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Spatial patterns and competition of tree species in a Douglasfir chronosequence on Vancouver Island
 Ecography
, 2006
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Spatial patterns and competition of tree species in a Douglasfir chronosequence on
On moment closures for population dynamics in continuous space
, 2004
"... A firstorder moment closure, the meanfield assumption that organisms encounter one another in proportion to their spatial average densities, lies at the heart of much theoretical ecology. This assumption ignores all spatial information and, at the very least, needs to be replaced by a secondorder ..."
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A firstorder moment closure, the meanfield assumption that organisms encounter one another in proportion to their spatial average densities, lies at the heart of much theoretical ecology. This assumption ignores all spatial information and, at the very least, needs to be replaced by a secondorder closure to gain understanding of ecological dynamics in spatially structured populations. We describe a number of conditions that a secondorder closure should satisfy and use these conditions to evaluate some closures currently available in the literature. Two conditions are particularly helpful in discriminating among the alternatives: that the closure should be positive, and that the dynamics should be unaltered when identical individuals are given different labels. On this basis, a class of closures we refer to as ‘power2’ turns out to provide a good compromise between positivity and dynamical invariance under relabelling.
Species association in a heterogeneous Sri Lankan dipterocarp forest.
 The American Naturalist
, 2007
"... abstract: We used point pattern analysis to examine the spatial distribution of 46 common tree species (diameter at breast height 110 cm) in a fully mapped m tropical forest plot in Sin500 # 500 haraja, Sri Lanka. We aimed to disentangle the effect of species interactions (secondorder effects) a ..."
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abstract: We used point pattern analysis to examine the spatial distribution of 46 common tree species (diameter at breast height 110 cm) in a fully mapped m tropical forest plot in Sin500 # 500 haraja, Sri Lanka. We aimed to disentangle the effect of species interactions (secondorder effects) and environment (firstorder effects) on the species' spatial distributions. To characterize firstorder associations (segregation, overlap), we developed a classification scheme based on Ripley's K and nearestneighbor statistics. We subsequently used heterogeneous Poisson null models, accounting for possible environmental heterogeneity, to reveal significant uniand bivariate secondorder interactions (regularity, aggregation and repulsion, attraction). Firstorder effects were strong; overall, 53% of all species pairs occupied largely disjoint areas (segregation), 40% showed partial overlap, and 6% overlapped. Only 5% of all species pairs showed significant secondorder effects, but about half of the species showed significant intraspecific effects. Significant plantplant interactions occurred mostly within 24 m and disappeared within 1520 m of the focal plant. While lack of significant species interactions suggests support for the unified neutral theory, species' observed spatial segregation does not support the assumptions of the neutral theory. The strong observed tendency of species to segregate may have supplementary effects on other processes promoting species coexistence. Keywords: coexistence, habitat association, paircorrelation function, plantplant interactions, point pattern analysis, tropical forest. A persistent challenge in ecology is to explain the high species diversity of tropical forests However, there is ample evidence that species are not equivalent but that speciesspecific differences in their traits and ecological strategies affect population dynamics and the functioning of the entire community. Plant species can have strong direct and indirect positive and negative effects on other species Several of the processes that have been hypothesized for explaining species coexistence and community structure have a strong spatial component. Examples include direct plantplant interactions, such as competition or facilitation In this article, we analyze data from the FDP at Sinharaja, Sri Lanka, to investigate whether, and at what spatial scales, positive or negative speciesspecies associations occur and whether they are significant. Methods for the spatial analysis of point patterns, that is, data sets consisting of mapped locations of plants as provided by the FDP plots, are well established However, studying speciesspecies association is not always straightforward because firstorder effects (i.e., habitat preference, where the occurrence of the species at the plot depends on altitude, shading, soil moisture, nutrients, etc.) may confound secondorder effects (i.e., direct plantplant interactions, such as competition or facilitation within or among species). Separation of firstand secondorder effects is also an important biological issue because habitat preference and plantplant interactions represent different hypotheses for explaining species coexistence and community structure. Specific methods are required to carefully separate both effects (e.g., In this article, we study all speciesspecies associations occurring between 46 frequent species at the fully censused 25ha FDP at Sinharaja, Sri Lanka. We first develop a method to roughly categorize the firstorder association of heterogeneous bivariate patterns and apply it to the resulting 2,070 species pairs. Next we use point pattern analysis in combination with heterogeneous Poisson null models to identify species and species pairs with significant secondorder effects. Finally, we investigate whether significant effects were correlated with the degree of univariate clustering and species abundances. Methods Study Site and Study Species The area studied is the 25ha Sinharaja FDP, a 500 # 500 m permanent study plot that is located in the lowland rain forest of the Sinharaja UNESCO World Heritage Site at the center of the everwet southwestern region of Sri Lanka, at 6Њ21Ј26ЈN and 80Њ21Ј34ЈE. The Sinharaja FDP is representative of the ridgesteep slopevalley landscape of the lowland and midelevational rain forests of southwestern Sri Lanka. The forest has been classified as a MesuaDoona community Species Associations in Tropical Forests E79 forest structure within the plot as a whole have been documented by Vegetation Sampling The established methodology of We analyzed the spatial pattern of nonjuvenile trees with DBH 110 cm. To obtain a sufficiently large sample size for the point pattern analyses, we used only 46 species with more than 70 nonjuvenile trees. Spatial Pattern Analysis We used univariate and bivariate paircorrelation functions to analyze the spatial pattern of individual species and the association of the patterns of two tree species at different spatial scales r (Stoyan and Stoyan 1994). The paircorrelation function is closely related to Ripley's K function (Ripley 1976); both are based on the distribution of distances of all pairs of points of the patterns. The bivariate K function K 12 (r) can be defined as the expected number of pattern 2 points within distance r of an arbitrary pattern 1 point, divided by the intensity l 2 of pattern 2 (Ripley 1976). The bivariate paircorrelation function g 12 (r) is related to the derivative of the K function; that is, g (r) p K (r)/(2pr) 12 12 Classification Scheme of LargeScale FirstOrder Association The tree species at the Sinharaja FDP are characterized by a high degree of habitat association To define the two axes we first needed to specify an appropriate spatial scale r L to calculate K 12 (r L ) and P 0 (r L ). We used a spatial scale r L that was slightly larger than the typical range of plantplant interactions in tropical forests (i.e., m; e.g., The two axes P and M are thus defined as L L We log transformed the K function to weight departures in both directions equally and scaled P and M to yield positive values if there were more pattern 2 points than expected (K function) or if the probability of having a nearest pattern 2 neighbor within distance r L was greater than expected (emptiness probability). Note that the P axis ranges theoretically between Ϫ1 and 0.45 (see appendix). The M axis may theoretically reach for complete M p Ϫϱ segregation and may have large values if the two patterns occupy the same small subarea. Our scheme allows four different types of bivariate associations. The two patterns are segregated for and P ! 0 (type I; ). For and , the association is character ), but the overlap of both patterns is large P ! 0 enough to make M positive. For and , the two P 1 0 M 1 0 patterns occur within the same area (mixing; type III; Null Models To identify significant speciesspecies associations occurring at the Sinharaja FDP, we proceeded in three steps. We first analyzed secondorder effects in the univariate patterns (analysis 1), next we analyzed secondorder effects in the bivariate patterns (analysis 2), and finally we tested for significant largescale association to find out how frequently species pairs share the same subareas of the FDP (analysis 3). In all cases, we used heterogeneous Poisson null models to account for possible firstorder effects. Heterogeneous Poisson Point Processes Firstand secondorder effects may interact Species Associations in Tropical Forests E81 not readily available with a fine spatial resolution. A shortcut based on separation of scales is to use heterogeneous Poisson point processes where smallscale effects are attributed to secondorder plantplant interactions and largescale effects are attributed to environmental heterogeneity We used an Epanečnikov kernel, a nonparametric method recommended by where d is the distance from a focal point and h is the bandwidth. For