### Abstract

- Top of page
- Abstract
- Introduction
- Experimental methods
- A density-dependent population model
- Discussion
- Acknowledgements
- References

**1.** Density dependence may act at several stages in an organisms life-cycle (e.g. on mortality, fecundity, etc.), but not all density-dependent processes necessarily regulate population size. In this paper I use a density manipulation experiment to determine the effects of density on the transition rates between different size classes of the clonal zoanthid *Palythoa caesia* Dana 1846. I then formulate a density-dependent matrix model of population dynamics of *Palythoa*, and perform a series of sensitivity analyses on the model to determine at what stage in the life-cycle regulation acts.

**2.** Seven of the 16 transition probabilities decreased with density, most of them being shrinkage (due to loss of tissue or fission) and stasis (the self–self transition) of medium and large colonies. The only probability to increase was for the stasis of large colonies. Recruitment was quadratically dependent on density, peaking at intermediate densities.

**3.** Equilibrium cover in the model was 84% and was reached in ≈40 years. To determine which density-dependent transitions were involved in population regulation, the strength of density dependence was varied in each independently. This sensitivity analysis showed that only changes in the probabilities of large colonies remaining large and producing medium colonies, were regulating.

**4.** These results suggest that regulation is primarily acting on fission of large colonies to produce intermediate-sized colonies, in combination with size specific growth rates. Fission rates decrease greatly with density, resulting in a greater proportion of large colonies at high densities and large colonies grow more slowly than small. Overall, this behaviour is very similar to that of clonal plants which have a phalanx type life history.

### Introduction

- Top of page
- Abstract
- Introduction
- Experimental methods
- A density-dependent population model
- Discussion
- Acknowledgements
- References

The question of how, and even if, populations are regulated has been a controversial one. There has been an extensive debate in recent years about how to detect density dependence in natural populations (e.g. Hanski, Woiwood & Perry 1993; Holyoak 1994a,b; Wolda, Dennis & Taper 1994; Fox & Ridsdill-Smith 1995; references therein). However, this debate has often overlooked the potential for experimental manipulations and modelling studies to provide insights that cannot be gained from simple analysis of population abundance over time. The utility of the experimental approach has been well demonstrated by Murdoch and colleagues in their study of population regulation in red scale (e.g. Murdoch 1994; Murdoch *et al*. 1995, 1996). The experimental approach has several major advantages over purely observational approaches, including its ability to determine the mechanisms of regulation and because conclusive results can be obtained in less than the tens of generations required for an observational approach.

In conjunction with well planned experiments, modelling can allow us to obtain a much greater understanding of the factors controlling abundance. The problem with many modelling studies to date, however, is their lack of grounding in real data. Many models assume certain functional forms for their parameters, even when they are based upon a particular species. For example, it is often assumed that the Beverton–Holt or Ricker functions are suitable descriptions of how certain parameters depend on density, apparently without ascertaining their adequacy (e.g. De Angelis *et al*. 1980; Levin & Goodyear 1980; Caswell 1989). It is thus not always clear how the results obtained from these models pertain to the real world, although they do provide valuable insights into potential population behaviour. In the best studied systems, observation and experimentation are combined with modelling, to investigate the mechanisms of population regulation (e.g. Watkinson, Lonsdale & Andrew 1989; Hubbell, Condit & Foster 1990; Murdoch 1994). The use of models can also help in determining if a density-dependent process (e.g. mortality) is, in fact, regulating (i.e. keeps population size within bounds). An intensive study of a particular organism may show that density dependence is acting at several stages of the life-cycle (e.g. Tanner 1997), but it may be difficult to tell which if any of these stages is regulating population size.

Insects are probably the most intensively studied group of organisms with respect to population regulation (e.g. Cappuccino & Price 1995), although they are by no means the only animal group that has been studied. There is also a wealth of data on modular plants, which have size-, rather than age-dependent vital rates, in contrast to most terrestrial animals (e.g. Harper 1977; Crawley 1990). There is another major group of modular organisms, however, which potentially have different dynamics to both unitary terrestrial animals and modular plants. These are the clonal marine invertebrates. In one of the few studies examining how changes in density affect this group of organisms, Karlson, Hughes & Karlson (1996) showed that the soft coral *Efflatounaria* behaves like many plants, because individual colonies showed a decreased rate of fission at high densities. The generality of this result is not known, however.

