229 15 Challenges in the Analysis and Simulation of Benthic Community Patterns Mark P. Johnson CONTENTS 15.1 Empirical and Theoretical Treatments of Spatial Scale in Benthic Ecology 229 15.1.1 Rarity of Spatially Explicit Models for Benthic Systems 230 15.2 Robust Predictions from Spatial Modeling 231 15.3 Comparing Markov Matrix and Cellular Automata Approaches to Analyzing Benthic Data 231 15.3.1 Nonspatial (Point) Transition Matrix Models 233 15.3.2 Spatial Transition Matrix Models 234 15.3.3 Comparison of Empirically DeÞned Alternative Models 235 15.4 Extending the Spatial CA Framework 236 15.5 Conclusions 239 Acknowledgment 240 References 240 15.1 Empirical and Theoretical Treatments of Spatial Scale in Benthic Ecology The composition and dynamics of benthic communities reßect the interplay of factors that operate at a range of scales. Variability at almost every scale of observation is likely to affect benthic species. For example, hydrodynamic gradients exist from the centimeter scale of the benthic boundary layer to ocean basin scale circulation patterns. The settlement of benthic species from planktonic life history stages will reßect both the large-scale and small-scale inßuences on propagule supply. Many benthic species have limited mobility as adults, so individuals may only interact with other individuals within a relatively short distance. However, population dynamic processes such as mortality may also be composed of elements at quite different scales. For example, mortality of barnacles can be caused by both the crowding effects of neighbors and mobile predators such as whelks or crabs. Given that there seems no basis for assuming any “correct” scale of observation (Levin, 1992), the empirical response has been to characterize variability at a number of scales. This form of pattern identiÞcation can be considered a prerequisite for subsequent studies on process (Underwood et al., 2000). Many studies of spatial scale have used nested analysis of variance (ANOVA, e.g., Jenkins et al., 2001; Lindegarth et al., 1995; Morrisey et al., 1992). This may reßect the familiarity of the ANOVA approach from experimental hypothesis testing in benthic ecology. Indeed, many experimental manipu- lations also include explicit considerations of scale and scaling effects (Thrush et al., 1997; Fernandes et al., 1999). Spatial autocorrelation and fractal analyses have also been used to characterize spatial pattern in benthic systems (Rossi et al., 1992; Underwood and Chapman, 1996; Johnson et al., 1997; © 2004 by CRC Press LLC 230 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Maynou et al., 1998; Snover and Commito, 1998; Kostylev and Erlandson, 2001). Although patterns may have been considered at different times, many studies of spatial scale have taken a snapshot view: spatial pattern at one point in time has not been mapped onto the spatial pattern at other times. The snapshot approach restricts further investigation into issues of turnover and dynamics. However, with the growing interest (Koenig, 1999) in deÞning the spatial scales over which population dynamics are linked (synchronous), there are likely to be more spatiotemporal studies of benthic populations and communities in the future. These studies of population synchrony are important as they deÞne the spatial structure of populations (Koenig, 1999; Johnson, 2001). The degree of synchrony between local popu- lations has implications for conservation biology. If local populations are asynchronous, then a local extinction event may be reversed by individuals supplied from another, healthy, population. Large-scale loss of a species is more likely where local populations are synchronous and no rescue effects occur (Harrison and Quinn, 1989; Earn et al., 2000). The inclusion of explicit treatments of spatial scale in much of the empirical research on benthic systems has not been paralleled by extensive theoretical work on the same systems. Inßuential models do exist for space-limited benthic systems (Roughgarden et al., 1985; Bence and Nisbet, 1989; Possingham et al., 1994) and patch dynamics on rocky shores (Paine and Levin, 1981). However, these models do not have an explicit treatment for space: the variables in the models are not differentiated on the basis of their relative locations (although see Possingham and Roughgarden, 1990, and Alexander and Roughgarden, 1996, for extensions of the framework to include spatial population structure along a coastline). Simulations of benthic communities with a one-dimensional representation of spatial location have been used to look at the development of intertidal zonation along environmental gradients (Wilson and Nisbet, 1997; Johnson et al., 1998a). Spatially explicit models of benthic systems on two dimensional lattices have generally shown interactions between processes at different scales. Local interactions can lead to a large-scale pattern (Burrows and Hawkins, 1998; Wooton, 2001a) and the predictions of spatially explicit and nonspatial models differ (Pascual and Levin, 1999; Johnson, 2000). 