a given location (x, y), the intensity l(x, y) is constructed by using a moving window with circular shape and radius h around location (x, y) and summing all points in the circle but weighting them with factor e h (d) according to their distance d from the focal location (x, y). Clearly, the intensity estimate depends on the bandwidth h: for large h, one obtains smooth intensity functions, and for small h, the estimated function is rough and may obscure the fundamental structure of the distribution (Stoyan and Stoyan 1994). We used a biological argument and defined the bandwidth h as the maximal scale at which secondorder effects are expected in tropical forests. Analysis 1: Univariate PlantPlant Interactions To reveal significant secondorder effects in the univariate patterns (i.e., regularity and aggregation), we constructed the intensity function l(x, y) based on the pattern of the species under study and selected for all 46 species a bandwidth m. This scale is slightly larger than typical h p 30 scales at which local pointpoint interactions have been analyzed in tropical forests (e.g., Analysis 2: Bivariate PlantPlant Interactions To reveal significant secondorder effects in the bivariate patterns (i.e., repulsion and attraction), we kept the location of the trees of the first species fixed and randomized the locations of the trees of the second species using a heterogeneous Poisson null model. Here the intensity function l 2 (x, y) was constructed based on pattern 2. Again, we used a bandwidth m and a spatial resh p 30 olution of 2 m. This null model allowed us to assess whether pattern 2 points were more or less frequently around pattern 1 points than expected by the intensity of pattern 2, which would indicate attraction or repulsion, respectively. Note that we tested all pairs, that is, species 1 versus species 2 and species 2 versus species 1, since we cannot assume that the interaction will be symmetric. Analysis 3: LargeScale SpeciesSpecies Associations Most species in tropical forests are clustered at some spatial scale (e.g., Monte Carlo Simulations For all analyses, we performed 99 Monte Carlo simulations of the null model and used the fifthlowest and fifthhighest simulated g(r) values as simulation envelopes. The 99 simulations generated sufficiently smooth simulation envelopes and were sufficient for our purpose given the high E82 The American Naturalist number of analyses required. Significant departure from the null model occurred at scale r if the empirical paircorrelation function was outside the simulation envelopes. However, because of simultaneous inference (i.e., we tested at several spatial scales r simultaneously), Type I error may occur if the value of g(r) is close to a simulation envelope (i.e., the null model may be rejected even if it is true; Loosmore and Ford 2006). We therefore combined the common simulation envelope method with a goodnessoffit test In short, the goodnessoffit (GOF) test collapses the scaledependent information contained in the paircorrelation function into a single test statistic u i that represents the total squared deviation between the observed pattern and the theoretical result across the distances of interest. The u i values were calculated for the observed data ( ) and for the data created by the i p 0 i p 1, … s simulations of the null model, and the rank of u 0 among all u i is determined. The observed p value of this test is Details can be found in work by Results Classification Scheme of Bivariate Associations Analysis 1: Univariate PlantPlant Interactions The 46 species showed diverse spatial patterns ( To obtain a rough estimate of the magnitude of scaledependent effects of regularity and aggregation, we counted for each scale r the number of species (using only species where the rank of the GOF test was 195) for which the paircorrelation function was above or below the fifthhighest or fifthlowest value of the paircorrelation function in the 99 Monte Carlo simulations. The frequency of aggregation peaked at scales between 0 and 8 m and was almost 0 at scales m ( Species Associations in Tropical Forests E83 Figure 2: Classification of largescale associations at the Sinharaja m forest dynamic plot. A, Allocation of the largescale association of 500 # 500 the 2,070 speciesspecies pairs based on the classification axes defined in equations (1). Axis P is positive (negative) if there are on average more (less) pattern 2 points at distance m from pattern 1 points than expected without firstand secondorder effects, and axis M is positive r p 30 L (negative) if the probability that a pattern 1 point has its nearest pattern 2 point within distance r L is larger (smaller) than expected. The classification of individual pairs is represented as gray open circles, and black open circles mark significant and positive largescale associations in analysis 3. The blue dots locate the four examples shown in BF, and the red dots locate the patterns shown in geneous Poisson process), the assumption of