There is considerable potential for colonial invertebrates, even those that rely on photosynthetic zooxanthellae, to behave differently to plants. For instance, invertebrates have no below-ground competition for nutrients, although they may still compete ‘above-ground’ for food (e.g. Buss 1979; Frechette, Aitken & Page 1992). Many invertebrates are also capable of active (contest) competition, whereas plants tend to compete passively (scramble competition). A few modelling studies have addressed population regulation and density dependence in marine invertebrates, although these either utilize separate models for each of a number of different densities (e.g. Harvell, Caswell & Simpson 1990; Tanner 1997) or have constant recruitment, meaning that as model population size increases, *per capita* recruitment decreases (e.g. Hughes 1990). To my knowledge, there are no studies where the full effects of density have been included in a population model of a clonal marine invertebrate. In fact, studies of density dependence and population regulation in clonal marine invertebrates are extremely rare, despite them being the dominant organisms in several well studied ecosystems, particularly coral reefs. If we are to fully understand how these marine communities function, it is essential that the mechanisms regulating the population sizes of their constituent species are thoroughly examined.

In this paper, I use a density manipulation experiment to determine the affects of density (percentage cover) on the transition rates between different size classes of the inter-tidal zoanthid *Palythoa caesia* Dana 1846. These transition rates are then used as the basis of a density-dependent size-classified matrix model of population dynamics (e.g. Caswell 1989). First I examine a model where all transition probabilities with significant density effects are density-dependent. I then go on to examine a series of altered models to determine which density-dependent transitions are regulating the population size of *Palythoa*. To determine the effect of variations in the strength of density dependence in each element, I also conduct a sensitivity analysis, where I make slight changes to the strength of the relationship between each transition probability and density while holding all other transition probabilities constant.

While the use of matrix models to examine population dynamics has become relatively popular (see Caswell 1989), they have rarely been applied directly to the study of population regulation. There is a large body of theoretical literature on density-dependent matrix population models (e.g. Leslie 1948; Charlesworth & Giesel 1972; Smouse & Weiss 1975; Bishir & Namkoong 1992; Takada 1995), but only a handful of studies which utilize empirical data to model the population dynamics of a particular species (e.g. DeKroon *et al*. 1987; Solbrig, Sarandon & Bossert 1988; Watkinson *et al*. 1989; Burgman & Gerard 1990; Gillman *et al*. 1993). None of these studies directly address the issue of population regulation and few are combined with experimental manipulations. While these types of models cannot be used to determine the mechanisms regulating the population (e.g. predation, parasitism, etc.) without further assumptions, they can be used to determine at what stage of the life-cycle regulation is acting. This knowledge can then shed light on the actual mechanisms of regulation and allow experimental work to be more tightly focused on the relevant stage of the life-cycle.

In a previous paper (Tanner 1997), I showed that the density of *Palythoa caesia* appears to be regulated at ≈ 80% cover. As density increased, the rate of fusion between colonies increased, while the rate of fission decreased. At the same time, the growth rates of large colonies decreased, while those of smaller colonies remained unchanged. As a result of these changes in fission, fusion and growth, the size-structure of the population changed with density, with a greater proportion of large, slow growing colonies at higher densities. Mortality showed no relationship with density, while larval recruitment peaked at intermediate densities, but was low overall. Combining these previous results with the results presented in this paper allows the main stage of the life-cycle regulating the population to be tentatively identified and suggests several mechanisms by which regulation may be acting. These mechanisms can then be the subject of further experimental work to verify how *Palythoa* populations are regulated.

