Notice: Wiley Online Library will be unavailable on Saturday 27th February from 09:00-14:00 GMT / 04:00-09:00 EST / 17:00-22:00 SGT for essential maintenance. Apologies for the inconvenience.
1 Ecological patterns in space and time have been well documented using interval variables such as plant density. In this paper, we use join Page: 1Is the text OK: join count statistics to examine spatio-temporal patterns in a binary variable, plant establishment.
2 Establishment along a belt transect is represented as a grid or lattice in which each cell denotes a particular quadrat in a particular year. A cell is ‘black’ if a plant established at that particular time and place; otherwise it is ‘white’.
3 Each pair of black cells is connected by a ‘join’ that represents the association between stems established at two different places and times. Join ‘length’ is the pair of factors (s,t) that describes the distance between stems in space and in time, respectively. Spatio-temporal pattern is detected by comparing the number of joins of each ‘length’ in a particular lattice with the expected number calculated from lattices generated by three different random models. Colonization rates can be estimated if there are particular join classes that occur more frequently than expected.
4 Artificially generated lattices were used to examine the effect of background noise on the ability to detect underlying pattern.
5 Field data were obtained from a Populus balsamifera clone that was colonizing a grassland. Three belt transects, 1 m wide and up to 13 m in length, were established, and ramets in each 0.5-m interval were aged by tree ring counts. Position and age were used to construct a lattice of stem establishment.
6 We were able to discern a pattern both in artificial lattices and in the colonization of P. balsamifera ramets. Ramets took between 1 and 2 years to advance 1 m into the grassland (equivalent to a rate of 0.5–1 m per year).
7 Describing plant establishment in space and time with two-factor join count statistics provides a way of measuring the rate of species movement at the scale of the individual.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
It has long been evident that in ecological systems spatial patterns change in time, often in predictable ways ( Watt 1947; Andrewartha & Birch 1954; Hanski 1996). Spatio-temporal pattern describes the tendency for events separated in space and time to be associated in a non-random fashion (either positively or negatively). For example, consider a string of light bulbs where the blinking of one light is immediately followed by blinking of the bulb to its immediate left. For any bulb at any time, blinking is not random, but associated with the blinking of a neighbouring bulb.
In plant ecology, the concept of spatio-temporal pattern goes back at least to Watt (1947), who recognized that certain vegetation types tend to occur close to each other in either space or time. More recently, Herben et al. (1995) examined species’ abundance across permanent plots over a period of 9 years and showed that predictable changes in species’ densities at different places and times can be related to the form of clonal growth.
Clonal growth forms are often described in terms of the spatial pattern of ramets ( Lovett Doust 1981), but under certain circumstances it may be better to use stem establishment in space and time. Of the possible growth strategies identified for clonal plants (phalanx and guerrilla), the phalanx type, which is characterized by the compact spatial structure of ramets ( Cheplick 1997), would display highly predictable stem establishment in space and time, as stems either advanced to increase the size of the patch or remained static to defend it ( Slade & Hutchings 1987). The guerrilla form, characterized by more loosely arranged ramets, could establish either unpredictably, as the military metaphor suggests, or in a predictable manner like the previously mentioned blinking lights. This growth form therefore might be better described by considering stem establishment. If the phalanx and guerrilla growth forms are thought of as the endpoints of a continuum ( Cheplick 1997), more general growth forms can be described on the basis of the spatial arrangement of ramets and predictability of ramet establishment in space and time (cf. Oborny 1994).
Methods used to elucidate spatio-temporal pattern have been developed from single-factor autocorrelation analysis to consider factors separated by two lags, i.e. intervals along axes of space and time ( Herben et al. 1995 ). One such method, Griffith’s Space–Time Index, is related to Moran’s I, but examines the value of a variable at different places and times ( Henebry 1995). This index is appropriate in the analysis of interval variables such as plant density, but if a binary variable such as presence/absence of individuals is considered then join count statistics ( Cliff & Ord 1981) must be used. Join count statistics compares the number of times that individuals are separated by a particular lag, i.e. distance in space and time, with that expected from randomness ( Sokal & Oden 1978).
Recently, two-factor join count statistics has been used to detect anisotropy in ecological patterns in two spatial dimensions ( Proctor 1984; Gray et al. 1986 ; Dale 1995). To our knowledge, such an approach has not been used in plant ecology to analyse for spatio-temporal patterns, but could prove informative because it describes patterns of where and when individual establishment occurs. In this paper, we focus on the establishment of individuals in space and time and analyse both field-collected and artificially generated data. Analysis of artificial spatio-temporal arrangements allowed us to determine the extent to which non-random pattern can be distinguished in the presence of background noise.
