Seagrass Macrofaunal Abundance Shows Both Multifractality and Scale- Invariant Patchiness

$Seagrass Macrofaunal Abundance Shows Both Multifractality and Scale- Invariant Patchiness$

Seagrass macrofaunal abundance shows both multifractality and scale- invariant patchiness.

Accepted MS of Marine Environmental Research 138: 84-95

R.S.K. Barnesa,b,* and H. Lauriec

aSchool of Biological Sciences and Centre for Marine Science, University of

Queensland, Brisbane 4072, Queensland, Australia

bBiodiversity Program, Queensland Museum, Brisbane 4101, Queensland, Australia

cDepartment of Mathematics and Applied Mathematics, University of Cape Town, Cape

Town, Western Cape 7701, Republic of South Africa

*Corresponding author's email: [email protected]

Richard Barnes http://orcid.org/0000-0002-5773-5994

Henri Laurie http://orcid.org/0000-0002-1100-0861

ABSTRACT

Spatial patterns of abundance of the whole macrobenthic assemblage and of its 10 most numerous species were examined across hierarchically nested scales within a 0.85 ha area of intertidal seagrass in subtropical Moreton Bay, Queensland. Multifractality characterised the assemblage and all ten dominant species across those scales (c. 33, 130, 530 & 2115 m2), with patchiness of assemblage numbers and those of at least some dominants exhibiting scale-invariance. The system displayed several abundance peaks, 12% of stations accounting for 88% of total variance, with many individual dominants showing a series of non-overlapping 'hot-spots'. Scale invariance and multifractality occurred notwithstanding low levels of species interaction consequent on maintenance at very low density. This suggests that critical self-organisation cannot be responsible for such patterning. Contrary to received wisdom, coefficient β of Taylor's power-law cannot form an index of aggregation, although it does indicate direction of change in dispersion pattern with changing numbers.

Keywords: abundance, macrobenthos, Moreton Bay, multifractality, patchiness, seagrass, self-similarity, spatial scale, Taylor's index of aggregation

1. INTRODUCTION

Natural habitats are mosaics of patches (MacArthur, 1971; Wiens, 1989; Grünbaum,

2012) and seagrass beds are no exception (Duffy, 2006). Indeed, their botanical simplicity, ease of access and sampling, and, often, markedly contrasting faunal ecology to that of adjacent areas of unvegetated sediment have rendered them an ideal system for the study of the effects of such habitat patchiness on the associated macrobenthic fauna (Eggleston et al.,

1998; Healey and Hovel, 2004; Maciá and Robinson, 2005; McCloskey and Unsworth, 2015;

Henderson et al., 2017). Changes in the biodiversity and abundance of seagrass macrofaunal assemblages across patch systems with differing environmental characteristics are commonplace (e.g. Barnes and Ellwood, 2012a; Barnes, 2017a; Magni et al., 2017), but patchiness of species richness, or indeed its absence (Barnes, 2016; Boyé et al., 2017), also occurs in areas where systematic environmental variation appears to be lacking. This can be more instructive in aiding understanding of the causes and drivers of benthic biodiversity patterns (Chang and Marshall, 2016). However induced, such heterogeneous distribution of individual species is of critical macro-ecological importance, being for example the primary cause of the rate of distance-decay phenomena (species turnover inducing a loss of assemblage similarity over distance), whilst actual levels of assemblage similarity are primarily influenced by relative species abundances (Anderson et al., 2005; Morlon et al.,

2008).

Faunal patchiness, like virtually all ecological phenomena in marine soft-sediment habitats, is well known to be affected by spatial scale (Morrisey et al., 1992; Underwood and

Chapman, 1996; Kraan et al., 2009). Therefore the processes likely to be involved in creating patchiness cannot be understood without an appreciation of how it varies across the scales concerned (Underwood et al., 2000). Variation in seagrass assemblage composition 4 across space has been investigated in these terms, including that of the benthic macrofauna

(Omena and Creed 2004; Barnes and Barnes, 2011; Barnes and Ellwood, 2012b; Magni et al., 2017). How such compositional variation then affects emergent assemblage properties, including overall abundance, however, remains unclear, and measurement of variability in organismal abundance followed by elucidation of the underlying causes has been indentified as a major goal in ecology (Denny et al., 2004; Dal Bello et al., 2015).

The effect of changing spatial scale on the patterns in which intertidal soft-sediment faunal abundance is dispersed, whether of individual species or of whole assemblages, has also received some attention, with seemingly rather variable results. Thus Barnes and

Barnes (2011) and Chertoprood and Azovsky (2013) found that the dominant individual benthic species in Queensland Zostera (Zosterella) capricornia beds and in White Sea soft sediments, respectively, showed random dispersion across relatively small spatial extents (or when at low population density) and only occurred patchily when assessed across larger sampling areas (or when present at high densities). In marked contrast, however, the ecologically equivalent species at relatively low densities across seemingly similar Z.

(Zosterella) capensis beds in South Africa each showed patchy distribution across all spatial extents investigated (1 m - >1 km) (Barnes, 2010, and see Schabenberger and Gotway,

2004). Very little work has been devoted to whether such patches of different species coincide in space or whether they are complementary, although relevant data have been collected at some sites since the 1950s (Bassindale and Clark, 1960; Edwards et al., 1992).