### Experimental methods

- Top of page
- Abstract
- Introduction
- Experimental methods
- A density-dependent population model
- Discussion
- Acknowledgements
- References

*Palythoa caesia* is a highly abundant zoanthid which occurs on the reef crest of many reefs on Australia's Great Barrier Reef. Colonies have an encrusting (two-dimensional) growth form and very rarely exceed 10 cm in diameter. Individual zooids are ≈8 mm in diameter (Burnett *et al*. 1997), with typical colonies at the study site having 20–40 zooids and probably never more than 100 (personal observation). For this study, I established thirty-six 1-m^{2} permanent quadrats on the reef crest at Heron Island (23°26′S, 151°55′E), in areas of 80–90% cover of *Palythoa*. These quadrats were subjected to a graded series of partial removals in June 1993, such that they contained between 0 and 90% cover of *Palythoa*. Six quadrats were completely cleared (0% cover), 24 were partially cleared, with six each having 12–17, 23–33, 37–52 and 55–72% cover, and six were left unmanipulated (80–90% cover). As well as clearing excess *Palythoa*, I removed all hard corals and macro-algae, which accounted for less than 5% cover. Clearing involved the complete removal of randomly chosen colonies, and was conducted with minimal disturbance to the remaining colonies. Immediately after the manipulation, and at 8-month intervals until June 1995, I photographed the inner 0·25 m^{2} of each quadrat from directly overhead. Only the inner portion of the quadrat was used to eliminate any problems with edge effects due to higher densities outside most quadrats. To determine the fate of each colony and to record larval recruitment, these photographs were traced at 64% of life size. The outline of each colony was digitized to determine the colony's size at each census. The density of *Palythoa* in each quadrat was determined by summing the areas of all individuals in the quadrat and expressing this as a percentage of quadrat area (0·25 m^{2}). Photographs of two quadrats at the second census were inadequate for analysis and these quadrats were excluded from the experiment. Further details on the experimental procedure used are given in Tanner (1997).

Competition among sessile invertebrates is generally considered to be either for space and/or food (e.g. Buss 1979) and, therefore, depends greatly on colony size. Any density measure must then account for both the number of colonies and their size, which the chosen measure, percentage cover, does in an easily understood way, despite large variation in colony size. Because the size distribution of colonies was very similar between densities (Tanner 1997), percentage cover was closely related to the number of colonies present (Poisson regression, *P* < 0·001, *R*^{2} = 0·775, Fig. 1).

### A density-dependent population model

- Top of page
- Abstract
- Introduction
- Experimental methods
- A density-dependent population model
- Discussion
- Acknowledgements
- References

To determine the effect of density on the individual elements of a size-classified Leslie matrix model of population dynamics for *Palythoa*, I first calculated transition probabilities for each quadrat over each census interval. Each matrix was based on four size classes, 0–1000 mm^{2}, >1000–2000 mm^{2}, >2000–4000 mm^{2} and >4000 mm^{2}. These size classes were chosen to give approximately equal numbers of colonies in each size class (see Table 1) and to give as close as possible to an even distribution among size classes for each quadrat (this later requirement led to a different boundary between size classes 3 and 4 than used in Tanner 1997). A total of 84 4 × 4 transition matrices were calculated (28 non-empty quadrats × 3 census intervals). These matrices describe the probability of a colony in size class *i* at the beginning of the census interval being in size class *j* at the end of the interval. For each of these matrices, I used the percentage cover of *Palythoa* in the quadrat at the start of the census interval as a measure of density. The relationship between each of the 16 elements of the matrix and density was then determined by performing logistic regression on untransformed data (i.e. using GLM with a binomial distribution). Logistic regression is the method of choice when the dependent variable is constrained to lie between 0 and 1 (e.g. Trexler & Travis 1993). Each of the 16 matrix elements were examined separately, so each regression was based on 84 data points. These 84 points are not independent, but the fitted equations are still unbiased (e.g. Seber 1977), and applicable within the modelling context in which they are used here. Due to the non-independent nature of the data, significance levels were calculated using randomization methods (Manly 1991). Specifically, the observed transition probabilities were randomly allocated without replacement to the observed densities 4999 times, with a logistic regression done each time. A two-tailed significance level was then calculated as the proportion of random trials that gave a more extreme regression coefficient than that for the observed data. The appropriateness of a linear fit was assessed by checking residual plots, and also by performing non-parametric regressions (generalized additive models, Hastie & Tibshirani 1990). Here, I use linear in the sense that no powers of density are involved. The results of the generalized additive models showed that a linear relationship was suitable for all transitions (in all cases the non-linear component was highly non-significant with *P* > 0·8).