15.1.1 Rarity of Spatially Explicit Models for Benthic Systems Despite the observation that “space matters” and an explosion of interest in spatial ecology (Tilman and Kareiva, 1997), there are a number of reasons why spatially explicit models of benthic communities may be uncommon. The lack of system-speciÞc models partly reßects the manner in which spatial theory has developed. Spatially explicit models tend to be caricature or generic models that attempt to capture the essential features of the system (Keeling, 1999). This approach improves the conceptual understand- ing of systems and allows numerical experiments that would be difÞcult or destructive in a real system (Keeling, 1999). The use of generic models improves communication between theoreticians as there can be clarity about techniques and general conclusions without debate on the individual nature of biological interactions in particular systems. The rarity of system-speciÞc models can also be explained by considering the problems associated with an imaginary spatially explicit model for a benthic community. The model is as realistic as possible, with a number of interacting species inßuenced by stochastic variation in processes such as recruitment. Simulation output resembles the patterns seen in the real system. However, formal testing of the model would involve collecting large amounts of detailed spatial data from the Þeld (independently from that used to derive the model). As the model contains stochastic processes, a large number of repeated simulations are needed to deÞne the potential behavior of the system. Given the range of potential outputs that the model may produce, it is difÞcult to envisage how a limited number of spatial data sets could be used to falsify the model. Both the collection of data and repeated simulations are time-consuming. More importantly, we are not likely to be interested in the detailed spatial arrangement of species in the benthic community. Only a subset of model predictions (such as the mean abundance of a species) is likely to be both of interest to applied research and testable. Hence the time required to develop a model for a speciÞc system may not be justiÞed in the end results. © 2004 by CRC Press LLC Challenges in the Analysis and Simulation of Benthic Community Patterns 231 15.2 Robust Predictions from Spatial Modeling There is a tension between the observation that spatial effects can be important and the difÞculties involved in testing detailed spatially explicit simulations. However, if our understanding of benthic community patterns is to be addressed, a way of resolving this tension is needed. There have been two approaches to this problem, which can be loosely classiÞed as theory based and data based. Theory-based approaches to spatially explicit modeling are extremely diverse and include reaction- diffusion and partial differential equations. It is difÞcult, however, to construct a mathematically tractable model that is also applicable to particular ecological systems such as different benthic communities (Tilman and Kareiva, 1997). In a recent development, researchers have used “pair approximation” techniques to provide analytically tractable models (Levin and Pacala, 1997; Rand, 1999; see Snyder and Nisbet, 2000, for a critique and alternative approach). The idea behind pair approximation is that the equation for a nonspatial process can be extended to a spatial system by using functions that approximate the average neighborhood structure in a spatial model. Hence, in contrast to a model where the same equation is repeated at a large number of locations, the small-scale spatial detail is included in a limited number of equations. The pair approximation approach therefore facilitates investigation of model behavior more efÞciently than would be the case in a simulation. As yet these models still tend to be generic, and may thus ignore important features of benthic systems. For example, a common assumption is that dispersal is a local process (Levin and Pacala, 1997; although see Pascual and Levin, 1999). This contrasts to the characterization of many benthic populations as open (Roughgarden et al., 1985; Caley et al., 1996): new recruits may be supplied by sources at some distance from the local population. In contrast to the development of generic descriptions in the theory-based approach, the data-based approach involves case studies of speciÞc systems. Ideally, a number of alternative models with different treatments of space will be tested against Þeld observations. This approach has the advantage that movement to more complex models is justiÞed only where there are improvements in predictive ability. The beneÞts of building a sequence of models are further outlined in Hilborn and Mangel (1997). As it seems impractical to develop a large number of spatially explicit models for different benthic systems, the challenge in the analysis and simulation of benthic populations is to combine the theory- based and data-based approaches to produce