In some instances total macrofaunal abundance per unit area has been shown not to vary significantly across local space (e.g. Barnes, 2013; Lefcheck et al., 2016), suggesting that individual species patches tended to cancel each other out. In other cases, however, marked spatial heterogeneity in overall macrofaunal abundance has also been observed in seagrass beds (as well as in adjacent areas of bare sand), notwithstanding that homogeneous species 5 density and diversity characterised the same systems (Barnes, 2014, 2017b). Where no significant differences occur across space in the number of assemblage individuals in unit sample (Barnes, 2013), this, by definition, will result in scale-invariant area-abundance relationships. But is this also the case for animal numbers in spatially heterogeneous seagrass systems?

In this study we focus on a range of relatively small spatial scales within which some populations of soft-sediment macrobenthic species have been demonstrated individually to show fractal-like patterning, i.e. within 1 ha (Azovsky et al., 2000; Warwick et al., 2006).

As is emphasised below, although (mono)fractal analysis may be appropriate for the study of two dimensional systems (e.g. presence/absence data), it is not so for systems exhibiting greater complexity, in which a single fractal dimension is insufficient to describe their dynamics; for example those involving spatial variation in abundance as well as in occurrence. These require multifractal analyses (Harte, 2001). In the present context, it needs stressing that in the ecological literature, the main characteristic of (mono)fractal systems usually emphasised is their spatial self-similarity; multifractality, however, implies no such scale invariance.

In an attempt to understand the manner in which per-unit-area abundance of a whole macrofaunal seagrass assemblage is patterned spatially, and to investigate whether the degree of patchiness itself varies across scales, the present research was therefore conducted via the use of indices of patchiness and, for the first time on a macrobenthic faunal assemblage, multifractal analyses. These were undertaken within a single 0.85 ha seagrass bed of homogeneous visual appearance (i.e. lacking unvegetated patches or areas differing in apparent percentage cover) and without obvious environmental gradients. The bed concerned, however, was one already known to support significantly patchy distributions of 6 individual dominant faunal taxa and patchy overall per-unit-area animal abundance (Barnes,

2014; Barnes and Hamylton, 2015).

2. METHODS

2.1 Study area and sampling protocol

Macrofaunal sampling was conducted over a period of 9 weeks during the 2017 austral spring at a site at the southern end of the Rainbow Channel coast of North Stradbroke Island

(aka Minjerribah) within the relatively pristine Eastern Banks region of the oligohaline, mesotidal, subtropical Moreton Bay Marine Park, Queensland (Dennison and Abal, 1999;

Gibbes et al., 2014). The lower portion of the intertidal zone of this coast supports seagrass beds dominated by dwarf-eelgrass, Zostera (Zosterella) capricorni [Nanozostera capricorni in the recent revision of the Zosteraceae by Coyer et al., 2013] but also with some Halophila ovalis and Halodule uninervis (Young and Kirkman, 1975; Abal et al., 1998). The precise site investigated, centred on 27°30'26"S,153°24'30"E in Deanbilla Bay, was a c. 125 x 200 m block of seagrass occurring from some MLWN down to an unvegetated LWS sand bar parallel to the shoreline, and partly separated from the beds to northwest and southeast by areas of bare sandflat except for vegetated corridors c. 50 m wide (Fig. 1). Typically in such conditions, the dwarf-eelgrass plants were of the small morphological forms characteristic of shallow areas (Young and Kirkman, 1975). In consequence, although occurring at the same or higher shoot density as at other local sites (see Barnes and Hamylton, 2013), percentage ground covers were only in the range 40-65% (sensu McKenzie, 2003). The site was that earlier investigated in respect of the spatial ecology of its sandflat/seagrass interfaces

(Barnes and Hamylton, 2013, 2016), biodiversity metrics (Barnes, 2014) and functional- 7 group structure (Barnes and Hamylton, 2015). The downshore slope was a maximum of

0.25 m over a distance of 120 m, across which assemblage abundance per unit area had previously been shown not to vary significantly (Pearson R = 0.01, P >0.9; from the database of Barnes and Hamylton, 2015).

As recommended by Morrisey et al. (1992), Fortin (1994), etc., samples were arranged in a lattice (16 x 16 samples square), permitting a hierarchically nested series of smaller component areas to be analysed. For ease of geospatial referencing, this lattice was oriented at c. 25° off alignment with the long axis of the shore. As most benthic seagrass macrofauna are known to be located within the top few mm of sediment [e.g. 98% in the top 5 mm in the study by Klumpp and Kwak (2005) at other sites in Queensland], individual samples comprising each column and row of the lattice were in the form of a core with a spatial grain of 0.0054 m2 and depth of 100 mm, each core centred on one seagrass plant, and they were unit 5.75 m apart (c. 0.2" of latitude and longitude). This sampling procedure therefore collects the smaller and more numerous members of the macrofauna that constitute the large majority of tropical invertebrate biodiversity (Bouchet et al., 2002; Albano et al., 2011).