Table 1. Non-density-dependent transition probability matrix from which the non-density-dependent terms in all models were obtained. I–IV denote size classes detailed in the methods and *N* is the total number of colonies in each size class. (Note: colonies which survived the entire 2 years are counted three times, once for each census interval) | I | II | III | IV |
---|

I | 0·596 | 0·157 | 0·0822 | 0·0693 |

II | 0·241 | 0·566 | 0·205 | 0·0997 |

II | 0·0178 | 0·258 | 0·638 | 0·349 |

IV | 0·00556 | 0·0187 | 0·165 | 0·701 |

*N* | 900 | 1284 | 1209 | 1022 |

Recruitment was analysed in a similar way to the transition probabilities, with the exception that a Poisson distribution (suitable for count data) was used in the GLM. As *Palythoa* has a planktonic larval phase (Babcock & Ryland 1990), population dynamics were modelled as being open, with recruitment occurring from external sources. As a result, there are no fecundity terms in the model.

The initial density-dependent model incorporated linear density dependence in all transition probabilities (as per Table 2), as well as quadratic density dependence in recruitment (*E*[recruits] = exp (−2·21 + 0·0589*density − 0·0538*density^{2}), *R*^{2} = 0·07, *P* < 0·001 for both coefficients), as found earlier (Tanner 1997). The model is:

Table 2. Effect of density on the individual transition probability matrix elements. Element (*i,j*) refers to the transition from size class *i* to size class *j.* Density dependence was only retained in those transitions marked with an asterisk. The model equations for density-dependent transition probabilities are of the form Element | Intercept (α) | Slope (β) | P | Proportion of variation explained |
---|

(1,1) | 0·459 | −0·429 | 0·338 | 0·03 |

(1,2) | −0·631 | −0·596 | 0·17 | 0·07 |

(1,3) | −4·716 | 1·51 | 0·205 | 0·06 |

(1,4) | −4·562 | 0·409 | 0·878 | 0 |

(2,1) | −1·508 | −0·441 | 0·442 | 0·02 |

(2,2) | 0·239 | −0·126 | 0·698 | 0 |

(2,3)* | −0·522 | −0·773 | 0·052 | 0·13 |

(2,4) | −3·934 | 0·394 | 0·722 | 0 |

(3,1)* | −1·728 | −1·266 | 0·052 | 0·12 |

(3,2)* | −0·650 | −1·182 | 0·015 | 0·18 |

(3,3)* | 0·843 | −0·482 | 0·1 | 0·09 |

(3,4) | −1·534 | −0·151 | 0·711 | 0·01 |

(4,1)* | −0·428 | −3·013 | 0·039 | 0·17 |

(4,2)* | −0·36 | −2·562 | 0·041 | 0·15 |

(4,3)* | 0·991 | −2·216 | 0·008 | 0·19 |

(4,4)* | −0·386 | 1·616 | 0·036 | 0·12 |

- (eqn 1)

