Aquatic Botany 95 (2011) 173–181 Contents lists available at ScienceDirect Aquatic Botany journal homepage: www.elsevier.com/locate/aquabot Desynchronizing effects of lightning strike disturbances on cyclic forest dynamics in mangrove plantations Markus Kautz a,∗ , Uta Berger b , Dietrich Stoyan c , Juliane Vogt b , Nabiul Islam Khan b , Karen Diele d , Ulrich Saint-Paul d , Tran Triet e , Vien Ngoc Nam f a TU München, Hans-Carl-von-Carlowitz-Platz 2, 85354 Freising, Germany TU Dresden, Pienner Str 8, 01737 Tharandt, Germany c TU Bergakademie Freiberg, Prüferstr 9, 99596 Freiberg, Germany d Leibniz Center for Tropical Marine Ecology, Fahrenheitstr 6, 28359 Bremen, Germany e Vietnam National University, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Viet Nam f Nong Lam University, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam b a r t i c l e i n f o Article history: Received August 2010 Received in revised form 20 May 2011 Accepted 23 May 2011 Available online 30 May 2011 Keywords: Can Gio Canopy gap Field-of-neighbourhood Individual-based modelling Interrupted point process KiWi model Matern cluster process Point pattern analysis Rhizophora apiculata a b s t r a c t Plantations released from management are vulnerable to transient oscillations until cohort dynamics are broken and the vertical and horizontal structures of the plantation are transformed to those of more natural forests Cohort-desynchronizing factors such as canopy disturbances are expected to accelerate this process Using well-established mangrove plantations in Can Gio (Viet Nam) as an example, we tested whether lightning gaps can affect transition dynamics of plantations to more natural forests by damping the amplitude or by shortening the period of oscillations in tree densities This was done by applying point pattern analyses to remotely sensed data, and by further combining statistical and individualbased modelling The occurrence of lightning gaps was biased by the forest matrix, which presented a challenge for the point pattern analysis This problem was solved by using the scattered forest area as a binary mask A Matern cluster process model was found to be suitable for describing the lightning regime This statistical model was incorporated into the individual-based mangrove model KiWi, and simulation experiments revealed that: (i) the evenly spaced distribution of the tree cohorts in the plantation supports non-linear transition behaviour, i.e oscillation of tree density, and (ii) the lightning regime in Can Gio damps the oscillation amplitude but is not sufficient to prevent the latter nor to decrease the length of the period of oscillations © 2011 Elsevier B.V All rights reserved Introduction The occurrence of cyclic dynamics in populations of annual plants, perennial plants and forests has attracted considerable attention over the last few decades (e.g Mueller-Dombois, 1991; Franco and Silvertown, 2004; Caplat et al., 2008) Both empirical studies (Pacala and Silander, 1990; Franco and Silvertown, 2004) and modelling approaches (Bauer et al., 2002; Caplat et al., 2008) revealed that oscillations in species abundance and biomass occur when factors that synchronize mortality and establishment at stand level predominate factors that desynchronize dynamics such as fire disturbances and storms (Remmert, 1991) Several factors including seed dormancy, germination success, plant fertility, or soil fertility can promote oscillations (e.g Pacala and Silander, 1990) Recent studies highlight the importance of spatial constellations (namely the spatial arrangement of neighbouring ∗ Corresponding author Tel.: +49 8161 714592; fax: +49 8161 714598 E-mail addresses: markuskautz@hotmail.com, kautz@wzw.tum.de (M Kautz) 0304-3770/$ – see front matter © 2011 Elsevier B.V All rights reserved doi:10.1016/j.aquabot.2011.05.005 trees) and the resulting neighbourhood interactions (Bauer et al., 2002; Caplat et al., 2008) and reveal that variability of plant vitality can alter both the rate with which a population converges to a stable size and the average duration of related oscillations (Franco and Silvertown, 2004) Mangrove forests serve as extraordinary examples of both synchronization and desynchronization effects On the one hand, sedimentation and storms frequently create new areas for colonization or secondary succession, maintaining the cyclic dynamics mentioned above (Fromard et al., 1998) On the other hand, individual tree mortality is comparatively high (e.g due to the static instability of larger trees in soft sediments or due to frequent lightning strikes) suppressing the development of