Neither meiofauna nor the much scarcer megafauna (here including the occasional

Stichodactyla, Fragum, Pinna and Holothuria) nor sessile animals attached to the seagrass leaves were included, however; since Warwick et al. (2006) have shown that different spatial patterning rules may apply to meiofauna and macrofauna, and likewise Davidson et al. (2004) to sessile versus mobile species. Collection and treatment of core samples followed the same procedure as earlier studies of macrofaunal assemblages associated with dwarf-eelgrass beds both within the North Stradbroke intertidal (e.g. Barnes and Hamylton,

2015; Barnes, 2017a) and elsewhere (e.g. Barnes, 2013, 2016). Trampling adversely impinges on seagrass systems (e.g. Eckrich and Holmquist, 2000) and hence great care was taken to cause minimal disturbance to the site, particularly to areas to be sampled on future 8 occasions. Cores were collected just before tidal ebb when the regions of the bed concerned were still covered by some 5-10 cm of water, and were gently sieved through 710 µm mesh on site. Retained material from each core: (i) was placed in a large polythene bag of seawater within which all seagrass was shaken vigorously to dislodge all but sessile animals and then discarded; (ii) was then re-sieved and transported immediately to a local laboratory, and (iii) was there placed in a 30 x 25 cm white tray on an A3 LED board in which the living fauna was located by visual examination using 3.5x magnifying spectacles until no further animal could be seen during a 3 min search period. The total number of individual animals in each core sample was recorded. To provide information on which, if any, particular species might be particularly responsible for establishing observed faunal patterns, the numbers of each taxon that earlier work at the site (especially Barnes, 2014; Barnes and

Hamylton, 2015) had shown to be a numerically dominant element of the local macrobenthic assemblage were also recorded, i.e. those known previously to have achieved mean densities of ≥33 m-2 and together to have comprised >80% of total individuals. All organismal nomenclature is as listed in the World Register of Marine Species (www.marinespecies.org), accessed December 2017, except, following Ozawa et al. (2009), for the separation of

Velacumantus from Batillaria.

2.2 Indices of patchiness and scale-invariance

Data on the total numbers of animals and on the 10 individual dominant taxa in the 16 x 16 matrix were analysed in terms of their spatial patchiness and for the occurrence of correlations between the abundance patterns of various taxa. Spatial dispersion at all scales was assessed by Lloyd's (1967) index of patchiness [1 + (variance/mean2 - 1/mean)], one of a family of indices advocated by Payne et al. (2005), Rindorf and Lewy (2012), Stanley et al. (2012), etc., with, following Hurlbert (1990), statistical significance of patchiness (= 9 contagion, heterogeneity, aggregation, etc.) determined by Morisita's original (1959, 1962) sample-size independent procedure (rather than by Smith-Gill's (1975) standardised version), using one-sided upper-tail χ2 (Morisita, 1962) [i.e. where χ2 = ((n Σx2)/N)-N, in which n = number of core samples, x = number of individuals in the i-th core, and N = total number of individuals, with n-1 degrees of freedom]. Hoel (1943) demonstrated that tests of significance using χ2 can break down when mean values are <5, as were the numbers of all the individual species at the smallest scale reported here. The problem is essentially one of type II errors (here, the failure to detect patchy distributions), and therefore it would not appear to affect those present conclusions at the level of the individual samples; the

'problem' disappeared at larger spatial scales of analysis and did not affect overall assemblage abundances. Data were also assesssed in terms of Taylor's (1961) power law, s2

= αmβ (where s2 is the variance, m is the mean value, and α and β are constants), of which β has been considered to form an 'index of aggregation'. Correlation analyses used Spearman's

ρ.

2.3 Multifractal analyses

Multifractals are a generalisation of fractals. Intuitively, fractals are shapes that are irregular at all scales, and to which ideas like length of a line do not apply. A classic example of a fractal is the coast of Britain, which as a fractal has no length definable independently of the scale at which it is measured (Mandelbrot, 1967). Similarly, a multifractal is a density to which the concept of an average does not apply, as it does not become uniform at any scale. Hence multifractals do for densities what fractals do for shapes: a multifractal shows contrast at all scales. Multifractals have previously been applied to ecological data (Laurie and Perrier, 2011; Savaria et al., 2012; Savaria, 2014; Dal Bello et al., 2015) and in ecological theory (Yakimov et al., 2014); here for the first time we apply 10 them to macrobenthic faunal assemblages. Prigarin et al. (2013) provide a useful summary of fractal and multifractal concepts and the numerical techniques that may be used to estimate the fractal and multifractal dimensions (note that they use the term “Rényi entropy” for the function ( ) below, more generally, as here, termed Rényi dimensions). Salat et al.

(2017) is a more𝐷𝐷 recent𝑞𝑞 technical review that is particulary helpful for understanding links among various equivalent or nearly equivalent concepts and methods.

A standard way of characterising a multifractal is via its Rényi dimensions

( ) (Laurie and Perrier, 2010, 2011; Prigorin et al., 2013; Salat et al., 2017) which are a generalisation𝐷𝐷 𝑞𝑞 of fractal dimension in that (0) is the fractal dimension the object would have if all non-zero densities were given the𝐷𝐷 value 1. Positive values of progressively emphasize the role of higher densities, so that for large the value of (𝑞𝑞 ) represents

(loosely speaking) the fractal dimension of higher densities.𝑞𝑞 As can be𝐷𝐷 deduced𝑞𝑞 from this, if a density is constant, then ( ) = (0) for all values of . Hence a simple criterion of multifractality is that the plot𝐷𝐷 𝑞𝑞of (𝐷𝐷) versus should not𝑞𝑞 be a horizontal line.