*n*(*t*) is a vector of length 4 describing the size structure of the population at time *t*. *A[d*(*t*)] is the 4 × 4 transition probability matrix, which is a function of density, and *r*[*d*(*t*)] describes recruitment into the first size class as a function of density. *d*(*t*) is the density (percentage cover) of the population at time *t*, and is calculated assuming that the mean size of colonies in each size class is the same as that observed at the start of the study, that is 610, 1474, 2866 and 5815 mm^{2} for sizes 1–4, respectively. The model was then simplified by sequentially eliminating density dependence in the terms where density was least significant, until doing so caused a change of greater than 10% in the prediction of one of equilibrium population size or the change in cover at any density in comparison to the full model, or caused a given cover to be reached more than 5 years earlier or later than in the full model. This elimination procedure resulted in linear density dependence in eight transition probabilities, those with *P*≤ 0·1, and quadratic density dependence in recruitment, being retained in the final model. The other eight transition probabilities were constant, and were calculated by pooling the transition matrices for all quadrats and census intervals, and then calculating a single density-independent transition probability matrix (Table 1). The retention of terms was based on their effect on model output rather than by setting a (ultimately arbitrary) level of significance a priori because it was felt that weak density dependence that was not statistically significant due to noise may be important in regulating the population. The effect of this noise (mostly spatial and temporal variability) on population dynamics will be examined in a later paper. Models where density dependence was only retained when significant at α = 0·05 (with or without adjustment for multiple tests), equilibrated at over 100% cover, meaning that important elements in the population's dynamics were missing. The eight transitions in which density dependence was retained were predominantly shrinkage and stasis of the two larger size classes, but also included growth of medium colonies (from size class 2 to size class 3). Seven of these eight relationships were negative, that is, as density increased the transition probability decreased. Thus, high densities mainly reduced the number of colonies shrinking (or undergoing fission) and remaining the same size, but did not decrease the number growing (with the exception of size class 2). On the other hand, stasis in size class 4 [(i.e. the (4,4) transition], increased as density increased. Thus, at high densities, large colonies showed less of a tendency to shrink (or undergo fission) and die, than at low densities. A simulation of the final model according to eqn 1, starting with no *Palythoa* present [i.e. *n*(0)′ = (0 0 0 0)], showed that the population reached an equilibrium at 84% cover after about 40 years (Fig. 2). Population growth for the first 20 years was approximately exponential, with cover remaining very low for the first 10 years, after which it increased steeply. As could be expected, the four size classes peaked in number sequentially from smallest to largest (Fig. 3). The equilibrium size structure predicted by the model is very similar to the size structure found in the unmanipulated quadrats at natural densities of 80–90% cover. Over the 2-year study period, there was a slight increase in the observed proportion of small and large colonies [size classes 1 (+10%) and 4 (+6%)], with a concomitant decrease in medium colonies [(size classes 2 (−4%) and 3 (−6%)]. For all size classes, the model predicted an equilibrium cover either between the values observed (size classes 1 and 3), or within 1% of these values (size classes 2 and 4). The equilibrium density predicted by the model (84%) is also very close to that suggested in Tanner (1997) on the basis that quadrats with cover less than 80% increased in cover and quadrats with cover greater than 80% decreased in cover. These two similarities between predicted population behaviour and observed behaviour suggest that the model is accurately describing the population under study. A further test of the model's accuracy is to compare the predicted change in cover for any initial cover to that observed in the field. The observed change in cover was accurately described by a quadratic function of initial cover (Fig. 4, Tanner 1997). Population growth at low densities is slightly underestimated by the model, while that at high densities is over-estimated (Fig. 4). This seems to be driven by the very low initial growth rates in the model, when cover is less than 10%, which is exactly the range of cover values for which no field data were available. Overall, however, the predicted and observed changes in cover for a given density were very similar and all model predictions fell well within the range of scatter of the field observations.

#### Effect of eliminating density-dependent terms

While there are nine terms (eight transition probabilities and recruitment) for which density has a significant impact on one or more aspects of the model behaviour, some of these terms may not be important in regulating population size. To determine which are the most important transitions for regulating model population size, I ran simulations in which density dependence was excluded in one term, while retaining it in the other eight. Non-density-dependent transition probabilities to replace the density-dependent probabilities were obtained from Table 1 and density-independent recruitment was set at the mean observed level of 0·3 recruits/m^{2}/8 months. This was repeated excluding density dependence in each term separately, i.e. I ran all possible combinations with eight of the nine terms density-dependent. When density dependence was dropped in a single term only, population size was still regulated at levels between 81 and 93% cover (Fig. 5). The single exception to this was when density dependence was dropped in the (4,3) transition (i.e. shrinkage from the largest to the second largest size class), which resulted in an unbounded exponential increase in population size, although growth in the first 30 years was slower than for most other models.

#### The effect of varying the strength of density dependence in individual elements

To determine the effects of changing the strength of density dependence, I altered the relationship with density for each transition probability, while keeping all other density-dependent terms as observed. These alterations consisted of multiplying the slope of the density-dependent relationship (i.e. β in Table 2), by a range of values between 0·5 and 2. For the majority of transition probabilities, changing the strength of density dependence had little affect on population dynamics, with stronger density dependence only slightly decreasing equilibrium population size and weaker density dependence only slightly increasing it (e.g. Figure 6). There were only two transition probabilities for which these moderate alterations substantially altered population dynamics, these being stasis in size class 4 (the only transition probability that increased with density), and shrinkage from size 4 to 3 (the only transition in which density dependence was required for the population to be regulated). Either decreasing the slope of the (4,3) transition by Ð44% (Fig. 7a), or increasing the slope of the (4,4) transition by Ð48% led to continued exponential growth (Fig. 7b). These changes correspond to increases in the transition probabilities at 50% cover from 0·47 and 0·60 to 0·58 and 0·68, respectively, which are biologically possible, although perhaps unlikely.

### Discussion

- Top of page
- Abstract
- Introduction
- Experimental methods
- A density-dependent population model
- Discussion
- Acknowledgements
- References

The main regulating agent in this population of *Palythoa caesia* appears to be acting on the shrinkage and/or fission of large (size class 4) colonies into size class 3. The (4,3) transition is the only probability that needs to be density-dependent in order for the model population to be regulated, and eliminating density dependence in any other element has very little affect on population dynamics (Fig. 5). This is consistent with my previous analysis (Tanner 1997), which showed density dependence in fission, fusion and the growth of large colonies. Putting the two results together suggests that population regulation is achieved by density dependence in the growth and fission of large colonies. Thus, as density increased, large colonies produced fewer size class 3 colonies by fission, probably because of a decrease in their growth rates. Decreased growth led to decreased fission because fission rates increased with colony size (Tanner 1997). At low densities, fission accounted for 86% of the production of size class 3 colonies from size class 4. This proportion decreased steadily as density increased to only 38% when cover was greater than 80%, with the remainder produced by shrinkage (either due to resorption or partial mortality).

Decreases in fission with increases in density are fairly common for clonal organisms which utilize fission as a major form of asexual reproduction. In the only comparable marine study, Karlson *et al*. (1996) found that stolon production (=fission) by soft corals decreased with density, as did mortality. Grasses can also behave in the same way (Harper 1977). Lovett Doust (1981) divided grasses and other clonal plants into ‘guerilla’ and ‘phalanx’ species, with the former having widely spaced ramets and spread rapidly, while the later have tightly packed ramets and spread slowly. Schmid & Harper (1985) showed that for a ‘guerilla’ species, mortality is density-dependent and birth of new modules (equivalent to fission) is density-independent. Conversely, *Palythoa*, with density-dependent birth of new modules and density-independent mortality (Tanner 1997), follows the pattern found for ‘phalanx’ species (Schmid & Harper 1985; Cain *et al*. 1995). This fits with *Palythoa’s* life history, as it is slow growing and forms dense monospecific clusters of colonies, much as phalanx plants do. Unlike many plant species which undergo a process of self-thinning at high densities (e.g. reviewed in Lonsdale 1990), density-dependent mortality appears to be uncommon in clonal invertebrates, including those that do not reproduce by fission (e.g. Harvell *et al*. 1990). In some clonal invertebrates, mortality even decreases with density (Karlson *et al*. 1996). At this early stage, it thus appears that clonal marine invertebrates do not use the same mechanisms to divide up the guerrilla/phalanx continuum as clonal plants do, although more species need to be studied to confirm this.