a set of methodological approaches that can be used to investigate and contrast different systems. Although this viewpoint is not novel, there remain few examples of synergy between theoretical and empirical approaches for benthic systems. A notable exception is the work of Wootton (2001b) on intertidal mussel beds. The approach taken in the Þrst section below mirrors that of Wootton (2001b) in that multispecies Markov models are used as the basis for comparing spatial and nonspatial models. A slightly different approach is taken in the second section, where a more complex model is used to suggest methods for distinguishing between alternative hypotheses using Þeld data. The analyses presented use a broad interpretation of “benthic” that includes the rocky intertidal. Rocky shores are generally considered more tractable than sandy or muddy systems. For example, it is far easier to Þx locations and organisms are generally not subsurface in rocky systems. On a conceptual level, however, there is nothing to prevent the application of spatial models to sandy or muddy systems (although the scales of processes such as adult mobility are likely to differ with increasing mobility of the sediment). 15.3 Comparing Markov Matrix and Cellular Automata Approaches to Analyzing Benthic Data What approaches are there available to move beyond generic models and statistical pattern identiÞcation in the analysis of benthic systems? A Þrst task is to recap on the potential shortcomings of theory-based and wholly empirical approaches. The generic nature of certain theoretical approaches has been detailed above. Potential limitations of statistical pattern analysis (e.g., spatial autocorrelations, nested ANOVA) are restrictions on generalization from results and a lack of sensitivity tests of conclusions. Assessment © 2004 by CRC Press LLC 232 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation of a pattern, once identiÞed, can be limited to rhetorical arguments about the interaction of processes at different scales. Follow-up experiments can be difÞcult to design as the alternative, spatially explicit, hypotheses are not always intuitive. By examining the consequences of different assumptions, models can extend experimental results to create appropriate hypotheses. Existing techniques for incorporating Þeld data into a modeling framework include Markov transition matrix models and cellular automata. Other techniques exist, probably dependent on the ingenuity of the investigator. Markov models and cellular automata, however, have several advantages. They are well known and relatively simple to apply. Hence, different investigators can use them and compare results in a common format. Markov models have the potential for sensitivity testing; they also form an appropriate nonspatial null model for comparison with data and spatially explicit alternative models. Cellular automata can be used to inves- tigate neighborhood effects and can be used to identify scaling properties of systems (e.g., power law relationships in patch geometry; Pascual et al., 2002). Here I emphasize cellular automata as they can be “twinned” with experimental procedures at the same scale: within shore patches and external forcing at the grid scale (cf. spatial replication between shores). Other techniques exist for spatial modeling, for example, applications of geographical information systems (GIS) at the landscape scale. However, GIS applications are probably closer to statistical pattern analysis in that the scope for sensitivity tests and experimental investigation of predictions is limited. Cellular automata (CA) and Markov transition matrix approaches have underlying similarities and yet they are generally used in completely different ways. Both approaches use a discrete description of time and state. Temporal dynamics in both frameworks are usually Þrst order: state at time t + 1 is dependent on state at time t. Such transitions may be entirely deterministic or occur with a speciÞed probability. Where the two approaches differ is that CA includes a discrete representation of space, typically visualized as a grid of square or hexagonal cells. The cells in the neighborhood of an individual location on the CA grid inßuence the transition between states at that location from one time step to the next. Applications of CA usually stress simplicity at the expense of biological realism (Molofsky, 1994; Rand and Wilson, 1995) but cite speed of computation and heuristic value (Phipps, 1992; Ermentrout and Edelstein-Keshet, 1993). In comparison, Markov transition matrix models are frequently derived directly from Þeld data and are used to examine characteristic processes in the observed communities (Horn, 1975; Usher, 1979; Callaway and Davis, 1993; Tanner et al., 1994). In theory, it is straightforward to reconcile the issues of spatial dependence and empiricism that transition matrices and CA, respectively, ignore. By constructing a CA using observed local transition probabilities, it is possible to compare models containing