𝐷𝐷 𝑞𝑞 𝑞𝑞 The definition of exact ( ) requires taking limits as , which obviously does not suit real data. An estimate𝐷𝐷 𝑞𝑞( ) ( ) is available by𝑟𝑟 using→ ∞ an approach similar to the estimate of fractal dimension via𝐷𝐷′ fit𝑞𝑞 ting≈ 𝐷𝐷a line𝑞𝑞 to a log-log plot. However, instead of one plots a sum namely versus . For example, in two dimensions one would embed𝑁𝑁 the 𝑁𝑁 𝑞𝑞 𝑖𝑖=1 𝑖𝑖 object in a simple shape∑ like𝜙𝜙 a rectangle.𝑟𝑟 The count ( ) is then the number of squares of area required to cover the rectangle without overlaps,𝑁𝑁 𝑟𝑟 is the proportion of the total 2 𝑖𝑖 density𝑟𝑟 that occurs in the -th square, and the number𝜙𝜙 of squares in the tesselation of the rectangle. As changes,𝑖𝑖 so does the slope𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 of the fitted line, and this yields the estimated

( ). 𝑞𝑞

𝐷𝐷′ 𝑞𝑞 11

To quantify the degree of multifractality, we use the scale-invariant, self-similar multifractal MFp1p2 (Perrier and Laurie, 2010). It has two parameters and , which

1 2 determine density variation between adjacent areas of equal size or 'cells'𝑝𝑝. Its 𝑝𝑝( ) depends only on the parameter = , which determines the strength of the contrast𝐷𝐷 in𝑞𝑞 density

1 2 between those adjacent𝑏𝑏 equisized𝑝𝑝 ⁄𝑝𝑝 areas. Without loss of generality we assume > , so

1 2 that larger corresponds to more pronounced multifractality. When = 1, 𝑝𝑝= 𝑝𝑝so there

1 2 is zero contrast𝑏𝑏 between adjacent areas and hence zero multifractality.𝑏𝑏 𝑝𝑝 𝑝𝑝

To apply the algorithm above, measures of abundance are needed at several scales. We alternately amalgamated pairs of adjacent cells across vertical and horizontal shared boundaries to obtain cells with area = 1, 2, 4, … , 256. For each of the resulting cells the abundance was calculated as the sum of the abundances of the two cells being amalgamated.

Thus all multifractal calculations were done on sets with 511 ( , ) pairs, where is the area of a given cell and its abundance. 𝐴𝐴 𝑁𝑁 𝐴𝐴

𝑁𝑁

3. RESULTS

3.2 Overall abundance and nature of the faunal dominants

Mean number of macrofauna per core sample was 13.8 ± 0.4 SE (Fig. 2) with a range of 0-51, equivalent to an overall faunal abundance of 2555 m-2, with the following 10 taxa displaying overall individual densities ≥50 m-2, and together comprising 72% of the total individuals: the microgastropods Calopia imitata (392 m-2), Pseudoliotia speciosa (165 m-

2), P. micans (52 m-2) and Circulus cinguliferus (61 m-2); tanaid Longiflagrum caeruleus 12

(299 m-2); amphipods Limnoporeia yarrague (200 m-2) and Eriopisella1 sp. nov. (137 m-2); decapod Enigmaplax littoralis (340 m-2); and burrowing polychaetes Malacoceros ?reductus

(114 m-2) and Dasybranchus caducus (74 m-2). The major functional group characterising the assemblage was that of biofilm-grazing microgastropods, with 29% of total numbers.

3.3 Patchiness and its spatial invariance

Macrofaunal abundance showed a number of peaks (Fig. 3A) and was dispersed significantly patchily across the site at all spatial scales (Table 1 and Fig. 4), as were the abundances of almost all individual numerically-dominant species at almost all spatial scales

(i.e. with the exception only of one infaunal polychaete species at the smallest scale and one peracaridan crustacean at the largest scale). Individual species, however, varied considerably in their degree of patchiness, again especially at the smallest spatial scale

(Table 1; Figs 3B-D), from low occupancy and extreme aggregation (e.g. Circulus, Fig. 3B) through to high occupancy and relatively even density (e.g. Enigmaplax, Fig. 3C).

Nevertheless, in all cases, their degree of patchiness appeared quickly to stabilise as spatial scale increased, with that of the overall macrofauna and of all dominants except Circulus exhibiting a very similar degree of patchiness at a scale of 530 m2 to that at 2115 m2 (Fig. 4).

The apparent extreme patchiness at the smallest spatial scales of the microgastropods

Circulus and the two Pseudoliotia species (Fig. 4) is a consequence of their low levels of occupancy at those scales, each occurring in <20% of the 256 cores (c.f. >30% for all other dominants) and in a mean of only 44% in the 64 tetrads (c.f. a mean of 85% for the other dominants). However their degree of patchiness is shown below in fact to be close to scale- invariant.