The regulating role of growth and fission in large colonies may well be related to food supply and this is currently being experimentally tested. It has been established in several other clonal organisms that large colonies have a lower particle capture rate per polyp than do small colonies (bryozoans: Bishop & Bahr 1973; soft corals: McFadden 1986; hydroids: Hunter 1989; scleractinian corals: Johnson & Sebens 1993; anthozoans: Anthony 1997), although this is not always the case (e.g. Okamura 1984, 1985 for bryozoans). Feeding rate is also likely to decrease with density (e.g. Anthony 1997). As a result, colonies are likely to experience decreased food capture rates at high densities, and this is likely to be exaggerated for large colonies. These decreased capture rates will presumably lead to decreased growth and fission due to fission being size-dependent. It is also possible that as colonies come into contact at higher densities, they expend resources on aggressive interaction, although no mechanisms of interference competition have yet been described for *Palythoa*. Another alternative is that large (presumably sexually mature) colonies divert resources from growth to sexual reproduction at high densities, as predicted by the strawberry-coral model of Williams (1975), although often the opposite occurs (e.g. Stocker & Underwood 1991). In *Palythoa*, the high rates of fission (up to 70%Tanner 1997) appear to be related to habitat, rather than being a species characteristic, because larger colonies (several square meters in area cf. <100 cm^{2} intertidally) occur in subtidal areas near the study site (personal observation). These colonies experience decreased water motion and increased feeding opportunities (due to being continually submersed) compared to intertidal colonies, suggesting that food supply is the more likely explanation of population regulation, or else that it is related to disturbance. The food supply hypothesis fits in with the suggestion by Cappuccino & Harrison (1996) that bottom-up regulation is much more common than top-down, although predation does appear to be important in some groups (e.g. small mammals: Sinclair & Pech 1996; fish: Hixon & Carr 1997).

In contrast to the previous paper (Tanner 1997), where it was suggested that an increase in fusion rates with density may also be important in regulating the population, the main transition probabilities corresponding to fusion were found to be of little importance here. Fusion primarily contributes to the (2,3), (2,4) and (3,4) transitions, none of which were found to be density-dependent. It thus appears that although fusion is density-dependent (Tanner 1997), this density dependence makes very little, if any, contribution to population regulation. This lack of an affect of density-dependent fusion is likely to be due to the low overall fusion rates, on the order of 5–10% in 2 years (Tanner 1997).

At larger spatial scales than that studied here, recruitment limitation may become an important component of population regulation. Larval recruitment rates of zoanthids are generally low (e.g. Karlson 1983, 1988; Tanner 1997), which may account for the patchy distribution of *Palythoa* at the scale of the entire reef. Thus, the conclusion that fission of large colonies is the primary agent of population regulation only applies to the within patch scale at which this study was conducted. At the between patch (metapopulation) level, larval recruitment is likely to become important in that it will limit the number of new patches colonized and thus the number occupied at any one time. The open nature of the population studied is supported by the high levels of gene flow measured for *P. caesia* over the entire length of the Great Barrier Reef (Burnett *et al*. 1994). While the larval duration of *Palythoa* is unknown, a closely-related species of *Protopalythoa* requires 17–19 days to become competent to settle and it has been suggested that this is close to the minimum larval duration for zoanthids (Babcock & Ryland 1990). Thus, it is unlikely that *Palythoa* recruits originate from the patch within which they settle and are instead likely to originate from other reefs.

The sensitivity analysis conducted indicates that the model is fairly robust to errors in estimating the relationships between density and the individual transition probabilities. For most of the transition probabilities, even major departures from the estimated parameters, as in doubling or halving the slope, produced very little qualitative effect on population dynamics. The only exceptions to this are for the (4,3) and (4,4) transitions, i.e. for the largest colonies either shrinking one size class or remaining in the largest size class. Only slight changes are needed in the transition rates of large colonies for population growth to become exponential. This is not unexpected, especially for the (4,3) transition, which is the main transition regulating the population. The transition probabilities for size class 4 are probably reasonably accurate as they are based on a total of 1022 transitions.

The primary regulating agent of *Palythoa caesia* appears to be a decrease in the fission rates of large colonies to produce smaller colonies as density increases, in combination with size-dependent colony growth rates. The primary evidence for this is that the (4,3) transition produces the strongest regulatory effects in the density-dependent model developed in this paper. That fission is the important component of the (4,3) transition is suggested by a previous study which shows that fission rates are closely linked to density (Tanner 1997). The changes in fission rates which are regulating this population of *Palythoa* are likely to be brought about by mechanisms intrinsic to the population, possibly competition for food, rather than by external forces such as predation or parasitism. Overall, the behaviour of this marine invertebrate very closely parallels that of clonal plants which have a phalanx type life history. These plants tend to have density-dependent birth of new modules (equivalent to fission in *Palythoa*), which is thought to be regulating and density-independent mortality rates (e.g. Schmid & Harper 1985; Cain *et al*. 1995). This is exactly the pattern I have found here for *Palythoa* (see also Tanner 1997).