senescent cohorts (Duke, 2001), and thus desynchronizing potential oscillations Due to their pronounced cohort structure and regular spatial constellation, plantations are particularly vulnerable to stand dieback and synchronized collapse Rotation systems usually prevent large stand collapses, but this factor which suppresses the occurrence of senescence stages is lost when management is abandoned Whether subsequent succession cycles emerge, how long 174 M Kautz et al / Aquatic Botany 95 (2011) 173–181 Fig Location of the study site within the UNESCO Biosphere Reserve (BR) Can Gio (based on SPOT Image 2007) they last, and to what extent natural processes are capable of desynchronizing such non-linear dynamics are the questions that arise We examine these issues in a case study of mangrove plantations located in the Can Gio UNESCO Biosphere Reserve approximately 60 km south of Ho Chi Minh City, in southern Viet Nam (Fig 1) More than 90% of this forest area was destroyed during the Second Indochina War (Tran et al., 2004; Arnaud-Haond et al., 2009) Re-plantation with Rhizophora apiculata began in 1978 and was aimed at sustained yield management of charcoal, poles, and firewood (FAO, 1993) The release of parts of the plantation to natural forest dynamics without further managed wood production poses a problem: large plots consist of old even-sized trees standing densely in regularly planted rows These trees are vulnerable to synchronized collapse, possibly induced by storms and hurricanes, or other disturbances Lightning strikes are the most frequent forces producing canopy gaps in Can Gio (Dijksma et al., 2010), usually killing several trees at the same time Dead trees remain upright for several years before they crash down providing light and space for re-colonization To evaluate the risk of a synchronized collapse it is important to know whether the frequency and intensity of these disturbances are sufficient to remix the regular cohort structure of the plantation, to accelerate its transformation into a more natural forest structure and thus to prevent cyclic dynamics of the vegetation cover The aim of our study was threefold: (1) to analyse the lightning pattern in Can Gio and to mimic it using a statistical model, (2) to simulate the dynamics of the Can Gio mangrove plantation by means of an individual-based model that includes the lightning model, and (3) to investigate whether canopy disturbances induced by lightning strikes can significantly affect expected oscillations in tree density We expected our analyses and experiments to reveal that the cohort structure of this plantation induces oscillations in forest development as known from terrestrial forests (e.g MuellerDombois, 1987; Itow and Mueller-Dombois, 1988; Elias and Dias, 2009) Furthermore it was hypothesized that canopy disturbances caused by lightning strikes have a damping effect on transient oscillation, accelerating the forest’s transition to a more natural and stable state Methods 2.1 Identification of lightning gaps In order to quantify canopy disturbances induced by lightning strikes in the study site, we used both satellite images and ground truth data The satellite images were selected according to image quality parameters (e.g high spatial resolution, low ratio of cloud coverage, sufficient spectral contrast) and availability of recent, multi-temporal data (time series) highly representative of the study area Based on these criteria, we used panchromatic SPOT images (time series: 2003/2005/2007 with a 2.5 m × 2.5 m pixel size) representing 86% of the reserve above minimum sea level including the entire core zone All three images were taken at similar tidal levels, which minimized the error when identifying the coastline The analysis was based on reference points obtained during the field surveys in December 2007 and September 2008, and from multi-spectral, high-resolution Quickbird satellite images (©Google Earth, 2008) with pixel size 1.0 m × 1.0 m being available for a small part of the core zone As a reference, the topographic situation (coastline, infrastructure, etc.) was incorporated into a geographical information system (GIS) based on the 2007 SPOT image An automated classification of the canopy disturbances caused by lightning was not possible since the quality of the satellite images was inadequate, with a high complexity of the disturbance signatures even at one time step Instead, two subplots were selected and the lightning gaps therein were identified manually (Fig 2) The size of each subplot was 4.0 km × 4.0 km Structural parameters that have been used successfully for identifying