether this is in fact an important concern. Furthermore, grouping patches in an appropriate way would reduce the computational cost of calculating patch values in the QSD. In systems with a large number of patches (e.g. the 75 villages considered in Beyer et al., 2012), this would be highly valuable. An alternative approach in this latter case would be to test to what extent approximations to the QSD that remain plausible for the number of patches under consideration provide accurate predictions of patch value.
Another area worthy of investigation is that of the extent to which reliance on the QSD concept, when intervening on a system is valid, especially below the persistence threshold. In this chapter, we have assumed that a system is initially in the quasi-stationary regime, and that it immediately falls into the new quasi-stationary regime of the modified landscape once a patch is removed. Obviously the first assumption may not be true in the real world, where even many long-standing systems may in fact be in a transient phase; in addition, recovery of the new QSD after patch removal may be fast or slow. None of these concerns