local interactions with nonspatial models. A problem with this approach is the data requirement needed to parameterize even a simple CA. For example, a cell in a system with four states and eight neighbors would have 4 8 (65,536) possible neighborhood conÞgurations. It would be practically impossible to empirically deÞne a transition prob- ability associated with each neighborhood conÞguration. However, given information about the important interactions in a system, effort can be concentrated on deÞning a limited number of transitions. An example of a relatively well studied system is the mosaic of macroalgal (mostly Fucus spp.) patches on smooth moderately exposed rocky shores in the northeast Atlantic (Hawkins et al., 1992). Spatial structure and patch dynamics in this system are thought to be driven by limpet grazing (Hartnoll and Hawkins, 1985; Johnson et al., 1997). Spatial autocorrelation studies have suggested an algal patch length scale of approximately 1 m in this system. Time series from a quadrat of similar dimensions to the patch scale show multiannual variations in algal cover, with limpet densities tending to lag these ßuctuations. The conceptual model developed for this system is based on the interaction between limpet grazing pressure and the recruitment of algae. Limpets are aggregated in clumps on the shore and the uneven spatial distribution of grazing pressure leads to the formation of new patches of algae in areas where there are few limpets. The spatial mosaic of algal patches formed by uneven grazing pressure is in grazing pressure and allows new patches of algae to be generated elsewhere on the shore. Older patches of algae do not regenerate, possibly because of the increased local density of limpets associated with them. Hence the shore is patchy, but the locations of patches change, creating the multiannual ßuctuations seen at the patch scale. © 2004 by CRC Press LLC dynamic (Figure 15.1). Adult limpets relocate to established patches of algae. This generates changes Challenges in the Analysis and Simulation of Benthic Community Patterns 233 The proposed mechanism for the patch mosaics on moderately exposed rocky shores in the northeast Atlantic implies that the effort in deriving spatial transition rules can be concentrated on deÞning how they are affected by the local limpet density. By constructing a traditional nonspatial transition matrix model it is possible to test if the system dynamics are at least a Þrst-order Markov process. Empirically derived CA rules with and without a local limpet presence can be tested to investigate whether limpets do actually affect local state transitions. Spatial and nonspatial Markov processes can be compared to test whether local interactions alter the projected dynamics of the system. 15.3.1 Nonspatial (Point) Transition Matrix Models Transition matrix models are deÞned by marking out Þxed sites, deÞning states, and recording the transitions between states in a deÞned time period. In work carried out in the Isle of Man (methods described more fully in Johnson et al., 1997) the Þxed sites were 0.01 m 2 square “cells” in permanently marked 5 ¥ 5 m quadrats (2500 cells per quadrat) and the time step was 1 year. If a cell contained algae, a distinction was made between “mature” and “juvenile” cells. A juvenile cell was one where algal frond lengths did not exceed 0.1 m and reproductive structures were absent. Barnacle cover outside algal patches was variable. If a cell contained no barnacles at all it was classed as bare rock. Coralline red algae were generally associated with small rock pools. If the areal cover of coralline red algae exceeded that of barnacles, a cell was classiÞed as “coralline red.” The presence or absence of adult limpets was recorded (shell diameter >15 mm) for each of the Þve basic classiÞcation states (barnacle, juvenile, mature, coralline red, and rock). Transition matrices take the form: (15.1) where p jk is the probability of transition from state k to state j with each time step. Transition probabilities are derived from a frequency table of state k to j changes. The frequency of each change from one state to another is divided by the column total to give the probability of each transition. Transitions are tested for interdependence (with the null hypothesis being that transitions are independent, i.e., random, and therefore the process is not Markovian) using a likelihood ratio test, with –2 ln l compared to c 2 with ( m –1) 2 degrees of freedom (Usher, 1979): FIGURE 15.1 Idealized cycle at the patch scale on moderately exposed shores in the northeast Atlantic. Spatial variation in limpet grazing pressure allows recruitment of juvenile algae to the shore. Patches eventually decay. The aggregation of limpets in aging patches of algae changes the spatial pattern of grazing pressure, allowing new patches to be formed elsewhere on the shore. Barnacles, no limpets (b -) (j -) (m -) (b+) (m+) Juvenile algae, no limpets Mature algae, no limpets Mature algae, limpets Barnacles, limpets A = Ê Ë Á