the shape and signature of lightning gaps in other mangrove forests (Sherman et al., 2000; Duke, 2001; M Kautz et al / Aquatic Botany 95 (2011) 173–181 175 occurrence of thunderstorms; (2) the description of lightning strikes produced by a thunderstorm A stationary space-time Poisson process (see Illian et al., 2008, p 429) of intensity thund was assumed for the occurrence of thunderstorms This means that the instants of thunderstorms in an arbitrary region of area A form a homogeneous Poisson process of intensity thund ·A on the time axis The locations of the thunderstorms in that region in a time interval of length T form a homogeneous planar Poisson process of intensity thund ·T The parameter thund had to be estimated from the data The lightning strikes produced by a thunderstorm are scattered around the thunderstorm centre Remote sensing analysis (Fig 2) revealed spatio-temporal clustering, which was also reported by Vietnamese foresters (pers comm.) We applied a Matern cluster process (see Illian et al., 2008, p 376) for the description of the spatial distribution of the lightning in a given time interval This model includes parent points (=thunderstorms), which form a planar Poisson process of intensity p (= thund ·T, where T is the length of the time interval) A random number of daughter points (=lightning) are scattered around each parent point This number has a Poisson distribution of mean c¯ The locations of the daughter points are uniform and random within a disc of radius R centred at the parent point The intensity of the point process of lightning is ¯ · thund light = c The parameters thund , c¯ and R had to be estimated based on the remote sensing data Statistically, this posed a challenge since lightning leaves traces in forest canopies, but not in tidal channels or creeks, on mudflats, and in shrub zones (Fig 2) Hence, the window of observation W for the lightning patterns only comprised the forest area within the km × km plots, rather than the entire square This area W, however, is so irregular that even the best software for point process statistics could not be applied Therefore, we applied the interrupted-point-process approach (Stoyan et al., 1995) This approach assumes that there is an original point process ˘ true in the whole area (=lightning strikes) and a random set X (=W), both of which are independent of each other Only those points of ˘ true that fall into X are visible, while all other points of ˘ true are removed The remaining point process ˘ obs is the interrupted point process To simplify the calculation, the random set X was assumed to be spatially homogeneous and isotropic Such a random set is characterized by two fundamental characteristics: Fig Subplot (a) and (b) chosen for the identification of lightning gaps (red crosses) The yellow areas mark open areas such as tidal channels and mud flats The mangrove forest (green) serves as a mask for pattern analysis (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.) Dahdouh-Guebas et al., 2005; Whelan, 2005; Zhang et al., 2008) were considered for use in the visual analysis, increasing the suitability of this method for our specific ecological application Size, frequency and spatial distribution of the lightning gaps were analysed statistically in order to gain all information necessary for modelling the effects of the lightning process on the forest dynamics by means of the individual-based mangrove simulator KiWi (see below) The gap sizes were measured based on the 2007 SPOT image A comparison with SPOT data obtained in 2003 and 2005 provided a rough analysis of the annual lightning frequency For practical reasons, a time interval of two years was used for identifying temporal frequencies irrespective of the slightly different times the images were taken (2003 and 2007: end of January; 2005: mid-April) A lightning process model was designed as a spatio-temporal point process consisting of two parts: (1) the description of the (1) area fraction f = proportion of total area covered by X, and (2) covariance or two-point correlation function C(r) = probability that two randomly chosen points of distance r both lie in X Once these characteristics are known, the fundamental characteristics of ˘ obs can be obtained If true is the intensity of ˘ true , and gtrue (r) its pair correlation function, then the intensity of ˘ obs is given by obs =f · true , (1) and its pair correlation function gobs (r) is gobs (r) = C(r) · gtrue (r) f2 (2) Eqs (1) and (2) enable determination of true and gtrue (r), when f, C(r), obs and gobs (r) are known true can be used to estimate thund , and gtrue (r) leads