1This is the first record of the genus Eriopisella from Australia. 13

Across spatial scales, the numbers of all ten dominant species (Table 2) and the of total macrofauna (Fig. 5) conformed to Taylor's power law. In a notion repeated and used hundreds of times in the literature, Taylor (1961) considered the coefficient β of his expression s2 = αmβ to form an 'index of aggregation', a value of β = 1 indicating random dispersion, β = <1 regular (= homogeneous), and β = >1 aggregation. Values of β in Table 2 range from 0.89 to 1.62, two of them being <1 (Circulus and Limnoporeia) whilst that for the whole assemblage is effectively 1 (Fig. 5). Yet, Table 1 shows those same β = ≤1 dispersions to be significantly patchy by Morisita's χ2, and Fig. 4 shows them to have values of Lloyd's index >1, with Circulus, for example, having a variance : mean ratio of at least 12

(at the notional scale of 2115 m2 a variance of 258 and a mean of 21) and the whole assemblage one of at least 3 (at the same scale a variance of 2652 and a mean of 883). This clearly suggests that β cannot function as Taylor suggested. Indeed, as has been pointed out before (e.g. George, 1974), it is obvious that when β = 1, s2 will equal m and hence indicate random dispersion of counts, only if α also = 1. Thus the patchiness of Circulus is consequent on a large value of constant α notwithstanding β being <1. In contrast, the patchiness of Enigmaplax is consequent on a large (>1) value of β, overturning the effect of the small (<1) value of α. What coefficient β does indicate, however, is the general direction of change in dispersion pattern with increasing total numbers (i.e. larger population densities or spatial scales of assessment). When coefficient β is close to 1 (as it is in both

Pseudoliotia spp and in the whole assemblage), s2 is directly proportional to m and hence the ratio s2/αm will be unchanging and scale invariant, not indicative of randomness or of any particular dispersion pattern. Values of β >1 (as here for Enigmaplax, Eriopisella,

Malacoceros and Dasybranchus) will result in s2 increasing faster than m with increase in numbers regardless of the value of α, and therefore indicate change in dispersion pattern 14 towards the patchy end of the spectrum (since few animal populations are regularly dispersed, in practice this equates either to a progression from random to patchy, or to increasing patchiness: a common phenomenon in animal populations (Taylor, 1961) with increase in counts); and values of β <1 will result in decreasing s2 relative to m with increase in numbers, and hence a progression towards the regular end of the continuum (most often in practice resulting in decreasing patchiness, as in the case here in Circulus and Limnoporeia).

Values of α and β in Table 2 are all such that variance > mean for all the species and for the total assemblage at all spatial scales assessed, and there was a strong correlation between the patchiness of a species, as indicated by its mean value of Lloyd's index, and α (ρ = +0.91; P

<0.0003), though unsurprisingly in light of the above not with β (ρ = -0.37; P >0.25).

The frequency distribution of macrofaunal abundance per unit area conforms to an approximately symmetrical lower portion with a number of outliers with exceptionally large densities (Fig. 2). The lower region is not normally distributed (Shapiro-Wilk statistic

0.984, P <0.02) but shows a low flat peak extending from some 8 to 16 ind core-1, having a mean value of 12 (SE 0.2) and comprising 58% of the samples. Some 15% of the samples formed tails symmetrically on each side of this plateau, with the series of outliers, comprising 12% of the total, beginning at some 22 ind core-1 (see Fig. 2). In 71% of these

≥22 ind core-1 cases, the hot spot was caused by unusually high abundances of just one of the locally dominant species although the identity of that dominant varied from core to core but most commonly was Pseudoliotia speciosa and Calopia (each five times), or

Longiflagrum (six); in two cases, two or all three of the species of Pseudoliotia present were co-dominant, whilst in the remaining nine cases no single taxon dominated (see Fig. 6).

Magnitude of the variances in abundance indicates that these hot-spot cores accounted for

88.4% of the total assemblage variance. If a hot spot is defined in terms of the abundance of any one species, rather than of the whole fauna, 23 samples contained a density of one 15 species of ≥10 sample-1 (all numerically dominant species showing such hot spots except the two polychaetes) but in no sample did more than a single such species reach this total, the highest total of any syntopic species was 6 and that only twice. The significantly patchy dominant species in Table 1 together also accounted for 83% of the overall variance, with the four species Pseudoliotia speciosa, Longiflagrum, Calopia and Circulus between them accounting for 59%. The individual densities of the dominants all also correlated significantly with the macrobenthic total, but especially those of (in descending order)

Longiflagrum, P. speciosa, Limnoporeia, Calopia and P. axialis (all ρ >+0.3; P <<0.00001).

The only significant negative correlation between the syntopic densities of the major functional groups present at the site  grazing gastropods, the omnivorous decapod

Enigmaplax, detritus & meiofaunal-feeding peracaridans (Wongkamhaeng et al., 2009;

Macdonald et al., 2010), and deposit-feeding polychaetes  was between Enigmaplax and the polychaetes (ρ = -0.24; P <0.0002); that between the polychaete and the gastropod groups was positive but weak (ρ = +0.13; P <0.04). There were no significant negative correlations within any feeding category, the only significant ones occurring being weak and positive, i.e. between members of the gastropod group, and between the peracaridans (ρ =

+0.13 to +0.18; P <0.04). There were no significant correlations of faunal abundance with either distance down or along the shore (ρ = -0.12 & 0.09 respectively; P >0.05).