Á Á Á Á Á ˆ ¯ ˜ ˜ ˜ ˜ ˜ ˜ ppp p ppp p ppp p ppp p n n n nn n nn 11 12 13 1 21 22 23 2 31 32 33 3 123 L L L MMMMM L © 2004 by CRC Press LLC 234 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation (15.2) where (15.3) n jk = number of transitions from state k to j in the original data matrix p jk = probability of transition from state k to j p j = sum of transition probabilities to state j m = order of the transition matrix (number of rows) The sum effect of all transitions over a time step is found by the multiplication: (15.4) where x (t) is a column vector containing frequencies of separate cell state at time t. With transition matrices, repeated multiplication by A generally causes the community composition to asymptotically approach a stable state distribution deÞned by the right eigenvector of A (Tanner et al., 1994). The temporal scales of processes can be investigated from metrics derived from transition matrices. For example, the rate of convergence to a stable stage structure is governed by the damping ratio, r (Tanner et al., 1994; Caswell, 2001): (15.5) where l j is an eigenvalue of the transition matrix. As matrix columns sum to one, the Þrst eigenvalue is always one. A convergence timescale is given by t x , the time taken for the contribution of the Þrst eigenvalue to be x times as great as the contribution from the second eigenvalue (Caswell, 2001): (15.6) 15.3.2 Spatial Transition Matrix Models Maps of adjacent 0.01 m 2 cells allow spatial transition rules to be deÞned. The effect of limpets on the transitions occurring in their neighborhood can be tested by deriving two separate transition matrices: one for transitions when limpets were present in at least one of the neighboring eight cells and one matrix for cell transitions occurring in the absence of limpets in surrounding cells. The signiÞcance of differences between “local limpets” and “no local limpets” transition matrices can be examined using (Usher, 1979; Tanner et al., 1994): (15.7) where L = number of transition matrices associated with limpet grazing effects (= 2) n jk (L)= number of k to j transitions recorded for matrix L p jk (L)= transition probability from k to j in matrix L p jk = transition probability from k to j if L matrices are pooled The likelihood ratio is compared to c 2 with m(m – 1)(L – 1) degrees of freedom and a null hypothesis that there is no difference between matrices dependent on the presence or absence of limpets in the eight cell neighborhood. -= Ê Ë Á ˆ ¯ ˜ == ÂÂ 22 11 ln lnl n p p jk jk j k m j m p n n j jk jk k m j m k m = == = ÂÂ Â 11 1 Ax x t t1 () + () = rl l= 12 / tx x = ln( ) / ln( )r -= Ê Ë Á ˆ ¯ ˜ === ÂÂÂ 22 111 ln ( )ln () l nL pL p jk jk jk L L k m j m © 2004 by CRC Press LLC Challenges in the Analysis and Simulation of Benthic Community Patterns 235 It is not possible to iterate the spatial model using matrix multiplication as the choice of transition probability is dependent on local conditions. Spatial transition matrices were therefore investigated using CA simulations. These simulations were based on 50 ¥ 50 square cell grids with periodic boundary conditions (cells on one edge of the grid are considered to be neighbors to cells on the opposite edge of the grid). As the CA rules are derived empirically from counts of 0.01 m 2 cells, spatial simulations represent an area of 25 m 2 . Cell state transitions at each time step were based on probabilities drawn from a matrix chosen according to the neighborhood state (“local limpets” or “no local limpets”). Simulations were stochastic as random numbers were used to generate cell state transitions based on the probabilities in the appropriate matrix (the spatial model was what is sometimes referred to as a “probabilistic CA”). 15.3.3 Comparison of Empirically Defined Alternative Models Point and spatial transition matrices were derived for three separate 25 m 2 quadrats at different sites in the Isle of Man (hereafter referred to as sites a, b, and c). At each site, likelihood ratio tests supported the application of Markov matrices to the observed transitions (Equation 15.2, p < 0.001 in all cases). Hence the matrices contain information about a nonrandom process of transitions at each site. An example point transition matrix is shown in Table 15.1. The pattern of transitions reßects parts of the patch cycle proposed by Hartnoll and Hawkins (1985). For example, the majority of barnacle-classiÞed cells became occupied by algae. Most cells classed as juvenile algae were recorded as mature algae in the following year. The predicted dynamics rapidly approached equilibrium, with convergence time scales ( t 10 ) of 4.17, 1.51, and 1.23 years for sites a, b, and c, respectively. This implies a high degree of resilience at two of the sites with recovery to the equilibrium state within 2 years of a perturbation. It is not clear what features make site a recover more slowly than the other sites. One possibility currently under investigation is that variation in dynamics reßects differences in surface topography. The spatial