The field site was at the interface of a forested area and grassland, where we observed that ramets of the clonal species Populus balsamifera L. decreased in size with increasing distance from the forest. This site thus showed an advancing front of establishing ramets. As well as showing clonal movement along a transect, the data also allowed us to determine the rate of advance in the front of ramet establishment. In this paper we aim to show that spatio-temporal pattern in the dynamics of plant establishment can be related to clonal growth strategy.
Materials and methods
Join count statistics is a way of measuring autocorrelation in nominal data distributed across a lattice or grid ( Cliff & Ord 1981). Here we deal specifically with binary data, where each cell of a lattice takes one of two values, black or white. A join is defined as a connection of particular length between pairs of predefined cell types (e.g. black–black), and for single-factor joins length is measured as the number of cells traversed by either rook’s, bishop’s or queen’s moves ( Sokal & Oden 1978). The extent of autocorrelation among cells is then determined by comparing the number of joins in an observed data set to the number expected from randomness. On a two-dimensional lattice representing a spatial area, if the observed number of black–black joins of length class d is greater than expected, then black cells are said to be aggregated or positively autocorrelated at distance d. Where the observed number is less than expected, black cells are said to be segregated or negatively autocorrelated at distance d.
In a spatio-temporal approach, however, a two-dimensional lattice represents a single spatial dimension (e.g. a transect of n quadrats) at m intervals of time, i.e. a time sequence for a one-dimensional transect, with each cell specifying a particular quadrat along it at a particular time. We chose to let a black cell signify the point of plant establishment rather than the longer term variable presence: the occurrence of several differently aged plants in the same quadrat can then be distinguished by placing multiple black cells in that column of the lattice. Join lengths are specified by a combination of two factors (s,t), each pertaining to an interval along one of the axes, so that join class (2,3) would signify joins of length 2 units along the spatial axis and 3 units along the time axis. Each pair of black cells is connected by a join of a particular class, and spatio-temporal association among establishment events is examined by comparing the observed number in a class to the number expected from randomness. Autocorrelation statistics (the deviation of the observed number of joins from that expected) can be calculated for each join class and plotted on the third axis of a correlogram, in which the first two axes represent the two components of the join class. If more joins of a particular class (s,t) occur than are randomly expected, then this indicates a tendency for stems to become separated by a distance of s spatial intervals in t time units, thus giving a measure of colonization rate.
As a further refinement, we were interested in only the first stem that establishes in each quadrat, and these initial colonizers are represented by a lattice with no more than one black cell in each column. Therefore, in addition to the ‘fully randomized cell model’ where the probability of any cell being black is p, and more than one black cell may occupy a column, two further models were used in which black cells are randomly arranged so that no more than one can occur in a column. In the ‘uniform row model’, a quadrat has probability r of being colonized, i.e. the probability that each of the n columns contains a single black cell is r, and each black cell in a column is distributed uniformly among the rows. Alternatively, in the ‘top-black model’, black cells may occur in any cell on the lattice with probability p, but only those in the highest row of each column (i.e. the earliest establisher) are retained.
For each join class (s,t) we generated 10 000 random lattices using each of the three random models (i.e. fully randomized cell, uniform row or top-black). These were used to calculate a frequency distribution, f (Qst), and expectation, E[Qst] =ΣQstf (Qst)/10 000, where Qst is the number of joins, Q, in join class (s,t). We preferred this approach to obtaining a frequency distribution for all join classes from the same 10 000 random lattices because join classes are not independent of each other.
The autocorrelation statistic of each join class is based on the deviation of observed from expected join frequencies. Deviations are calculated by allocating each observed count to a tail of f(Qst) on either side of E[Qst]. This is assigned a percentile value based on the number of randomly generated lattices remaining in the tail (equivalent to a one-sided hypothesis test) and is expressed as a P-value. Two-factor correlograms displays the results by overlaying circles, whose size represents the P-value, on an array representing join classes.
Generation of artificial data
Four artificial 15 × 15 lattices were created to represent random and patterned spatio-temporal arrangements. Lattice A is a random lattice created by assigning black to each cell with a probability P = 0.2. Lattice B is analogous to a string of blinking lights, in which there is a black cell at the top left corner and subsequent black cells occur at an interval of s = 1, t = 1 from the previous one. Two types of noise were imposed on the pattern in lattice B. In C, we added random background noise by allowing other cells to become black with a probability of 10% ( Fig. 1). In D, each black cell in pattern B was shifted by a time interval determined by the integer component of the Gaussian random variable, N(0, 0.5) ( Fig. 1).