3.3 Multifractality

Overall assemblage abundance as well as those of each of the 10 dominant species all showed quite marked multifractality. Figure 7 illustrates four cases: the highly aggregated Circulus, the relatively evenly distributed Enigmaplax, the intermediate

Longiflagrum, and the overall macrofaunal assemblage. The > 0 part of the plot in all

𝑞𝑞 16 cases provides a very good fit between observed and model Rényi dimensions, and this was observed for all 11 fits that were performed (i.e. for the whole assemblage and for each of the ten dominant species).

The values of for the least-squares fit between observed and model ( ) are reported in Table 3. Clearly,𝑏𝑏 the values vary. However, while we are confident𝐷𝐷 that𝑞𝑞 these differences are not mere noise, there is no theoretical basis for calculating their statistical significance. It may be of interest that the overall assemblage abundance appears less strongly multifractal than those of any of the individual species, but we cannot quantify the probability that the difference is significant. Log-log plots of abundance versus cell area are also given in Figure 7, because they furnish the data from which observed ( ) was calculated, but they are interesting in their own right. For example, it is remarkable𝐷𝐷 𝑞𝑞 that

Circulus presents basically the same frequency distribution of counts over a 32-fold range of areal comparisons. This is contrary to what is expected under MFp1p2, but nevertheless the

> 0 fit is good.

𝑞𝑞

4. DISCUSSION

The basic nature of the Deanbilla seagrass fauna appears remarkably homogeneous over time in terms both of species composition and of overall abundance. Out of a known macrobenthic fauna of some 200 genera, the truncatelloids Calopia and Pseudoliotia, spionid Malacoceros, macrophthalmid Enigmaplax, and peracaridans Limnoporeia and

Longiflagrum were six of the seven most abundant animals in 2011 (Barnes and Barnes,

2012), in 2013 (Barnes, 2014) and in 2014 (Barnes and Hamylton, 2015), between them then comprising 61-63% of the 2300-2600 m-2 total individuals present. In 2017, they were also 17 six of the seven most abundant macrofauna, again comprising 62% of a total of 2555 ind m-

2. Both number of species and the proportional importance of each functional group have also been shown to be spatially homogeneous per unit area (Barnes, 2014; Barnes and

Hamylton, 2015). This general uniformity of faunal composition and density renders the area ideal for an analysis of processes structuring spatial patterns. The present study has shown that to that broad uniformity of faunal composition and species density can now be added multifractality and effective spatial invariance of degree of patchiness in abundance both of the whole macrofaunal assemblage and of several of its dominant species, 'effective' because as stressed by Halley et al. (2004) few if any ecological phenomena can ever be truly or exactly scale invariant.

In comparison with other equivalent seagrass beds in other latitudes — i.e. intertidal areas dominated by species of dwarf-eelgrass, Zostera (Zosterella), in sheltered, semi- enclosed though fully saline, conditions — Deanbilla is rich in species but very poor in overall abundance; areas of cool-temperate Z. noltei and warm-temperate Z. capensis support >30 times as many macrofaunal individuals (Barnes and Ellwood, 2011). Some of these other systems have also been reported to show patchy distribution of abundance

(Barnes and Ellwood, 2011; Barnes, 2013), but none has been investigated in detail in that respect. In the Deanbilla Z. capricorni bed, although overall and individual macrofaunal patchiness extended across all spatial scales investigated and across all the major species, patches of individual species rarely coincided: in 77% of cases individual hot spots were created by abnormal concentrations of just a single species or genus, not by the fauna as a whole, and wherever one species attained >10 individuals per sample no other achieved >6 individuals. 18

Amongst crucial questions in ecology is whether such patchiness is intrinsic to the organisation of assemblages, regardless of the nature of their environment (Solé and

Manrubia, 1995; Brown et al., 2002; Seuront and Spilmont, 2002), or is instead a direct organismal response to a patchy habitat (Wiens, 1976; Eggleston et al., 1998; Niebuhr et al.,

2015). For reasons mentioned above, previous investigations of spatial self-similarity have involved monofractal analysis of presence/absence data rather than the more complex issue of patterns in animal abundance, but at least in that respect a number of processes have been identified as potentially able to give rise to approximately scale-invariant patterning of organisms, one of which is a response to equivalent fractality in their habitat. Others include: diffusion-limited aggregation, random recruitment but spatially aggregated mortality

(or vice versa), self-organised criticality, and certain combinations of random processes operating at different resolutions that together statistically yield fractal-like outcomes (Halley et al., 2004).

The present results are relevant to two of these potential processes. Are the hot spots of high macrobenthic abundance observed at Deanbilla direct responses by the species concerned to the local habitat at those sites, for example to concentrations of microphytobenthic resources, perhaps themselves distributed scale-invariantly as they are known to be at least over small scales of assessment (Seuront and Spillmont, 2002; Dal Bello et al., 2015); or could they be part of the spatial heterogeneity generated by self-organisation independent of the local nature of the habitat (Saravia et al., 2012)? Some members of seagrass assemblages have certainly been shown to possess considerable powers of dispersal and redistribution within beds (Edgar, 1992; Boström et al., 2010), especially direct developers (which have the greatest need of means of redistribution since they can only disperse as benthic organisms) and detritivores. At Deanbilla, however, those species with marked local concentrations of abundance were either burrowing or tube-dwelling 19 peracaridans (albeit feeding at the surface) or microgastropods that can be considered unlikely to be capable of homing in on resource concentrations over considerable distances through the seagrass 'jungle' (Barnes in prep.). It seems almost certain to be the case at the study site in question that animal densities are being maintained well below carrying capacity (Barnes, 2017b), most likely as a result of top-down predator control including by the many juvenile shrimp, prawns and to a lesser extent fish that use the area as a nursery

