The sample of *Dizygocrinus* from the Monteagle Limestone consists of 161 individuals that show no evidence of infestation and 37 (19%) infested individuals; the latter either have a drillhole or snail at the base of the tube. Using the width of the radial plate as a proxy for crinoid body size, uninfested specimens average 3.67 mm (SD 0.58) and infested crinoids average 3.95 mm (SD 0.46) (Fig. 4). The difference in means (0.28 mm) is significant (*P* < 0.01 using *t* test).

The size distribution of *Dizygocrinus* from the Monteagle Limestone indicates that individuals infested by platyceratids are on average larger than the uninfested conspecifics. Although the difference in their average size is small, about 7.5%, it is nevertheless statistically significant. This result might appear to contradict the kleptoparasitic hypothesis, suggesting that infesting gastropods benefited their hosts instead of impacting them negatively. To explore this further, we present a heuristic for hosts and infesters assuming neutrality, that is, no negative or positive effect on host growth. The heuristic shows that under realistic, empirically based assumptions, the average size of infested and uninfested crinoids from the same population will not be the same, as one might intuitively expect, but rather that the average size of infested crinoids will be larger than uninfested ones for reasons we explain below. In addition, we present a computer model to demonstrate that reducing the size difference to the 7.5% value observed in the Monteagle Limestone *Dizygocrinus* requires a substantial decrease in the growth rates of infested individuals relative to their uninfested conspecifics. This leads us to conclude that the observed pattern is consistent with platyceratids having a detrimental effect on their hosts by reducing the hosts’ growth rates as predicted by kleptoparasitism.

### Platyceratid–crinoid association: a conceptual model

Evaluating fossil populations of crinoids and platyceratids is complicated by the fact that we must rely on cross-sectional data in our analyses. Rather than evaluating single individuals over time or comparing individuals of the same age, our data consist of overlapping cohorts even under the best of circumstances. Thus, the expected size distribution is highly dependent upon not only the nature of the biotic interaction (e.g., commensalism vs. parasitism), but also the timing of infestation relative to the age of the hosts.

The traditional hypothesis of coprophagy for the platyceratid–crinoid interaction infers that the infesting gastropods have no negative or positive effects on their hosts. Thus, two crinoids from the same cohort, whether they be infested or uninfested, should be the same size regardless of when in its life history the infested individual was colonized. However, a coprophagous scenario does not necessarily predict an absence of difference in the average sizes of infested and uninfested subsets from the same population of crinoids. Such a pattern would only be expected if infestation were congenital, occurring only among juveniles and never subsequently. Assuming that the population is stable (e.g., birth rate = death rate and the proportion of infesters is constant), the size distribution of infested and uninfested subsets of the population will be the same only under a model of strict congenital infestation and commensalism (neutrality).

A strict model of congenital infestation of crinoids by platyceratids is unrealistic and unsupported by the fossil record (see Baumiller 2002). Crinoids are likely to be subjected to infestation throughout their lives, and infestation of larger individuals would be especially favored. The larger individuals in a population are more likely to be infested by platyceratids for at least two reasons. First, larger crinoids can generally be assumed to be older individuals, and the longer an individual persists, the more likely it will be infested. Second, larger crinoids provide larger targets (greater surface area) for settling platyceratid larvae than do smaller members of the same population. Therefore, infested and uninfested subsets from the same crinoid population should not have similar average sizes, but rather the subset of infested crinoids should be considerably larger than the subset of uninfested crinoids.

The larger average size of the infested subset of the population will be maintained if infesters do not affect the growth rates of their hosts, as is the case of coprophagy (neutral; commensal). On the other hand, gastrophagy would lead to a decrease in the average size of infested crinoids, which would reduce the size difference between the two subsets. Nevertheless, the average size of infested individuals could still remain greater than that of the uninfested conspecifics. Only when the negative effect on the hosts’ growth rate by the infesters is large enough, will the average sizes of the uninfested subset be equal to or even surpass that of the infested subset. The larger size of uninfested individuals is, therefore, the clearest indication of a detrimental impact of the infester on the host, and was the pattern found by Rollins and Brezinski (1988) and Gahn and Baumiller (2003), which led these authors to conclude that in the populations examined, infesting platyceratids had a detrimental effect on their crinoid hosts.

As mentioned above, the published studies of cross-sectional analyses of platyceratid–crinoid interactions revealed patterns that are most consistent with parasitism (Gahn and Baumiller 2003; Rollins and Brezinski 1988). However, data for the Monteagle Limestone *Dizygocrinus* cannot be so easily interpreted, as the small, but significantly greater, size of the infested subset of the population could imply either a positive, neutral or a detrimental relationship. Therefore, we developed a numerical model to further explore platyceratid infestation among these tubed camerates.

### Platyceratid–crinoid association: a numerical model

#### Model description

In modelling the association between platyceratids and crinoids, we made some simplifying assumptions, and relied on what is generally accepted about platyceratids and crinoids. We selected model parameters that allowed us to reproduce the Monteagle Limestone population as observed.