separation of scales may not hold. Analysis 2: Bivariate SmallScale PlantPlant Interactions We performed a total of bivariate point 46 # 45 p 2,070 pattern analysis for all pairs of the species. The n p 46 GOF test detected significant associations for 122 species pairs (p5.8%); in 54 cases, the smallscale association was positive (attraction), and in 68 cases, the smallscale association was negative (repulsion). To obtain a rough estimate of the magnitude of scaledependent effects at different scales, we counted for each scale r the number of species (using only species where the rank of the GOF test was 195) for which the paircorrelation function was above or below the fifthhighest and fifthlowest value of the paircorrelation function of the 99 simulations. Repulsion occurred somewhat more frequently than attraction, and pair frequencies peaked at the 2m scale with 32 and 17 pairs, respectively ( Several types of significant smallscale associations occurred at the Sinharaja FDP ( E84 Species Associations in Tropical Forests E85 The next two types of association included cases where both species did not show largescale effects in the paircorrelation function (and the K function) but did show significant smallscale repulsion or attraction. However, these types were rare. We found only one significant example (X. championiiCullenia ceylanica) for smallscale repulsion where the paircorrelation function approximated a value of one for larger scales ( Finally, the two species may show largescale segregation but significant smallscale repulsion or attraction. This case is exemplified by the species pair S. trapezifoliaM. nagassarium, which showed segregation together with repulsion ( To find out whether the significance of our results was dependent on the number of stems of the species pair or on the univariate spatial structure of the component patterns, we calculated for all 2,070 species pairs the Spearman rank correlation between the rank u 0 and the number n 1 of stems of species 1, the number n 2 of stems of species 2, and the value of the paircorrelation function at scales , 2, 6, 10, 20, and 40 m. The rank u 0 correlated weakly r p 0 and positively with the number of stems of species 2 ( ; ) and with the number of stems of r p 0.16 p ! .01 Sp species 1 ( , ). Thus, the significant effects r p 0.05 p ! .05 Sp detected did not primarily depend on the sample size, although as expected, significant effects tended to be more frequent for larger sample sizes. This result also suggests that ignoring species with low abundance should not severely bias our results. We also found a negative correlation of the rank of u 0 with the paircorrelation function of species 1 at scales 040 m with , which peaked r ! Ϫ0.1 Sp at scale 20 m ( ; ). This indicated that r p Ϫ0.14 p ! .01 Sp we were less likely to find significant smallscale association with species that were strongly clumped at scales of about 20 m. This is reasonable, since a strong clumping of species 1 makes it less likely that points of the second species will be in contact with points of species 1 because pattern 1 points are clustered. From the 46 species studied, only eight (Dillenia retusa, Hopea jucunda, Hydnocarpus octandra, Mallotus fucescens, Palaquium thwaitesii, Shorea stipularis, Urandra apicalis, Agrostistachys hookeri) did not show any significant association to another species. Four of these showed a very high degree of univariate clustering (A. hookeri, D. retusa, H. jucunda, and M. fucescens), and the remaining four species did not show significant univariate effects. On the other hand, cases where a species showed significant association with more than five other species were rare. The abundant subcanopy species M. dactyloides, which was scattered throughout the plot ( Analysis 3: LargeScale SpeciesSpecies Association We now investigate how often pairs of species shared the same habitat. Analogous to analysis 2, we analyzed all 2,070 possible species pairs, but now we used a heterogeneous Poisson null model where the randomization of trees of species 2 was based on the intensity function of species 1 (determined by a kernel estimate with bandwidth h p m). Note that this null model explicitly assumed a 50 positive largescale association (i.e., species 2 followed the intensity of species 1) and that the null model will be met E86 The American Naturalist if the area occupied by species 2 is within the area occupied by species 1 (see appendix). The GOF test for distance interval 50250 m revealed significant largescale association for 68 species pairs (3.3%). The black open circles in In total, 42 of the 46 species analyzed showed significant largescale association with at least one other species. Fifteen species were in only the role of pattern 1, 13 were in only the role of pattern 2, and 14 appeared in the roles of pattern 1 and pattern 2. The species X. championii ( Discussion We performed a comprehensive spatial pattern analysis of thousands of speciesspecies associations at a fully mapped 