(Alongi, 1989; Dall et al., 1991; Tibbetts & Connolly, 1998; Skilleter et al., 2005; Meynecke et al., 2008). If so, is the level of abundance at any given point then purely stochastic? Were the Deanbilla assemblage to be released from that external and possibly indiscriminate pressure, perhaps amounting the loss of 99+% of those young stages settling out from the plankton in their first few weeks or months (Bachelet, 1990), could it potentially occur much more commonly at the levels seemingly attainable in its hot spots? The observed maximum at Deanbilla of 51 ind. core-1 is equivalent to some 9,500 m-2, which would be regarded as a near average macrofaunal density in some of the South African Z. capensis beds studied by

Barnes and Barnes (2014) and to be a very low abundance for the macrofauna of North Sea

Z. noltei beds (Barnes and Ellwood, 2011). In other words, re-running Margalef's (1968,

1975) hypothesis of a ladder of ecosystem maturity, severe top-down control might be expected to prevent the attainment of order and a self-organised state. This would appear a likely outcome if assemblages are reduced to levels at which the potential species interactions that have been considered necessary for self-organsation (Solé and Bascompte,

2006) can no longer operate. Under such circumstances, alternative hypotheses to self- organisation seem more appropriate. What such processes are in the present case is much less evident, particularly if potential microphytobenthic food is present to excess. Predation is one obviously relevant process, especially since most potential prey individuals are small and widely, albeit usually thinly, dispersed. Surprisingly little research has concerned the 20 magnitude of or variation in epibenthic predation across spatial scales (Thrush, 1999), although Hammerschlag et al. (2013) did investigate predation effects across two spatial scales in Bahamian seagrass beds, finding no difference between the two in its impact. As yet, however, there are no relevant data available in respect of spatial aggregation across the scales concerned of either recruitment or mortality, predator-induced or otherwise (see

Vasconcelos et al., 2014). Neither is it known whether scale-invariant patterning of habitat characteristics occurs in seagrass beds (e.g. of particle size spectra, water velocities or epiphyte loads), or whether they might support anything analagous to random walks leading to aggregation.

Fractality in nature often manifests as a complex system of many superimposed individual monofractal sets, each with its own scaling exponent (fractal dimension); multifractality therefore integrates individual scale-dependencies. Thus whereas each monofractal set does imply a specific pattern of scale-invariance under a specific power law, multifractal structure does not imply any such overall scale-invariance, although the whole

Deanbilla assemblage and each dominant species clearly do conform to specific individual power laws. In spite of multifractality and scale invariance being separate phenomena, the present results do indicate that the ratio between the abundances of adjacent areas of equal size is approximately scale-invariant, because the scale-invariant parameter provides a good fit between the observed and expected Rényi dimensions. Hence (unexpecte𝑏𝑏 dly) our multifractal analyses clearly tend to confirm the scale-invariance of the whole Deanbilla assemblage and of several of its individual component species that are suggested by the indices of patchiness and by the values approximating unity of exponent β in the Taylor's power-law relationship. At this stage, however, there are many unexplained aspects to the observed multifractality. We simply do not know whether, for example, modifying MFp1p2 so that varied with scale would lead to detectable difference, whilst for some of the data it

𝑏𝑏 21 is clear that MFp1p2 does not hold in detail, for example the more or less constant frequency distribution of abundance over a 32-fold change in cell area in the case of Circulus.

It may be equivalent to other intertidal dwarf-eelgrass systems in respect of the spatial dispersion of its species richness (Barnes, 2014), but the extent to which the Deanbilla macrobenthos is typical of seagrass systems in general in terms of its multifractality and of its scale-invariant patchiness cannot be judged on current information, and this study was restricted to a relatively narrow band of spatial scales, spanning only 10s of metres.

Nevertheless, these scales can be regarded as being those most significant in relation to the sizes, behavioural and ecological nature of at least the benthic stages of the organisms concerned (c. 2-40 mm vs a hypotenuse of 130 m) (Dal Bello et al., 2017). That individual and overall abundances are patchy across a number of spatial scales is hardly surprising, but that the distribution of these patterns of abundance shows both multifractality and a system of patchiness that is close to — or actually is — scale invariant is both more remarkable and more puzzling. Many questions therefore arise. In that regard, further study at even smaller scales relevant to the lives of the sedentary seagrass macrobenthos and to their more mobile juvenile stages should prove rewarding. Equivalent analyses of other seagrass beds, especially those contrasting with the Z. capricorni ones in Moreton Bay — for instance those with more abundant and/or more temporarily variable inhabitants — would also be most interesting.