In the model, at time 0, a population of crinoids consists of 198 individuals of different ages and sizes as in the population of Monteagle Limestone *Dizygocrinus*. In every subsequent time step, each crinoid can die (probability *μ*) or survive to the subsequent time step (probability 1 − *μ*). If the crinoid dies, it is replaced by an uninfested juvenile—this assumes that the population is stable. Uninfested crinoids, regardless of their ages or sizes, are subject to becoming infested by juvenile platyceratids with a probability *i* set so that the frequency of infested crinoids is about 19% the observed population. Crinoid and gastropod growth are governed by the von Bertalanffy (1938) equation with parameters selected to generate a range of sizes that reflect the sampled population of Monteagle Limestone *Dizygocrinus.*

Crinoid mortality in the population, regardless of cause, is time homogenous; that is, the probability of death in any time unit, *μ*, is constant, independent of age. The average age of crinoids in the populations is, thus, 1/*μ*.

In the model, gastropods infest crinoids as juveniles and remain sedentary, fixed to the host on which they settled, for the duration of the life of their host. Only a single gastropod can settle on each crinoid.

Each simulation was allowed to run for 500 time steps, but because the results reached a quasi-equilibrium after about 20 steps (see below), only a single time step was randomly selected to represent steps 100–500. At each time step, the following data were recorded for each live crinoid: age, size, and the presence or absence of an infesting platyceratid. Snail size was also recorded for live and dead snails. At each time step, the following population parameters were calculated for both live and dead crinoids: (a) the number of infested individuals; (b) the average age and size of infested individuals; (c) the variance in age and size of infested individuals; (d) the number of uninfested individuals; (e) the average age and size of uninfested individuals; and (f) the variance in age and size of uninfested individuals.

#### Numerical model results

The population structure of the live crinoids stabilized after about 20 time steps; that is, the average size and variance of infested and uninfested crinoids reached a quasi-equilibrium. The same was true for the frequency of infestation and the size structure of live gastropods. The absolute numbers of dead crinoids and snails continued to increase with each time step, but the population structure of the death assemblage also reached a quasi-equilibrium after about 20 time steps. The population parameters of the live and death assemblages were nearly identical.

To model the coprophagous (neutral) scenario of the association between platyceratids and crinoids, characterized by no detrimental effect on the growth rate of the host by the infester, 1000 simulations of the model were run with the same growth parameters for infested and uninfested crinoids. As the conceptual model predicted, the numerical model results indicate that infested crinoids are on average larger than uninfested crinoids: The size ratio of infested–uninfested crinoids was 1.24 (SD 0.05). Infested crinoids were larger, on average, in all 1000 simulations. Figure 5 shows the size distributions of the infested and uninfested crinoids from one of the simulations of the neutral scenario. Even by eye, it is clear that the infested part of the population (dashed line) is shifted substantially towards larger sizes. When compared to the observed size distributions of infested and uninfested *Dizygocrinus* (Fig. 4), the differences between the two are notable. Although infested crinoids are significantly larger than uninfested conspecifics in both the observed and simulated distributions, the ratio of infested–uninfested *Dizygocrinus* is only 1.07 in the observed data, much smaller than in the modelled neutral scenario (1.24). Again, this can be seen visually by comparing the differences in the modes of the infested and uninfested populations in Figs. 4 and 5. In the former (*Dizygocrinus* observed), the modes are much closer to each other than in the latter (*Dizygocrinus* modelled). This indicates that the growth rates of the infested Monteagle Limestone *Dizygocrinus* are much lower than expected given the behavior of the model. Therefore, despite the average size of infested *Dizygocrinus* being larger than their uninfested conspecifics, the observed pattern is consistent with a reduction in growth rates among infested crinoids as predicted by kleptoparasitism. To explore this further, we conducted several experiments by running simulations in which growth rates of infested crinoids were progressively reduced relative to uninfested conspecifics.

The results of those experiments (Fig. 6), representing neutrality and five instances of increasingly severe parasitism, are shown as the ratio of the average size of infested–uninfested subsets of crinoid populations (*y*-axis) against the ratio of their modelled growth rates (*x*-axis). Under the neutral scenario (solid circle), where the growth rates are the same (ratio = 1), the size ratio of infested–uninfested crinoids is 1.24, as discussed above; the vertical bars represent the ± 1 standard deviation of the ratio obtained from 1000 simulations. The average size ratio under the neutral scenario is, thus, significantly higher than the size ratio observed for the Monteagle Limestone *Dizygocrinus* (1.07, represented by the solid horizontal line in Fig. 6).

The size ratio of infested–uninfested Monteagle Limestone *Dizygocrinus* is consistent with simulated growth rates of infested crinoids that are only 0.67–0.5 of growth rates of uninfested conspecifics (Fig. 6). This suggests that the platyceratids infesting the Monteagle Limestone *Dizygocrinus* exacted a substantial toll on their hosts, decreasing their growth rates by 33–50%. Whether this can be attributed entirely to gastrophagy by the platyceratids is uncertain as these gastropods are likely to have had other deleterious effects on their host as discussed previously. Nevertheless, this analysis does support the hypothesis that the interaction between Monteagle Limestone platyceratid gastropods and *Dizygocrinus* is better characterized as parasitism than as commensalism. As the model developed here is quite general, applicable to any non-congenital association between a host and an infester, we are eager to have it applied to other platyceratid–crinoid associations for which population sizes are sufficiently large and other assumptions are met (see “Model description”).