25ha forest dynamics plot of a speciesrich tropical forest in Sinharaja, Sri Lanka. Our analyses were motivated by the notion that coexistence mechanisms operating in a forest should leave a spatial signature that can be detected by analyzing explicit maps of individual tree locations . In this spirit, several studies have analyzed neighborhood effects and negative density dependence in plant performance, for example, for recruitment of abundant species It is well known that, simply because of the high number of species, only a few of the neighbors of an individual tree (110 cm DBH) are likely to be conspecific, even among the most common or most highly aggregated species , where is the inten sity of species i and S 0 is the total number of species present in the area sampled; He and Legendre 2002.) Given the additional effect of partial overlap or segregation, it is clear that only a few species have the chance to develop specific interactions with other species and that in most cases, individuals of a given species will be associated by a different set of neighbors To verify the above hypothesis of "diffuse neighborhoods," we placed an arbitrary point inside the higherdiversity area (it had coordinates 185, 180) and determined the number of species that occurred in a circle with radius of 10 m around this point. Next we determined the number of species that occurred inside of 10m circles but were located (on a 1m grid) distances d away from the focal circle and determined the number of species shared with the focal circle. We found that the number of shared species declined linearly up to a distance of 20 m ( , n p 75 ) and then stabilized around a constant value. Thus, one characteristic of speciesrich tropical forest with strong habitat association and clustered individual species patterns, as found in Sinharaja, is that the set of species neighbors encountered by individuals of a given species is quite variable and not predictable for the individual. This is a sort of "spreading of risk" with respect to the neighbors. The more different species an individual may encounter in its neighborhood, the higher the chance that some individuals will have "favorable" neighbors and survive. From this perspective, it is not surprising that only about 6% of all speciesspecies pairs showed significant nonneutral, smallscale associations (i.e., attraction or repulsion), as revealed by our detailed pointpattern analysis of direct plantplant interactions (analysis 2). However, considering that our goodnessoffit test had a 5% Type I error rate, this may account for some of the apparently significant results so that the proportion of "truly" significant associations may even be lower. Interestingly, the analogous univariate analysis revealed that direct plantplant interactions among conspecifics produced significant smallscale aggregation for about half of the species. How Species Associations in Tropical Forests E89 Figure 6: Examples for largescale association. In the distribution maps, species 1 is indicated by black and species 2 by red. The heterogeneous Poisson null model used an Epanečnikov kernel estimate of the intensity of the pattern of species 1 with bandwidth m (a circle with a 50h p 50 m radius is shown in C), whereas the locations of species 1 remained fixed. The ring width for estimation of the paircorrelation function was 10 m; grid size was m. Other conventions are as in figure 5. 5 # 5 ever, the 6% figure is a surprisingly low figure when considering the increasing evidence for neighborhood effects on plant performance, even for larger trees. For example, This apparent contradiction raises the question why such nonneutral processes, which should also operate at Sinharaja, did not leave a signature in the spatial pattern. One reason could be that the studies investigating neighbor effects on tree performance and our study measured somewhat different things. Growth and mortality are dynamic processes that are measurable only using a number of snapshots (at least two) several years apart, whereas our analysis used only one snapshot. Moreover, effects of size were not included in our analysis (except that we analyzed only trees 110 cm DBH). To detect effects of size on spatial patterns, a more complex analysis would be required, for example, using the markcorrelation function Another possibility for the contrasting results would be that the processes that shaped the patterns were nonneutral, but the patterns that finally emerged were predominantly neutral. There are two possible interpretations for this. First, nonneutral processes in tree performance may not leave a detectable spatial signature in the map of individual tree locations. The spatial pattern of mapped tree locations may therefore not be used to assess the strength of bivariate plantplant interactions. A similar argument was recently made regarding nonneutral processes, such as niche structure, that may not leave a signature on the rank abundance curve E90 The American Naturalist dent