to estimates of c¯ and R A basic assumption of the interrupted point process model is independence of ˘ true and X Since ˘ true is unobservable, we tested an aspect of this independence property by means of a Monte Carlo test as described in Illian et al (2008, Section 6.11.3) For this test, the channels were approximated by a point pattern with the result that we obtained a bivariate point pattern, with two types of points, where = channel point and = gap The test checked spatial 176 M Kautz et al / Aquatic Botany 95 (2011) 173–181 Table Model description following the ODD protocol Overview Purpose of the model The model was used to investigate the long-term dynamics of mangrove forests considering tree-to-tree interactions, environmental settings and disturbances State variables and spatial scales Individual trees are described primarily by their stem position, stem diameter (dbh), and age Other descriptors such as stem height or the dimension of the field-of-neighbourhood (FON), used to describe local neighbourhood competition among trees, are derived from the dbh as shown in the growth function (see below) Species-dependent tree growth is calculated annually The spatial dimension and shape of the forest stand are variable Plot sizes of km × km were used in this study Process overview and scheduling The following processes occur each year: establishment of new saplings, growth of existing trees, and tree mortality The stem diameters of all trees are updated synchronously and the derived parameters such as tree height and FON radius are re-calculated Design concepts Emergence Interactions Population dynamics emerges from the life processes modified by tree-to-tree competition, e.g oscillation dynamics of tree densities and basal areas, or variations in the spatial distributions of trees ranging from clumped, to random to regular, among others Trees compete for all spatially distributed resources (not explicitly specified) via their FONs Sensing Trees “sense” the distance, size and explicit location of their neighbours by their overlapping FONs Stochasticity Saplings establish randomly, depending on local conditions (not in the initialization of the plantation) Tree mortality due to disturbances (here: lightning) is described by random functions Observations The model provides yearly tracking of all state variables and derived parameters for all trees Details Initialization Input Submodels Description of a single tree Empty plots of km × km were established as a plantation with a tree-to-tree distance of m New trees colonizing canopy gaps were considered as saplings once they had reached a minimum height of 1.27 m Yearly recruitment rates define the establishment of new saplings Abiotic factors such as topography, inundation height, inundation frequency, pore water salinity and nutrient availability can be addressed explicitly by user-supplied maps corresponding to the simulated forest stand; but for the purpose of this study they were considered to be optimal for the whole forest A tree is described by its stem position (x,y), stem diameter (dbh), and field-of-neighbourhood (FON) The latter describes the area which a tree influences its neighbours and is influenced by its neighbours The radius R of the FON increases with dbh: within √ R = a · 0.5 · dbh The intensity of competition is calculated as FON(r) = e−c(r−(0.5·dbh)) Recruitment and establishment Seedling growth is not explicitly modeled due to the lack of field data Seedling growth and mortality, however, are implicitly included in the sapling recruitment rates Saplings can establish if tree density and the resulting intra-specific competition are below a certain threshold at the potential, randomly chosen location This threshold mimics a given shadow tolerance of the species Tree growth The model uses a JABOWA-type growth function, where the annual stem increment is a function of dbh and stem height H: dbh = G·dbh·(1−dbh·H/Dmax ·Hmax ) 274+3b2 ·dbh−4b3 ·dbh2 · CFON , with H = 137 + b2 · dbh − b3 · dbh2 This function is parameterized for optimal growth conditions The growth multiplier CFON corrects the stem increment depending on tree neighbourhood competition Competition The intensity of the FONs of all neighbouring trees on the FON of a focal tree is taken as a measure of the competition strength the focal tree suffers This value is related to the area of the FON of the focal tree, assuming that the influence of larger trees on smaller ones is stronger than vice versa Mortality The model considers two different sources of mortality (1) Due to a prolonged period of growth depression: Since there is no