3.3 Results
3.3.1 Simulation results
Figure 3.2: There is a strong correlation of abundance and aggregation of species (top-canopy trees∆; middle canopy treesO; understory trees∇; shrubs∗). The green line shows the mean result of 500 runs of the individual-based model (for population sizes of 50, 100, 500, 1000, 2000, 5000, 10000, and 30000) with a mean dispersal distance of 2.8 m, the orange line the respective simulation results for a mean dispersal distance of 30 m, and the blue line for 152 m. The whiskers mark the 10 and 90 percentile of the simulation results. As reported by Muller-Landau & Hardesty (2005) species at BCI have mean dispersal distances between 2.8 m and 152 m, all but two species have Ω0,10,2005 values that simulations with a mean dispersal distance within that range can well explain.
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0 10 20 30 40 50 60
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Difference of mean log aggregation
Generation
Figure 3.3: The graph shows the mean difference between the log10(1 + Ω0,10)value of a simulated population with changing local abundance, and a population with the same but stable abundance. Depicted is the mean of 1000 runs of the basic individual-based model (with a mean dispersal distance of 10 m). It was initialized with 100 (dotted) or 500 (solid) individuals in the population. Then, in the population with changing local abundance, the population was decreased (lines going below zero) or increased (lines going above zero) by 10% (black) or 2% (gray) per generation for a total of 10 generations. After 10 generations (vertical dotted line), the abundance was kept constant and one can observe how the mean log10(1 + Ω0,10)relaxes back to the expected value for a stable population of that abundance (horizontal dashed line).
are likely to be more aggregated and declining populations are likely to be less aggregated than stable populations. As expected the effect on the aggregation strongly depends on the magnitude of the abundance change, with small changes yielding small differences from a stable population; and in all cases, after the population stops growing/declining there is a lag phase before the aggregation converges on theΩ0,10,t value for a stable population of the same abundance.
3.3.1.1 Locally density dependent mortality
Using the LDD IBM with a negative influence of local density on survival, we find that for stronger density dependence the model predicts a stronger effect of changes in local abun- dance on Ω0,10,t in shrinking populations (see Figure 3.4). This is because local density dependence preferentially removes individuals that have crowded neighborhoods, leading to a rapid reduction in aggregation when deaths outweigh births; and therefore in the short-term, the difference between the aggregation of a decreasing population and a stable population is widened even further. However, in the extreme limits of strong locally density dependent mortality, the population may become spatially segregated to an extent that further mortality
Figure 3.4: The graph shows the effect of changes in local abundance on aggregation, de- pendent on the strength of density dependent mortalityf. Depicted is the mean difference between thelog10(1 + Ω0,10)value of a simulated population with changing local abundance, and a population with the same but stable local abundance after 10 generations. All simu- lations are based on at least 100 repetitions, with a starting population size of 100, a mean dispersal distance of 10 m, and 10% change in abundance during the first 10 generations. The upper line is based on the results of growing populations, the lower line on those of shrinking populations.
cannot lead to any further reduction of aggregation. In these cases the neighborhoods of most individuals are already empty with respect to conspecific individuals.
3.3.1.2 Effect of different dispersal kernel
In order to test the sensitivity of our model to the shape of the chosen dispersal kernel we also ran our simulation model using a Cauchy kernel, which has a location parameter that we set to 0, and a single scale parameter. The Cauchy kernel has a fatter tail than the negative exponential distribution, i.e. very long dispersal events are more likely. A consequence of the fatter tail is, for the same mean dispersal distance, the Cauchy distribution will produce more very short and more very long distance dispersal than an exponential dispersal kernel.
We use a Cauchy kernel with a scale parameter of 2.5, but all other details of the individual- based model reported in the main text remain the same. This kernel has a slightly larger mean dispersal distance than the exponential kernel with a mean dispersal distance of 10 m that we use to generate the results in Figure 3.3, but still produces a more aggregated pattern, as even though long distance dispersal events are more likely, most dispersal events are over shorter distances.
The tail of a dispersal kernel can be important for the speed with which species can colo-
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Figure 3.5: The graph shows the mean difference between thelog10(1+Ω0,10)value of a sim- ulated population with changing local abundance, and a population with the same but stable abundance. Depicted is the mean of 1000 runs of the basic individual-based model (100 runs for the Cauchy kernel with a starting population of 500). Each run was initialized with 100 (dotted) or 500 (solid) individuals in the population. Then, in the population with changing local abundance, the population was decreased (lines going below zero) or increased (lines going above zero) by 10% per generation for a total of 10 generations. After 10 generations, the abundance was kept constant and one can observe how the meanlog10(1 + Ω0,10)re- laxes back to the expected value for a stable population of that abundance (horizontal dashed line). Black lines show the results for the negative exponential kernel with a mean dispersal distance of 10 m, gray lines show the results for a Cauchy kernel with scale parameter of 2.5 (the expected aggregation of the simulation with the Cauchy kernel is higher than for a expo- nential kernel with 10 m mean dispersal distance, even though the mean dispersal distance is larger for the Cauchy kernel).
nize new areas, but our results show that it is not important in producing the local aggregation pattern at the scale at which we are analysing our data. Figure 3.5 compares the results for the aggregation of species with changing abundances in our simulation model using the Cauchy or the negative exponential kernel. There is no qualitative difference in our results for the two kernels.