Quantifying spatial and temporal variability of spatially correlated disturbances

We propose three metrics to quantify the spatial and temporal structure of events within spatial dynamical systems. Although the original intention was to develop metrics to separately quantify spatial and temporal variability or clustering, our ultimate conclusion is that it is impossible to completely separate space and time. The spatial structure observed within a system depends on the temporal scale over which the measurements were taken, and the temporal structure observed depends on the spatial scale of measurements. We explore two variations of the basic contact process lattice population model where, instead of individual sites becoming empty independently, larger-scale disturbance events simultaneously affect several sites; these sites may either be arranged within a contiguous block, or sampled from within a larger region. We adjust the rate of such disturbance events in such a way that the per-site disturbance rate is held fixed. Changing the spatial scale of disturbances while controlling the per-site rate induces changes in the temporal structure of disturbances as well (depending on the spatial scale at which it is measured), because the larger-scale disturbances occur less frequently. Analytical results are obtained for our particular model using the hypergeometric distribution, but our spatiotemporal metrics can also be applied numerically to observations of events in simulations or field data to quantify the level of spatial and temporal variation in spatially structured systems. We find in our model that measurements of disturbances at intermediate spatial scales are most useful at indicating the success of the population; measurements at either the micro- (single sites) or macro- (entire landscape) scales miss fundamental features of the system. Although the two versions of our model go extinct for widely different scales of disturbances, our metrics have very similar critical values for population persistence among the two models.