field data available on that process, the model describes it phenomenologically A tree dies if its mean stem increment over a specified time range (here years) is less than half of the average increment under optimal conditions This occurs when the stem diameter approaches its maximum, or results from salinity stress, nutrient limitation, or competition among neighbouring trees This procedure assures that a tree has a chance to recover when conditions improve, e.g when a neighbouring tree dies (2) Due to lightning: For its description, we fitted a spatio-temporal point process model to the gap data observed at the study site The implementation of the lightning process comprises five steps: (1) choice of a random location of a thunderstorm centre within the forest; (2) choice of cluster type: small cluster (probability p1 ) or large cluster (probability p2 = − p1 ); (3) choice of the number of lightning strikes within the cluster according to a Poisson distribution with mean c¯ and c¯ respectively; (4) choice of the random position of each lightning strike in a radius R1 or R2 around the centre of the cluster; (5) removing all trees in the gap radius RGap surrounding the chosen position The value for RGap was chosen from size classes, according to the frequency of their occurrence measured for Can Gio (see Table 2) The gap shape was assumed to be circular correlations between the 1- and 2-point positions The bivariate Lfunction L12 (r) was estimated for the data and then compared with L12 (r)-functions of randomly generated bivariate patterns, where the 1-points were the original ones while the 2-points were randomly shifted, with the same shift vector for all points 2.2 Analysis of the impact of canopy disturbances on plantation structure In order to understand the influence of canopy disturbances on the dynamics of the mangrove forest plantation, the individualbased mangrove model KiWi, developed for simulating neotropical mangrove forests (Berger and Hildenbrandt, 2000), was parameterized to the Asian mangrove species R apiculata and combined with the lightning process model The KiWi model provides a spatially explicit description of a virtual mangrove forest Each individual tree is characterized by its stem position, stem diameter (dbh) and a circular field-of-neighbourhood (FON) which specifies both the zone of interaction with neighbouring trees and the strength of competition exerted by the tree at each location inside its FON (Berger and Hildenbrandt, 2000) The size of the FON increases with dbh The annual increment of dbh depends on the current size (dbh and height) of the tree, the competition strength of neighbouring trees, pore water salinity and nutrient availability For the simu- M Kautz et al / Aquatic Botany 95 (2011) 173–181 Table Parameters used for the simulations The KiWi model was parameterized to Rhizophora apiculata based on own data, according to the procedure described in Fontalvo-Herazo et al (submitted for publication) The lightning submodel parameters were obtained by statistical modelling of the observed gap pattern (see Section 3) Parameter Description KiWi model a Scaling factor of the FON b Scaling factor of the FON Minimum value of the FON Fmin G Growth constant Dmax Maximum dbh (cm) Hmax Maximum height (cm) b2 Constant height-to-dbh relationship b3 Constant height-to-dbh relationship th Tree mortality threshold Lightning submodel Intensity of lightning process (ha−1 year−1 ) light Intensity of thunderstorm process (ha−1 year−1 ) thund p1 Probability of small lightning clusters p2 Probability of large lightning clusters c¯ Mean number of gaps within a small cluster c¯ Mean number of gaps within a large cluster R1 Radius of small lightning cluster (m) R2 Radius of large lightning cluster (m) RGap Radius of lightning gaps (m), and corresponding probabilities of occurrence Value 0.0176 0.0075 0.38 0.62 2.01 2.56 62 380 ≤ (0.11) ≤ 14 (0.53) 14 ≤ 18 (0.26) 18 ≤ 23 (0.10) lation experiments we assumed pore water salinity not exceeding 58 ppm and optimal nutrient availability, so that tree growth is not reduced by these two factors (Chen and Twilley, 1998) Competition strength between neighbouring trees is expressed by the intensity of overlapping FONs We considered the fact that larger trees exert a stronger effect on smaller trees than vice versa by normalizing the FON overlap related to the FON of the focal tree The original KiWi model without lightning process has already been described in detail in previous publications (e.g Berger and Hildenbrandt, 2000; Berger et al., 2002, 2008) Here we thus provide a standardized ODD (=Overview, Design