Populus balsamifera encroachment data
The Ministik Bird Sanctuary south-east of Edmonton, Alberta, Canada, contains forest patches, generally greater than 1 ha, set amongst mowed grassland. Patches consist primarily of white spruce (Picea glauca) and balsam poplar (P. balsamifera), and when mowing at the edges ceases trees may colonize the grassland area. We selected an area where P. balsamifera was colonizing, and counted the stems in 0.5-m quadrats along three 1-m wide belt transects that extended no further than 13 m from the patch edge. Stems were cut, labelled and then aged by ring counts in the laboratory. The data were represented as three spatio-temporal lattices ( Fig. 2).
For analyses where comparison was made with the uniform row and top-black models, the observed spatio-temporal lattices were manipulated by deleting all occupied cells in a column (i.e. within a quadrat) except the one representing the first to occur (i.e. the top-most black cell). We used the proportion of black cells on the observed lattice to obtain an estimate of p, the probability that black cells occur at random, for use in the fully randomized cell and top-black models. For the uniform row model, we used the number of occupied quadrats on the observed lattice to estimate r, the probability of a quadrat being occupied at random.
Analysis of the random, artificial lattice A, using the fully randomized cell model, showed that no join class had more than a small deviation from expectation (data not shown). Similar results were obtained when only the topmost black cell was left in each column of lattice A and the resulting pattern was analysed with either the uniform row or top-black models. Analysis of the regular artificial lattice B, as expected, showed a large excess of joins in class (1,1) ( Fig. 3). Large resonance deviations occurred at multiples of this class, e.g. (2,2) (3,3). Similar results were obtained when the uniform row or top-black models were used.
When the regular pattern was obscured by the addition of 10% background noise (as in lattice C; Fig. 1), the pattern was still detectable using the fully randomized cell model ( Fig. 3), but the P-values of join class (1,1) and subsequent resonance peaks were smaller than without background noise ( Fig. 3). When the pattern was disrupted to produce lattice D ( Fig. 1), the fully randomized cell model still showed a large positive deviation at the original pattern (1,1), again with smaller resonance peaks at multiples of the pattern ( Fig. 4a). Similar results were found using the uniform row and top-black models, but the deviations were larger ( Fig. 4b,c).
Analysis of the three transects of P. balsamifera using the three random models is shown in Fig. 5. In the first transect, analysis using the fully randomized cell model showed significant positive deviations at join classes (1.5, 2) (2.0, 2) and (2.0, 3), and almost so at (1.5, 3) ( Fig. 5). A peak also occurred at (3.5, 5). We interpreted the further large peak at (3.5, 5) as a resonance peak because it was exactly twice (1.75, 2.5) the value of a point centred in the middle of the group of four smaller peaks. Analyses using the uniform row and top-black models showed a peak at (2.5,6) ( Fig. 5), but apart from a peak at (1.5, 2) in the uniform row model none of the peaks seen in the fully randomized cell model was detected.
A slightly different pattern was evident in the second transect. Analysis using the fully randomized cell model gave positive deviations at (0.5, 1) (0.5, 3) (2.0, 4) and (2.5, 5) ( Fig. 5). Using the other two models, however, the largest positive deviation occurred at (1.0, 2) ( Fig. 5), with peaks at (2.0, 4) and (2.5, 4) in the uniform row model. As in the first transect, these may have been resonance peaks, supporting the conclusion that a pattern occurs at a smaller scale, in this case at (1.0, 2).
Analysis of the third transect, with the fully randomized cell model, showed large positive deviations for all joins for which t = 1 or 2 ( Fig. 5). This difficulty of interpretation was the primary reason that led us to develop the other two models. Analyses using either of these showed a large peak at around (1.0, 1) and another at (3.0, 2) ( Fig. 5).
Our method of examining the patterns between stems established at different times and places allowed us to estimate colonization rates in a P. balsamifera clone. Values of approximately 1.75 m in 2.5 years were determined for transect 1 compared with 1 m in 2 years for transect 2, and 1 m in 1 years for transect 3. Another method of measuring colonization rate would be to fit a line of best fit for a lattice and obtain the rate from its slope. This method, however, would not allow us to determine whether establishment was random, nor would it allow accurate characterization of a more complex but biologically realistic pattern such as that which we now discuss.