processes should create spatial structure, except in cases where for some reason the different spatially dependent processes equilibrate. This is the second interpretation. Thus, we hypothesize that a specific characteristic of speciesrich tropical forests is that nonneutral processes in tree performance equilibrate and in most cases produce neutral bivariate patterns in the spatial distribution of trees. This can be interpreted as a strong argument in favor of neutral theory. The zerosum dynamics of unified neutral theory (Hubbell 2001) captured this characteristic by simplifying the entire regeneration process from seed dispersal to adulthood (within the range of influence of large trees) into the single step of replacement where the new individual is selected with probability from the local 1 Ϫ m community and with probability m from the metacommunity. Thus, nonneutral processes regarding tree performance (recruitment, growth, mortality) are lumped within this step and the zerosum dynamics assumes that there are regulating mechanisms that guarantee that the outcome of this step would be a neutral spatial pattern. This viewpoint is also supported by another result of our detailed analysis. We found that nonrandom plantplant interactions, if present, were quite local; their range did not reach farther than about one canopy tree crown radius (!20 m). This result is in accord with other studies (e.g
T.: Drought induces lagged tree mortality in a subalpine forest
 in the Rocky Mountains, Oikos
, 1983
"... Extreme climatic events are key factors in initiating gradual or sudden changes in forest ecosystems through the promotion of severe, treekilling disturbances such as fire, blowdown, and widespread insect outbreaks. In contrast to these climaticallyincited disturbances, little is known about the m ..."
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Extreme climatic events are key factors in initiating gradual or sudden changes in forest ecosystems through the promotion of severe, treekilling disturbances such as fire, blowdown, and widespread insect outbreaks. In contrast to these climaticallyincited disturbances, little is known about the more direct effect of drought on tree mortality, especially in highelevation forests. Therefore projections of droughtinduced mortality under future climatic conditions remain uncertain. For a subalpine forest landscape in the Rocky Mountains of northern Colorado (USA), we quantified lag effects of drought on mortality of Engelmann spruce Picea engelmannii, subalpine fir Abies lasiocarpa, and lodgepole pine Pinus contorta. For the period 19102004, we related death dates of 164 crossdated dead trees to earlyseason and lateseason droughts. Following earlyseason droughts, spruce mortality increased over five years and fir mortality increased sharply over 11 years. Following lateseason droughts, spruce showed a small increase in mortality within one year, whereas fir showed a consistent period of increased mortality over two years. Pine mortality was not affected by drought. Low predrought radial growth rates predisposed spruce and fir to droughtrelated mortality. Spruce and fir trees that died during a recent drought (20002004) had significantly lower predrought growth rates than live neighbour trees. Overall, we found large interspecific differences in droughtrelated mortality with fir showing the strongest effect followed by spruce and pine. This direct influence of climatic variability on differential tree mortality has the potential for
Spatial distribution of anopheline larval habitats in western Kenyan highlands: effects of land cover types and topography. Am J Trop Med Hyg 73
, 2005
"... Abstract The distributions of anopheline larval habitats were aggregated in valley bottoms in Kenya in both the rainy and dry seasons, although the degree of aggregation was higher in the dry season than in the rainy season. Larvae of the Anopheles gambiae complex larvae were found more frequently i ..."
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Abstract The distributions of anopheline larval habitats were aggregated in valley bottoms in Kenya in both the rainy and dry seasons, although the degree of aggregation was higher in the dry season than in the rainy season. Larvae of the Anopheles gambiae complex larvae were found more frequently in habitats in farmlands and pastures. However, An. funestus larvae were found more frequently in natural swamps and pastures. Canopy cover was the only variable significantly associated with the occurrence of the An. gambiae complex and An. funestus. The average canopy cover was significantly less in the habitats with the An. gambiae complex and An. funestus larvae than those without the anopheline larvae. Thus, land cover types and topographic features showed important effects on the distribution of anopheline larval habitats. These results suggest that clearing riparian forests would improve growing conditions of the An. gambiae complex and An. funestus larvae in Kenyan highlands.