concepts, Details) protocol following the recommendation of Grimm et al (2006, 2010) in Table For the description of lightning pattern, we used the spatiotemporal point process model described above This leads to positions of lightning strikes during the simulation time Then all trees in the gap radius RGap surrounding the chosen position are removed The value for RGap was chosen from four size classes, according to the frequency of their occurrence measured for Can Gio The gap shape was assumed to be circular All parameter values used for the simulations in this study are given in Table Simulation experiments were conducted for three disturbance regimes: (1) without lightning, (2) with the lightning gap frequency determined for our study site (Can Gio: 0.02 ha−1 year−1 ), (3) with the highest lightning strike frequency reported so far (Florida: 0.1 ha−1 year−1 ; Huffines and Orville, 1999; Shafer and Fuelberg, 2006), using the Matern process model For each disturbance regime we carried out 10 replications For a description of the transient oscillation, we analysed the evolution of tree density over time We then measured (i) the time t between consecutive maxima in tree density, (ii) the logarithmic decrement of consecutive amplitudes (A1 and A2) A1 A2 = ln , (3) t General information Location (xy coordinates of the NW corner; UTM/WGS ‘84) Total area Subplot Subplot 699000/1165000 704000/1168000 km × km (1600 ha) 71% (1137 ha) 116 km × km (1600 ha) 70% (1128 ha) 110 617 40 219 0.22 964 49 249 0.24 49 8988 0.022 0.040 44 9753 0.020 0.043 29 8756 0.013 0.039 36 10,622 0.016 0.047 To determine whether lightning regimes have different effects on the development of the simulated forests, a Kruskal–Wallis test was carried out for selected time steps In the case of significance, a Post Hoc test according to Siegel and Castellan (1988) revealed which scenarios differ from each other The satellite images were analysed using ArcGIS 9.0 (ESRI Inc., 2004) All statistical analyses were carried out with R (R Development Core Team, 2011): we used the R package SpatStat (Baddeley and Turner, 2005) for point pattern analyses and the R packages EMD (Donghoh and Hee-Seok, 2009) and pgirmess (Giraudoux, 2010) for the analysis of the simulated tree density time series 2.3 Sensitivity analysis To trace the contribution of uncertain model parameters to the forest dynamics, a sensitivity analysis was performed using the extended Fourier amplitude sensitivity test (eFAST), a variancebased global sensitivity method (Cukier et al., 1978; Saltelli et al., 1999, 2000) It quantifies the contribution of the individual input parameters to the variance of the output variables It reveals both the main effect of each parameter on the model output and the sum of the effects resulting from higher-order interactions of each parameter with the other parameters (Saltelli et al., 2000; Saloranta and Andersen, 2007) The model output was analysed based on discrete Fourier transformation using the software package SimLab (2011) The eFAST sensitivity analysis was performed using R (R Development Core Team, 2011) Following its specific sampling procedure, different parameter sets were estimated with a −10% to +10% range for each parameter described in Table The parameters of the lightning submodel were not considered here in order to reduce the complexity of this analysis This procedure seems to be adequate since this submodel could be fitted directly to the observed lightning patterns in Can Gio Multiple simulation runs (73 × parameters = 657 simulations) were carried out The tree densities after 100 years were registered Results 3.1 Identification of the lightning gaps and (iii) the damping coefficient = Table Lightning gap data obtained for Can Gio by satellite image analysis Mangrove area Total number of lightning gaps Gap size Maximum (m2 ) Minimum (m2 ) Average (m2 ) % of disturbed mangrove area Frequency of lightning gaps # of new gaps in 2007 Total gap area in 2007 (m2 ) # gaps 2005–2007 (ha−1 year−1) Disturbed area 2005–2007 (% year−1 ) # of new gaps in 2005 Total gap area in 2005 (m2 ) # gaps 2003–2005 (ha−1 year−1 ) Disturbed area 2003–2005 (% year−1 ) 0.654 0.1 281.25 80 3000 71.575 0.44734 0.2 177 (4) Table summarizes the lightning gap characteristics obtained in both subplots via the analyses of the satellite images Both 178 M Kautz et al / Aquatic Botany 95 (2011) 173–181 Fig The empirical bivariate L12 (r)-function demonstrating the independence of lightning gaps and channels in subplot with the 95% confidence bands (dashed) subplots include comparable portions of mangrove vegetation but differ slightly in number of gaps and gap sizes A comparison of gap frequencies between the two time steps 2003–2005 and 2005–2007 shows the annual variability of this process For implementation into the lightning submodel an average frequency (0.02 gaps ha−1 year−1 ) computed over both subplots and time steps was used The exact value was 0.017 ± 0.002 (mean ± SE), see Table 3.2 Analysis of lightning gap patterns Analysis of the spatial correlations between the gaps and the water channels reveals that both patterns are spatially noncorrelated: the empirical L12 -functions for both subplots are completely within the envelopes of L12 -functions obtained by 99 simulations of non-correlated bivariate patterns, as shown for subplot in Fig Thus we can assume that the gap locations are not related to the position of the channels The second-order analysis eventually yields the pair correlation function g(r) of the lightning process First, we estimated the pair correlation functions gobs (r) of the gaps for the whole subplots These were then corrected to obtain gtrue (r) by considering the forest matrix as observation window, using Eq (2) and statistically determined f and C(r) It should be noted that the estimated area fraction was f = 0.7 as a mean for both subplots, specifying that around 70% of the area was covered by trees The gtrue (r) and gobs (r) functions for subplot are shown in Fig 4a The values above for distances r < 800 m of both pair correlation functions indicate strong clustering of the gaps The smaller values of the corrected function gtrue (r) in comparison to gobs (r) indicate that the true degree of clustering is somewhat smaller than the degree of clustering observed The results for subplot (not shown) were similar For subsequent calculations the pair correlation functions for both subplots were aggregated by averaging these, using Eq (4.7.6) in Illian et al (2008) This is based on the assumption that the difference between the pair correlation functions of the two subplots is only a sampling effect 3.3 Modelling the lightning process Based on the observed lightning gap pattern, an adequate statistical model had to be developed to implement the disturbance Fig (a) Pair correlation functions of the lightning gaps in subplot 1, calculated for the whole km × km area gtrue (r) (solid), and corrected by an observation window specified by the forest matrix gobs (r) (dashed) (b) Empirical function gmean (r) (solid) representing the aggregated pair correlation function of the gaps observed in both subplots, and the derived model function gmodel (r) (dashed) Values above (horizontal line) indicate clustering regime into the mangrove forest simulations The aggregated pair correlation function gmean (r) appears to consist of two components: one representing small, highly concentrated clusters (the large values of gmean (r) for r < 100 m) and the other representing larger, thinner clusters (values above one in the range of 100 m < r < 300 m) (Fig 4b) Therefore, the statistical model assumes that there are two classes of clusters: large and small Both cluster classes are characterized by their radii R1 and R2 , their mean numbers per cluster c¯ and c¯ , and their probabilities p1 and p2 (Table 2) These parameters were estimated by a least squares estimation based on the theoretical pair correlation function of the generalized Matern cluster process and gmean (r) (Illian et al., 2008, p 371) Finally, we estimated the space-time intensity thund of the Poisson process of thunderstorm centres The data used for this purpose are the numbers of lightning gaps in the four-year period 2003–2007 within both subplots These are 78 and 80, respectively This yielded an estimate for the intensity light of lightning: light = 78 + 80 = 1.76 km−2 year−1 = 0.0176 ha−1 year−1 · × · · 0.7 M Kautz et al / Aquatic Botany 95 (2011) 173–181 179 Table Results of the Kruskal–Wallis test applied to tree densities simulated for the three different lightning scenarios (see also Fig 6) at selected time steps The right side of the table presents results of the Post Hoc test to specify the differences between particular scenarios: e.g the first column specifies a statistical difference (p < 0.05: TRUE) of tree densities in the scenario without lightning strikes in comparison to the scenario with a lightning frequency observed in Can Gio until time step III; this effect disappears in the time step IV Time step I II III IV Kruskal–Wallis statistics Post Hoc test results for differences between the lightning scenarios Chi-squared p-Value Without—Can Gio Without—Florida Can Gio—Florida 28.64 27.27 72.75 4.67