Consider a situation where the vegetation consists of multiple patches that grow and eventually coalesce ( Greig-Smith 1964; Yarranton & Morrison 1974; Dale & MacIsaac 1989). A hypothetical transect through such a community could give a spatio-temporal lattice as shown in Fig. 6. The major difference between this lattice and lattices like the regular lattice B is not the rate of establishment, which is still one quadrat in one time lag, but the spatial pattern in the initial time period. In Fig. 6 there is a regular pattern once every eight quadrats at the initial time period, which is not present in any of the variants of lattice B ( Fig. 1). Such initial spatial structure could be incorporated into our method if horizontal joins (i.e. join classes where t = 0) were also included. Horizontal joins represent spatial associations between stems of the same age, and their inclusion in analyses would show any spatial pattern in either initial or subsequent time periods. Inspection of our P. balsamifera data showed no such spatial structure and horizontal join classes were therefore excluded from the analyses presented.
Most reproduction in P. balsamifera is vegetative, involving root stems or suckers, and at the field site stem age decreased away from the centre of the clone, as documented visually for other Populus species by Barnes (1966). Based on an association between above-ground plant patterns and root patterns in the upper soil layer ( Pecháccaron;kováet al. 1999 ), there are two possible explanations for this depending on whether initial establishment occurs from new root growth or from roots that are already present. If lateral roots of the clone were present in the grassland prior to the last mowing, then the pattern of stem establishment times along transects could be caused by a reduction in photosynthate translocated through the root system as distance from the clone centre increases, allowing closer stems to establish first. Although some evidence suggests that distance from parent tree is not a regulating factor, the relation between sucker development and the parent root system is not well understood ( Peterson & Peterson 1992).
Alternatively, ramet establishment may follow root growth. Clones are presumed to produce adventitious roots in the grassland. Sucker development, which is stimulated by higher soil temperatures ( Peterson & Peterson 1992), may occur progressively from adventitious roots as they reach the warmer conditions under the surrounding grassland.
Analysis of the clonal growth form of P. balsamifera showed that ramet establishment was highly predictable in both space and time. High spatio-temporal predictability is expected where there is little uncertainty as to the location of resources. For this species, light, which is an important limiting resource ( Peterson & Peterson 1992), is abundant in the grassland. The postfire expansion of P. balsamifera ( Brodie et al. 1995 ) and the beachward advance of sand dune grasses such as Ammophila breviligulata ( Maun 1985) are similar to the clonal expansion reported here.
Time of establishment was measured from tree rings rather than a long-term observational study. The results from the field data should therefore be interpreted with caution because we assumed that the oldest stem observed in a quadrat was the first to colonize, and that no mortality has occurred. This assumption would be violated if an initial colonizer was replaced by a late-coming, superior competitor.
Two-factor join count autocorrelation analysis discerned the pattern in all the non-random artificial lattices tested, although this ability diminished as background noise was added or the pattern was jumbled. Resonance peaks, which occurred at multiples of the underlying pattern, were used to confirm the character of pattern, particularly in the field data. Interpretation of spatio-temporal pattern depended on the model used for comparison because each model produced a different random lattice. Overall, the uniform row and top-black models were best for distinguishing pattern in noisy spatio-temporal arrangements, partly because analysis using these models gave larger deviations than the fully randomized cell model. Another disadvantage of the fully randomized cell model is that it can result in such a large number of significant deviates that any interpretation is obscured. Such a problem is encountered in transect 3 ( Fig. 5a) where there is a cluster of 2- and 3-year-old stems in quadrats 12–20 ( Fig. 2c). Such clusters give more joins in classes with t = 1 than are expected by the fully randomized cell model. The number of significant deviates tends to be much lower in the uniform row and top-black models because we removed the cluster and operated only on the initial colonizer of each column. Although the two latter models produced slightly different results, both helped interpretation.
It is well known that observational scale, i.e. extent and grain size ( Wiens 1989), affects pattern interpretation. The extent of a spatio-temporal investigation depends on both transect length and the time that has elapsed since the initiating disturbance event, and is therefore determined by the size of the spatio-temporal lattice. Grain size, however, depends on quadrat size, quadrat spacing and the time interval between observations. When relatively large spatial and temporal scales are examined, interval variables, like plant density, might be appropriate and analysis could be performed using Griffith’s Space–Time Index (cf. Henebry 1995). At smaller scales, such as that of an individual plant, two-factor join count statistics can, however, prove more useful. At these scales, we believe that the spatio-temporal pattern of stem establishment provides a good descriptor of plant behaviour, and that a better understanding of plant stem interactions could result from considering patterns of establishment and mortality in both space and time.
The Natural Sciences and Engineering Research Council of Canada gave support in the form of a postgraduate scholarship to L. R. Little. We also gratefully acknowledge funds from the Canadian Circumpolar Institute in the form of a C/Bar Grant, and permission from Alberta Environmental Protection, Public Lands Division to work in the Ministik Bird Sanctuary. L. Haddon, E. Pharo and colleagues provided many helpful comments.
Received 3 March 1998revision accepted 3 December 1998