transition matrices for the “no local limpets” and “local limpet” cases were signiÞcantly different at all three sites (Equation 15.7, p < 0.05). This supports the hypothesis (Hartnoll and Hawkins, 1985) that the spatial pattern of limpet grazing affects interactions on the shore. There was some variation between sites, but the transition frequencies reßected the inßuences of limpets on transitions to algal cover. For example, at the site with the largest difference between matrices, 63% of all transitions were to algal occupied states in the “no local limpets” matrix compared to 53% in the “local limpets” case. As has been shown elsewhere (Wootton, 2001b), predictions of the matrix models Þt the observed state G tests show that the Þt of the models is closer than would be expected for randomly generated frequencies, The discrepancy between predicted and observed frequencies was generally not reduced by using a spatial rather than a point model. In addition, the predictions of spatial and point models were not signiÞcantly different for site c. Despite the detection of spatial effects associated with limpets, the increase in model complexity from point to spatial models was not justiÞed by a better Þt to the data. TABLE 15.1 Matrix of Transition Probabilities for Quadrat a Surveyed in the Isle of Man b+ b– j+ j– m+ m– cr+ cr– b+ 0.024 0.036 0.000 0.015 0.028 0.010 0.100 0.039 b– 0.040 0.105 0.021 0.024 0.028 0.030 0.100 0.078 j+ 0.079 0.042 0.021 0.039 0.056 0.035 0.050 0.024 j– 0.333 0.224 0.128 0.119 0.139 0.055 0.050 0.083 m+ 0.095 0.072 0.149 0.124 0.250 0.199 0.000 0.044 m– 0.397 0.468 0.670 0.671 0.500 0.662 0.200 0.150 cr+ 0.000 0.004 0.000 0.003 0.000 0.005 0.050 0.044 cr– 0.032 0.048 0.011 0.006 0.000 0.005 0.450 0.539 Note: Cells are classiÞed as barnacle occupied (b), juvenile Fucus (j), mature Fucus (m), and coralline red algae (cr). Bare rock was not recorded in cells at this site. + or – modiÞers indicate the presence or absence of limpets in the cells. © 2004 by CRC Press LLC frequencies reasonably well (explaining between 45 and 97% of the variation in frequencies; Figure 15.2). but that there were still departures between model predictions and observed frequencies (Table 15.2). 236 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation The spatial model may still have some heuristic value if it generates a dynamic pattern of states in simulations. Techniques for investigating spatiotemporal pattern include calculating correlations between sites at different distances from each other (Koenig, 1999). An alternative approach used in scaling investigations of spatial models (De Roos et al., 1991; Rand, 1994; Rand and Wilson, 1995) is to compare the dynamics of cell frequencies in “windows” of different sizes on the simulation grid. For any probabilistic CA, cell state frequencies will ßuctuate with time. The standard deviation of a time series taken from a window of L ¥ L grid cells will decrease with increasing L (tending to zero at very large window sizes). For a stochastic process, the reduction in standard deviation with window size will generally be proportional to 1/ L (Keeling, 1999). However, if a model contains coherent patch structures, there will be deviations from the 1/ L line predicted for a stochastic process. If the patches are long-lived structures with respect to the time series, then standard deviations taken from windows smaller or equal to the patch scale will be less variable than expected. The expected scaling behavior is seen in time series drawn from a probabilistic version of the point model (transitions occur to populations of L ¥ L cells with probabilities drawn from and window size was the same in probabilistic point and spatial models (ANCOVA, p no difference between slopes > 0.5). Hence there is no evidence that patch structures are formed at any scale in the spatial model. 15.4 Extending the Spatial CA Framework The derivation of a spatial matrix model demonstrated that the local density of limpets affected the transitions between states on the shore. However, the empirically derived CA failed to generate spatial FIGURE 15.2 Comparison of observed and predicted cell state frequencies in 25 m 2 sampling quadrats. Observed frequencies are the average of separate annual samples. Predicted frequencies are from point or spatial transition matrix models. 