Acknowledgements

RSKB is most grateful to the Quandamooka Yoolooburrabee Aboriginal Corporation, the

Quandamooka Aboriginal Land and Sea Management Agency, and the Queensland Parks 22 and Wildlife Service for permission to conduct field research within the native title area of the Quandamooka People and within a Habitat Protection Zone of the Moreton Bay Marine

Park, under permit QS2014/CVL588. He also warmly thanks Ian Tibbetts and all the staff of the Moreton Bay Research Station for their help and support, and Alan Myers and Jim

Lowry for confirmation of the generic identity of the amphipod Eriopisella. Support for the fieldwork was received via the MBRS Distinguished Researcher Award from the Centre for

Marine Science, for which RSKB is most grateful. We thank the reviewers of this paper for their constructive suggestions for improvement of the text.

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Table 1. Patchiness across spatial scales of the whole macrofaunal assemblage in the

Deanbilla Bay seagrass bed and of its ten most numerous species: Morisita's χ2 values, with significance at P <0.001, <0.01 and <0.03 indicated by ***, ** and *, respectively; ns =

>0.05.

per nested unit of: 20 sample 22 samples 24 samples 26 samples

notionally representing: 33 m2 130 m2 530 m2 2115 m2

n = 256 64 16 4

Total macrofauna 896*** 246*** 52*** 9**

Circulus cinguliferus 4786*** 229*** 249*** 37***

Pseudoliotia speciosa 2233*** 476*** 154*** 29***

Pseudoliotia micans 1421*** 328*** 64*** 23***

Longiflagrum caeruleus 1232*** 461*** 121*** 36***

Calopia imitata 835*** 244*** 101*** 55***

Eriopisella sp. nov. 722*** 255*** 124*** 75***

Limnoporeia yarrague 535*** 165*** 30* 4ns

Malacoceros ?reductus 327** 139*** 77*** 51***

Enigmaplax littoralis 302* 106*** 42*** 32***

Dasybranchus caducus 274ns 101** 51*** 24***

Table 2. Values of Taylor's power-law constants α and β for the abundance of individual dominant macrofaunal species across spatial scales. See also Fig. 5.

species α β R2

Calopia imitata 2.01 1.40 0.998

Pseudoliotia micans 5.31 1.06 0.991

Pseudoliotia speciosa 8.31 1.04 0.999

Circulus cinguliferus 17.96 0.90 0.997

Enigmaplax littoralis 0.71 1.51 0.997

Limnoporeia yarrague 2.51 0.89 0.995

Eriopisella sp. nov. 2.80 1.52 0.999

Longiflagrum caeruleus 4.50 1.21 0.999

Malacoceros ?reductus 1.49 1.62 0.999

Dasybranchus caducus 1.49 1.49 0.999

Table 3. Values of best-fit b for the least-squares fit between observed and model Rényi dimension ( ). Larger values correspond to more pronounced contrast between adjacent areas of equal𝐷𝐷 𝑞𝑞extent.

species b

Circulus cinguliferus 5.84

Pseudoliotia micans 2.74

Pseudoliotia speciosa 2.21

Longiflagrum caeruleus 2.03

Limnoporeia yarrague 1.84

Eriopisella sp. nov. 1.75

Calopia imitata 1.73

Enigmaplax littoralis 1.52

Dasybranchus caducus 1.45

Malacoceros ?reductus 1.42

Whole assemblage 1.37

Fig. 1. Study location, including a Google Earth Pro image centred on

27°30'26"S,153°24'30"E in Habitat Protection Zone 2 of the Moreton Bay Marine Park,

showing topography at low tide of the investigated stretch of North Stradbroke Island

intertidal seagrass and the region sampled by the lattice of 16 x 16 evenly spaced stations.

the north-east quadrant are no longer there).

Fig. 2. Frequency of different numbers of macrofaunal animals per unit 0.0054 m2 sample

across the sampling lattice (n = 256).

Fig. 3. Choropleth diagrams of variation in numbers per 0.0054 m2 sample across the 256-

station lattice of: (A) the whole macrofaunal assemblage and (B-D) three representative

numerically-dominant species; (B) one showing low occupancy and extremely patchy

abundance (the tornid microgastropod Circulus); (C) one showing high occupancy and

much less, though still significant patchiness (the small macrophthalmid crab

Enigmaplax); and (D) one showing an intermediate state (the parapseudid tanaid

Longiflagrum).

Fig. 4. Change in degree of patchiness of the whole macrofaunal assemblage and of each of

its ten most abundant species across spatial scales, as indicated by Lloyd's index of

patchiness.

Fig. 5. Taylor's power-law relationship between mean and variance of the total numbers of

the whole macrofaunal assemblage across the range of spatial scales of assessment.

Exponent b ≈1 indicates spatial invariance of dispersion pattern (see text), in this case of

patchiness.

Fig. 6. Dispersion across the site of population hot-spots (≥22 ind. core-1) of individual

numerically-dominant species: C – Circulus; Ca – Calopia; Li – Limnoporeia; Lo –

Longiflagrum; Pa – Pseudoliotia axialis; Pm – P. micans; Ps – P. speciosa; P – a

combination of all three Pseudoliotia spp. (i.e. cores in which the named taxon was >2 as

abundant as any other and comprised >33% of the total fauna). o – hot-spots where no

one taxon was dominant.

Fig. 7. Log abundance per spatial cell versus log cell area (column 1) and Rényi dimensions

( ) (observed and least-squares fit model, column 2) for the three representative

𝐷𝐷numerically𝑞𝑞 -dominant species of Fig. 3 (Circulus, Enigmaplax and Longiflagrum), and

for the whole macrofaunal assemblage.