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Our results have implications in coffee agroecological management, as this system provides important biocontrol ecosystem services. Less examined are the many cases of pattern formation in animal populations, most relevant to sessile species. The important issue of cluster scale remains understudied generally, especially at large scales. What would be that critical clustering scale, and would it make a difference in terms of the observed spatial pattern? Might we be forced to conclude that pattern formation would be endogenous at one cluster scale but exogenous at a different scale?
While our interrogations are cast in the context of one species of ant, however, we note that they are relevant to any sessile organism. In this work, we take on these questions using a spatial dataset spanning 9 years in a permanent 45 hectare plot. Our approach first used spatial descriptive statistics to estimate the spatial relationship between nests.
We fitted nest patterns from each year to point process models using two sets of spatial covariates representing hypothesized endogenous and exogenous contexts. Nest distributions in these models were alternately modelled under assumptions of spatial independence a Poisson process or dependence a cluster process between nests. The utility of adding the clustering process to our models was that it parametrized and fitted the pattern formation process within the spatial correlation analysis.
Our expectations were: i if endogenous processes played an important role in spatial pattern, nest distribution would have a significant density-dependent relationship over space and time, and ii if A. We compare the information explained by the fitted trends of the endogenous and exogenous covariates, interpret the fitted clustering parameters and the performance of the cluster process models, and discuss our results in the context of A. Like many areas of the tropics, this region experiences annual wet and dry seasons.
Our data were collected annually in the wet season, between the months of May and July from to Because A. Between each year, trees were removed or added to the dataset when they died or were cut down, or had grown large enough to be included. Both tests provide a population-wide summary of how individual points cluster with respect to other individuals as a function of an analysis scale r.
We used inhomogeneous point process models to fit nest patterns against exogenous and endogenous spatial variables that we hypothesized could have played a role in spatial distribution.
Our application of this cluster process to ant colony distribution is novel but appropriate, given the cluster-forming behaviour of A. We fitted separate endogenous and exogenous models for each year of the study. Model variables were represented by spatial grids covering the entire plot. The endogenous models included the year-lagged ant nest density nest and its second-order term nest 2 to accommodate nonlinear density-dependent effects.
The exogenous models fit tree density trees , elevation elev , slope and topographical wetness index wet. Owing to the similar hill aspect across most of the plot, sun exposure was considered generally uniform throughout and was not included in the analysis. Densities of trees and nests were calculated with a Gaussian kernel. However, it should be noted that while our decision was based on some experience and corroborated by our PCF results, the issue of optimal analysis scale and its implications in A.
The slope grid was calculated as a function of the neighbouring values of each cell in the digital elevation model. To aid interpretation of the fitted model coefficients, these variable grids were scaled to have an approximately unit standard deviation across the plot and between all years of data, giving the coefficient estimates a uniform scale in terms of their variances across the study.
We first modelled spatial variation in nest intensity by fitting an inhomogeneous Poisson process IPP model to nest points, using the endogenous or exogenous sets of spatial covariates. The IPP assumes no interaction between points, so any variation in intensity was attributed to spatial heterogeneity of the variables.
We compared the relative information explained by each IPP model to a homogeneous Poisson process null, which assumed a uniform mean intensity of nests for the entire plot. The Thomas cluster process was an intuitive choice, because it uses an isotropic Gaussian shape to disperse offspring points around each cluster centre. Tree population remained constant until tree thinning began in Exact nest and tree populations are given in the electronic supplementary material, S1.
The amount of clustering in the graph is quantified as the difference from a theoretical random Poisson distribution defined by the mean point intensity over the entire plot. PCF statistic for annual nest distributions, plotted as the difference from the theoretical average expected value, represented by the dotted line.
Qualitatively, the shape of the curves changed as well. Nests remained clustered within this structure, however, as the existing patterns were still significantly different from random reallocations within each year's tree distribution. Maps of nest locations are provided in the electronic supplementary material, S2. Because the Thomas process was fitted to the IPP trend, the coefficients for both models are the same, but the uncertainty of the ITCP increases to account for clustering.
Coefficients are the same between models but have different confidence intervals. The p -values are reported in the parentheses below each coefficient, with the IPP p- value listed on top. When assuming no nest clustering in the endogenous models IPP , elevation was significant in and with a positive coefficient; slope was significant in , and with a negative coefficient; and wetness was significant in with a negative coefficient. The ITCP exogenous models did not assign significant coefficients to any abiotic variables. After , nest occurrence became more likely with an increasing number of surrounding trees.