0 200 400 600 800 1000 1200 1400 1600 Observed Point Spatial 0 100 200 300 400 500 600 State frequency (0.01 m 2 cells occupied) 0 200 400 600 800 1000 Barnacles + Barnacles - Juveniles + Juveniles - Mature + Mature - Coralline + Coralline - Rock + Rock - Site a Site b Site c © 2004 by CRC Press LLC the nonspatial matrix for site a; Figure 15.3). The relationship between standard deviation of time series Challenges in the Analysis and Simulation of Benthic Community Patterns 237 pattern or improve model predictions of state frequencies when compared to a nonspatial model. Wootton (2001a) in a study of intertidal mussel beds also found that empirically derived CA simulations with local interactions (but without locally propagated disturbances) did not produce patterning. The absence of spatial pattern in the CA models may reßect that spatial structures are sensitive to the stochastic nature of transitions between states (Rohani et al., 1997). The spatial transition rules for the Fucus mosaic and mussel bed were deÞned from Þeld data. This implies that it is not possible to scale up from observations at small scales to patterns at large scales. There are, however, two reasons this conclusion may be premature. It may be that the CA framework is too crude a method to characterize the local interactions in the intertidal. The CA models also did not include “historical” effects, despite the TABLE 15.2 Comparisons between the Observed Frequencies of Different States, the Predictions of Point and Spatial Models, and Community Frequencies Generated Randomly Observed Point Model Spatial Model Site a Point model 2005.41 Spatial model 1883.29 22.17 Random 3533.15 6120.27 5907.50 Site b Point model 753.58 Spatial model 784.86 39.36 Random 6350.90 5576.49 4895.45 Site c Point model 42.01 Spatial model 56.41 4.77 Random 3166.67 3826.65 3766.11 Note: G tests (Sokal and Rohlf, 1995) are used as measures of goodness of Þt (Wootton, 2001b). Scores for the random model communities are averages of 250 independently generated tests. Lower G test values imply a better match between the frequencies being compared. Numbers in bold indicate signiÞcant differences between the frequencies being compared. FIGURE 15.3 Standard deviation of mature algal frequencies in time series collected at different spatial scales. Observation window length scales range from 4 to 256 cells. The common slope is a statistically signiÞcant regression passing through the origin. 1/(observation window length scale) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Standard deviation of time series for frequency of mature algal state 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Stochastic spatial model Stochastic nonspatial model Common slope © 2004 by CRC Press LLC 238 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation observation that history can intensify local interactions in probabilistic CA, leading to pattern formation (Hendry and McGlade, 1995). In the context of Markov transition matrices, historical effects are modiÞcations to the transition probabilities based on the state of the system at lags exceeding one time step (models include second and higher order processes, Tanner et al., 1996). Hence the age of particular states can affect their transition probabilities. For example, not all mussel beds are equivalent. Waves are more likely to remove old, multilayered beds (Wootton, 2001a). In the Fucus mosaic, patches of algae persist for 5 years before they break down (Southward, 1956). Tanner et al. (1996) demonstrated that historical effects could be detected in coral communities, although these effects did not affect overall community composition in comparison to Þrst-order models. Incorporating a more sophisticated representation of local grazing interactions and historical effects into CA simulations requires a framework variously known as mobile cellular automata, lattice gas model, or artiÞcial ecology (Ermentrout and Edelstein-Keshet, 1993; Keeling, 1999). Time, space, and state are still discrete, but the artiÞcial ecology formulation allows simulated organisms to move around the grid. This is a more ßexible method of representing aggregations of mobile organisms than a conventional CA. An artiÞcial ecology for the Fucus patch mosaic can be based on the spatial effect of individual limpets on the probability that new patches of algae will be formed. This relationship can be deÞned from maps of limpet and algal location. The maps previously used for transition matrices have a minimum spatial scale below the average distance that limpets forage from their semipermanent home scar (0.4 m; Hartnoll and Wright, 1977). Hence the grazing effects should extend over several 0.01 m 2 cells. Stepwise logistic regression using increasing distances from the target cell was used to deÞne the strength and the range of limpet effects on the probability of a cell containing juvenile algae (Johnson et al., 1997). This information was then used to simulate the Fucus mosaic in a 50 ¥ 50 cell grid, equivalent to the scale of the maps made in the Þeld. Each time step the distribution of limpets deÞned the probability of juvenile Fucus establishing in any unoccupied cell on the grid. As on the shore, limpets potentially relocated to new home scars each year, creating a dynamic pattern of grazing pressure. Simulations of this artiÞcial ecology created realistic mosaic patterns (Figure 15.4). A fuller description of the model, including investigation of the roles of limpet movement and habitat preferences is given in Johnson et al. (1998b). An advantage of the empirically deÞned rules for the artiÞcial ecology is that the scales in the simulations are clearly deÞned. This facilitates more demanding confrontations with data than is possible with more generic spatial models. For example, the spatial autocorrelation produced in simulations gives FIGURE 15.4 Screen grab of simulation output from the artiÞcial ecology of the limpet–Fucus mosaic. The spatial plots show (A) Fucus distribution (white–empty, gray–juvenile, black–mature) and (B) limpet occupancy (white–empty, gray–one limpet, black–more than one limpet). The time series (500 time steps) of algal cover (C) shows records from the patch scale (black line) and the grid scale (gray line). © 2004 by CRC Press LLC [...]