Figure 1.
Experiment: effect of pinning at the subpopulation level, averaged over four replicate metapopulations. (A) The mean FI of the pinned subpopulation was significantly less than the mean of the remaining eight subpopulations. (B) There was no difference in the average FI of the pinned group (the pinned subpopulation and its two immediate neighbors) and the two neighboring groups on either side (No Pin1 and No Pin 2). This suggests that the stabilized subpopulation could not stabilize the dynamics of the pinned group vis-à-vis the two neighboring groups. Error bars indicate standard errors around the mean in this and all subsequent figures.
Figure 2.
Experiment: effect of pinning at the metapopulation level, averaged over four replicate metapopulations. (A) Metapopulation stability and (B) subpopulation stability were measured as the fluctuation index (FI) over 21 generations. (C) Synchrony among nearest neighbors was measured as the cross-correlation at lag zero of the first differenced ln-transformed values of population sizes. Due to the high rates of migration, the subpopulations were found to be in synchrony, as demonstrated by the positive cross correlation coefficients. (D) Average subpopulation size. There was no difference among the pinned and the control metapopulations in any of the panels, indicating that pinning had no detectable effect on metapopulation dynamics.
Figure 3.
Simulations mimicking experiment: effect of pinning at the subpopulation level. (A) The FI of the pinned subpopulation was lower than the mean of the remaining eight subpopulations, over a substantial range of the intrinsic growth rate parameter, r. (B) There was no difference in the average FI of the pinned group and its two neighboring groups. These observations agree with the experimental results (cf. Fig 1), implying that the experimental findings are non-Drosophila specific. All data points in this, and subsequent simulation figures, represent average of 10 independent runs. See text for details of simulations.
Figure 4.
Simulations mimicking experiment: effect of pinning at the metapopulation level. Ricker based simulations predicted no difference in (A) metapopulation stability, (B) subpopulation stability, and (C) synchrony amongst nearest-neighbors, between the control and pinned metapopulations. (D) The simulations suggested a slight decrease in subpopulation size for low values of r, which was not picked up by the experiment. Overall, these results agree with the corresponding experimental findings (Fig 2), indicating that they are likely to be applicable to other species.
Figure 5.
Simulations relaxing experimental assumptions: effect of pinning density and magnitude on stability. (A) There was no effect on metapopulation FI due to pinning greater number of patches for r<3. When r>3, increasing the proportion of pinned patches generally increased FI, although there were no consistent patterns. (B) Varying the magnitude of pinning had no effects on metapopulation stability. These suggest that the empirical results are robust to departures from the conditions of the experiment.
Figure 6.
Simulations relaxing experimental assumptions: effect of migration rate on stability. Various rates of migration did not have a differential effect on the stability of the (A) control and (B) pinned metapopulation, again indicating the robustness of the experimental findings.
Figure 7.
Simulations with no extinctions: effect of pinning at the subpopulation level. (A) The FI of the pinned subpopulation was higher than the mean of the remaining eight subpopulations only for r>2.6. (B) The average FI of the pinned group tended to be higher than the two neighboring groups for r>2.6, although this difference was significant only for a comparatively narrow parameter range. Both these results were contradictory to the observations from the experiments (Fig 1) and the simulations mimicking the experiments (Fig 3), indicating that the effect of pinning at the subpopulation level interacts with the extinction probability
Figure 8.
Simulations with no extinctions: effect of pinning at the metapopulation level. Although there were qualitative differences in the shapes of the profiles compared to the case when extinction probabilities were incorporated (Fig 4), there were no systematic differences between the control and the pinned treatments in terms of (A) metapopulation FI, (B) subpopulation FI, and (C) subpopulation synchrony. However, the average subpopulation size (D) of the pinned subpopulations was predicted to be similar to the controls for r<3, which agrees with the experiments (Fig 2D), but not the earlier simulations (Fig 4D). Taken together it can be said that even in the absence of extinctions, pinning is unlikely to affect metapopulation dynamics.
Figure 9.
Simulations with no extinctions: effect of pinning density and magnitude on stability. (A) When there are no extinctions, increasing the number of pinned patches was generally found to destabilize the metapopulation dynamics, similar to the experimental scenario (5A). (B) Changing the strength of pinning, however, did not affect the metapopulation stability, although there was a change in the FI profile relative to the earlier simulations incorporating extinctions (cf. Fig 5B).
Figure 10.
Simulations with no extinctions: effects of migration rate on stability. In the absence of extinctions, there were no major differences in the FI of the (A) control and (B) pinned metapopulations. However, there was a change in the profile of the metapopulation FI (cf. Fig 6), indicating that migration rate can interact with the levels of extinction, although this is not expected to interact with pinning to alter the empirically observed patterns of metapopulation stability.
Figure 11.
Empirically observed extinction probabilities at different population sizes. This shows the fraction of times a population went extinct in generation t+1, when the population size in generation t fell within a particular range in the controls.