This corresponds with the initiation of the tree felling campaign on the farm that decreased the tree population and maximum tree density found on the plot. This could reflect an increasing dependence of nest density on tree availability, although this relationship may be indirect, as the projected nest density remained much lower than the predicting tree density. Predicted density was derived from the intensity value predicted by the models. All spatial covariates were held at the plot mean except for the x -axis variable. The plotted range of each year reflects the actual range for that year.
The shape of these curves support the idea that nest growth and survival are nonlinear and density-dependent, as the predicted nest occurrence is greater than the diagonal line at lower lagged nest densities, but switches over i. This threshold increased from around 30 in the 3 years before tree thinning — , to a significantly greater peak above 40 in the model only predicted density increase, as the nest density levels were not sufficient to cross the projected threshold , and returned to a lower level around 35 in — At very low levels of previous nest density, the predicted nest density also fell below the previous year's density every year, suggesting an Allee effect for very isolated nests.
In the first 5 years, exogenous models added relatively little information to their respective null models, compared with the last 3 years' models, which improved AIC by at least Endogenous models decreased improved in AIC from the null model by much more—approximately for each model. Comparison of AIC values of the exogenous and endogenous inhomogeneous Poisson process models, and a homogeneous Poisson process null model. Datasets are shared within years, so AIC values are comparable by column.
Relatively lower AIC values indicate that more information is explained by that model. In some years, these differences were significant. A MAD value below 0. Low goodness-of-fit p- values for the MAD test indicated that the endogenous models were significantly different from the actual original nest distributions, and thus not appropriate representations of clustering. From plotting simulated L- test results of both models against the actual pattern electronic supplementary material, S3 , it was evident that the endogenous ITCP models over-predicted clustering, whereas exogenous models encompassed actual clustering patterns.
This may have been because the Thomas process in the endogenous models conflicted with the lagged nest density covariates, which already accounted for aspects of the nest clustering process. On the other hand, the Thomas process of the exogenous models was able to account for all aspects of multigenerational nest distribution, and could thus simulate clustering accurately. For this reason, we consider only the clustering parameters of the exogenous ITCP models in our discussion. Our results demonstrate how Poisson and cluster point process approaches, previously limited to vegetation studies in ecology, can provide insights into the spatial distribution and clustering of other sessile organisms such as ants.
We determined a relevant scale of clustering for A. However, both these methods assumed a simplified interpretation of nest dispersal, ignoring multigenerational processes or approximating their patterns as only offspring nests, though actual distributions were a combination of offspring and survivors.
The uniformly and lushly treed environment of this organic shade coffee farm may have originally formed an effectively homogeneous environment for the arboreal A. However, our study reveals the changing nature of the interactions in this system and the emerging significance of environmental context.
Two consistent patterns emerged from our point process modelling: exogenous models indicated that tree density became an increasingly important positive predictor of nest density, whereas endogenous models suggested a changing density-dependent relationship between nests. However, the discrepancy in AIC between the two models indicated that endogenous effects still explained more of the present patterns.
Our models also suggested an Allee effect at very low nest densities, although it is unclear why small, isolated clusters would experience higher mortality.
This result opens a line of questioning that has not yet been considered by the rich body of empirical work done in the system. That study demonstrated through cellular automata modelling that escape from a more dispersal-limited natural enemy presents a plausible hypothetical mechanism for nest increase.
From the point of view of natural enemies, tree reduction would decrease nest density at a local scale and increase distances between nests, potentially hindering movement and counter-balancing a response to increases in plot-scale nest density. This study found empirical evidence that the negative control on nest densities did indeed change in character, whatever its species identity.
The threshold nest density above which negative control became dominant in the subsequent year i. This threshold peaked after the start of the tree thinning programme and returned close to initial levels in later years, a trend that could be indicative of a transition in the controlling regime. However, we did not see evidence of increased nest dispersal that would support the dispersal-limited natural enemy hypothesis in our Thomas cluster models.
The fitted clustering parameters suggested that the dispersal distance of the nests within nest clusters did not change significantly, even in later years when management on the farm changed. Rather, only the intensity of cluster centres increased. The hypothesis was also contradicted by the significant positive relationship between nest density and tree density in later years, as we would have expected more nests with lower tree density, because this would have been a condition for greater internest distances. We attempted to make an informed choice in the scale of our correlations by basing spatial variables on previous experience in the system, a choice that was supported by the PCF results.
However, the assumption that this scale best represents how a colony interacts with conspecifics and the environment remains untested explicitly.