... the predictions of the different methods may provide a fuller understanding of any community than application of a single approach © 20 04 by CRC Press LLC 240 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Markov transition matrices appear to produce reasonable Þrst approximations of community composition This may reßect the relatively open nature of many benthic... through the habitat: it provides increased possibilities © 20 04 by CRC Press LLC 2 54 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation of attachment, shade, and hiding places Quantitative measurement of structural complexity greatly enhances the possibilities of attributing patterns of epiphytal or epilithic assemblage composition to features of that complexity (e.g., Davenport... power function line © 20 04 by CRC Press LLC 262 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation 1E + 0 1E – 2 1E – 4 1E – 6 1 10 100 1000 10000 A 10 y = 20.2x –1.67 1 R 2 = 0.9998 0.1 0.01 0.001 1 10 100 1000 Rank B FIGURE 17 .4 The addition of noise to a power law on log–log plots (A) The thin diagonal line is a power law (x–1.333) Each thick black line emerging to the... this binning and that they effectively © 20 04 by CRC Press LLC 2 64 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Fluorescence 0 0.1 0.2 0.3 0 Depth (db) 1000 2000 3000 40 00 A 0.03 0.05 0.07 3200 Depth (db) 3300 340 0 3500 B FIGURE 17.7 A fluorescence profile from the Bermuda Atlantic Time Series (BATS) (A) The entire upper 40 0 m of the profile The right brace indicates... measured by the “walking dividers” method and construction of a Richardson plot (it could equally have been determined by the boundary-grid technique) The dividers were walked with alternation of the swing (i.e., clockwise then anticlockwise © 20 04 by CRC Press LLC 252 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation rotation) to avoid bias A total of Þve replicate perimeter... chap 4 © 20 04 by CRC Press LLC 242 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Koenig, W.D., Spatial autocorrelation of ecological phenomena, Trends Ecol Evol., 14, 22, 1999 Kostylev, V and Erlandson, J., A fractal approach for detecting spatial hierarchy and structure on mussel beds, J Mar Biol., 139, 49 7, 2001 Law, R., Herben, T., and Dieckmann, U., Non-manipulative... use of methods, including fast Fourier transformation, to provide power spectra The implicit assumption of a Gaussian distribution is rarely tested and unless one is extremely fortunate, the data set usually requires some massaging (e.g., despiking and detrending) before the analysis can be performed In addition, exactly 257 © 20 04 by CRC Press LLC 258 Handbook of Scaling Methods in Aquatic Ecology: Measurement,. .. diversity of animal community,22 and/or greater relative abundance of smaller animals.8,20–23 The utility of such studies is discussed in more detail later © 20 04 by CRC Press LLC 250 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation TABLE 16.1 Fractal Dimensions (D) of Perimeters of Images of Four Macroalgae from Sub-Antarctic South Georgia Measured over Various Scales Step... electron microscopy) so that cross-frond D could be estimated Two-dimensional images for estimate of perimeter D were obtained from each plant (or part of plant) by combinations of direct photocopying of plant material (using both enlarging and shrinking as appropriate), microscope/camera lucida drawings of plant pieces or projected 35 mm slides in the case of whole /part Macrocystis plants or whole... estimates of competition coefÞcients in a montane grassland community, J Ecol., 85, 505, 1997 Levin, S.A., The problem of pattern and scale in ecology, Ecology, 73, 1 943 , 1992 Levin, S.A and Pacala, S.W., Theories of simpliÞcation and scaling of spatially distributed processes, in Spatial Ecology, Tilman, D and Kareiva, P., Eds., Princeton University Press, Princeton, NJ, 1997, chap 12 Lindegarth, . 3 123 L L L MMMMM L © 20 04 by CRC Press LLC 2 34 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation (15.2) where (15.3) n jk = number of transitions from state k to j in the original. important, the patches cycle independently of Fucus abundance at the large scale (Figure 15.2C). In 240 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Markov transition. Oxford, U.K., 1999, chap. 4. © 20 04 by CRC Press LLC 242 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Koenig, W.D., Spatial autocorrelation of ecological phenomena,