#### Caveats

In the numerical models, we used a range of values for most of the parameters, including the mortality rate, *μ*, infestation rate, *i*, as well as the growth parameters in the von Bertalanffy equation. The values used were constrained by some of the properties of the observed Monteagle Limestone *Dizygocrinus*, such as the total size of the crinoid population (198), the proportion of infested crinoids (19%), and the size ranges of the crinoids and their infesters. However, there are very many combinations of the values of these parameters that can satisfy the constraints, and we have explored but a few. Nevertheless, in the combinations we examined, we discovered no results for the neutral (commensalistic) or detrimental (parasitic) scenarios that were substantially different from what we reported.

One assumption critical to modelling the platyceratid–crinoid system, or any other infester–host system, is the mode and timing of infestation. As we discussed in the section highlighting the conceptual model, strict congenital infestation of commensalistic platyceratids on their hosts leads to identical size distributions among infested and uninfested crinoids. Based on the conceptual model and empirical evidence, we argued that the association between platyceratids and crinoids is not congenital; thus, infested crinoids should be larger than their uninfested conspecifics. In the numerical simulations, we chose to model infestation as random with respect to the size of the crinoid—the probability of platyceratid infestation was size independent. While random infestation is certainly non-congenital, it is but one of many non-congenital infestation scenarios. For example, infesters may prefer either smaller or larger hosts.

We chose a size-independent infestation model for two reasons. First, a model of random infestation requires the fewest assumptions. Second, an analysis of the relative sizes of the Monteagle Limestone *Dizygocrinus* and their platyceratid hosts were not strongly correlated. If the Monteagle Limestone platyceratids preferentially selected small crinoids, there should be a tight correlation between the size of a host and its infester: Small crinoids should only be associated with juvenile platyceratids and large crinoids with adult platyceratids. Admittedly, because of taphonomic limitations, our sample size is relatively small (*N* = 8), but in the data available from the Monteagle Limestone, the relationship between hosts and infesters is positive (Fig. 7), although the slope of the regression is not significantly different from 0 (*P* = 0.056).

A further test of host selectivity on the Monteagle Limestone *Dizygocrinus* was conducted by running the model under three scenarios: (a) preferential infestation of smaller crinoids (*i* α 1/crinoid size); (b) preferential infestation of larger crinoids (*i* α crinoid size); and (c) random infestation (*i* independent of crinoid size). We ran 1000 simulations for all three scenarios, generating 95% prediction intervals (PI) for each scenario (Fig. 7). The observed data for the Monteagle Limestone *Dizygocrinus* overlap all three sets of prediction intervals, but they do not fit either of the host selectivity scenarios as well as that of random infestation, as they all fall within the 95% PI only for the latter scenario. While this test is clearly limited given the small size of the sample, it nevertheless guided our decision to use a size-independent infestation model.

In our analyses we implicitly assumed that the Monteagle Limestone data are not biased in ways that would impact the results. However, it is important to consider potential taphonomic and sampling biases. As discussed above, most of the specimens in this study were collected as loose calyxes despite being preserved as relatively complete specimens. This allowed us to see snails and drillholes that might have been otherwise obscured by matrix or the crinoids’ arms. However, the severe compression characteristic of many of the specimens, both crinoids and platyceratids, and the time they sat weathering on the surface, likely resulted in the disassociation of some platyceratids and their hosts. One reason we were not able to examine the size relationships between many of the crinoids and their infesters is that the snails were so badly fragmented that we could not adequately measure them. Snails were likely disassociated from their hosts as they weathered on the surface, as indicated by the many isolated platyceratids in the deposit, in addition to disassociating from their hosts during the trauma associated with the obrution event. However, it is unlikely that this would have led us to misidentify infested crinoids as uninfested, given the signatures of their presence left on the hosts.

Compression of the fine-grained matrix entombing the crinoids also produced potential biases. The degree of calyx compression and associated disarticulation is not random with respect to size. Splitting the Monteagle Limestone *Dizygocrinus* into two groups based on mean radial plate width reveals that crinoids below mean body size exhibit a significantly lower degree of compaction and disarticulation than their larger conspecifics (*P* < 0.05, see Gahn and Baumiller 2004 for methods). If this resulted in larger infested crinoids becoming dissociated from their infesters and being misidentified as uninfested, that would artificially reduce the observed difference in average size of infested and uninfested crinoids, making the neutral scenario easier to reject. On the other hand, infestation may bind the host’s plates and make infested specimens less prone to disarticulation, which would artificially increase the observed difference in average size of infested and uninfested crinoids, making the neutral scenario more difficult to reject. We are currently unable to quantitatively assess these two sources of plausible bias.