MINI REVIEW
, 2004
"... Minireviews provides an opportunity to summarize existing knowledge of selected ecological areas, with special emphasis on current topics where rapid and significant advances are occurring. Reviews should be concise and not too wideranging. All key references should be cited. A summary is required. ..."
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Minireviews provides an opportunity to summarize existing knowledge of selected ecological areas, with special emphasis on current topics where rapid and significant advances are occurring. Reviews should be concise and not too wideranging. All key references should be cited. A summary is required. Spatiotemporal development of forests / current trends in field methods and models
Longterm satellite NDVI data sets: Evaluating their ability to detect ecosystem functional changes
 in South America. Sensors 2008
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Tree invasion of a montane meadow complex: temporal trends, spatial patterns, and biotic interactions
"... Questions: Do spatial and temporal patterns of encroachment of Pinus contorta and Abies grandis in a montane meadow suggest strong biotic controls on the invasion process? Location: Forest–meadow mosaic, 1350m a.s.l., ..."
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Questions: Do spatial and temporal patterns of encroachment of Pinus contorta and Abies grandis in a montane meadow suggest strong biotic controls on the invasion process? Location: Forest–meadow mosaic, 1350m a.s.l.,
Characterizing the relative spatial structure of point patterns
 International Journal of Ecology
, 2012
"... We generalize Ripley's K function to get a new function, M, to characterize the spatial structure of a point pattern relatively to another one. We show that this new approach is pertinent in ecology when space is not homogenous and the size of objects matters. We present how to use the functio ..."
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We generalize Ripley's K function to get a new function, M, to characterize the spatial structure of a point pattern relatively to another one. We show that this new approach is pertinent in ecology when space is not homogenous and the size of objects matters. We present how to use the function and test the data against the null hypothesis of independence between points. In a tropical tree data set we detect intraspecific aggregation and interspecific competition.
Dealing with virtual aggregation— a new index for analysing heterogeneous point patterns,”
 Ecography,
, 2008
"... Point pattern analyses such as the estimation of Ripley's Kfunction or the paircorrelation function g are commonly used in ecology to characterise ecological patterns in space. However, a major disadvantage of these methods is their missing ability to deal with spatial heterogeneity. A heter ..."
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Point pattern analyses such as the estimation of Ripley's Kfunction or the paircorrelation function g are commonly used in ecology to characterise ecological patterns in space. However, a major disadvantage of these methods is their missing ability to deal with spatial heterogeneity. A heterogeneous intensity of points causes a systematic bias in estimates of the Kand gfunctions, a phenomenon termed ''virtual aggregation'' in the recent literature. To address this problem, we derive a new index, called K2index, as an extension of existing point pattern characteristics. The K2index has a heuristic interpretation as an approximation to the first derivative of the gfunction. We estimate the K, gand K2functions for six different types of simulated point patterns and show that the K2index may provide important information on point patterns that the other methods fail to detect. The results indicate that particularly the smallscale distributions of points are better represented by the K2index. This might be important for testing hypotheses on ecological processes, because most of these processes, such as direct neighbour interactions, occur very locally. When applied to empirical patterns of molehill distribution, the results of the K2analysis show regularity up to distances between 0.1 and 0.4 m in most of the study areas, and aggregation of molehills up to distances between 0.2 and 1.1 m. The type and scale of these deviations from randomness agree with a priori expectations on the hillbuilding behaviour of moles. In contrast, the estimated gfunctions merely indicate aggregation at the full range from 0 to 7 m (or even above). Considering the advantages and disadvantages of the different methods, we suggest that the K2index should be used as a complement to existing approaches, particularly for point patterns generated by processes that act on more than one scale.