Social tolerance in Octopus laqueus—A maximum entropy model

Octopus laqueus is a small tropical octopus found in Okinawa, Japan and the greater Indo-Pacific. Octopus are often viewed as solitary animals but O. laqueus live in close proximity in the wild, and will potentially encounter one another on a regular basis, raising the possibility of social tolerance. Adopting shared den occupancy in aquaria as a potential measure of social tolerance in O. laqueus, we studied the animals’ preference for shared dens over solitude. We characterized dependence of sharing preference on sex, den availability and den occupancy density. We designed two simple social tolerance assays in aquaria with a total of 45 daily measurements: (i) Pots Equal, with equal numbers of octopuses and dens and (ii) Pots Limited, with a 3:1 ratio of octopuses to dens. We found that O. laqueus will socially tolerate other individuals by sharing tanks and dens and with typically no loss to cannibalism or escape. However, animals also exhibit significant levels of social repulsion, and individuals often chose a solitary den when given the option. The patterns of den occupancy are observed to be consistent with a maximum entropy model that balances seeking shelter against avoiding other animals. The model accurately captures and predicts the data and can be generalized to other organisms and their social interactions. Overall, in O. laqueus the preference for a den is stronger than the preference to be solitary. The animals are tolerant of others with a mixture of sizes in the tank and even in a den, a reported first for octopuses outside mating. The relaxed disposition and social tolerance of O. laqueus make it a promising species to work with in the lab to explore social and potentially other behaviors in octopuses.


Introduction
Octopuses are traditionally viewed as solitary animals that do not form social aggregations, have relatively few and simple reciprocal interactions, and rarely make physical contact outside aggression and mating [1][2][3][4][5][6][7]. Further, species are known to be cannibalistic in the laboratory abundance and proximity of O. laqueus in the field raised the possibility that it is a social octopus, leading us to wonder whether O. laqueus would tolerate conspecifics in a den or tank under dense conditions, and to design a series of experiments to investigate this question. Our aim is to investigate functional and predictive-as opposed to descriptive-characterisations of sociality. Following Mather [15,30] we define as 'asocial' non-interacting animals, e.g. animals that ignore each other in a specified context. The typical number of such animals that share common dens sets a reference level for neutrality. Animals showing levels of sharing larger than neutral are then considered as 'social' whereas animals showing smaller levels of sharing are 'anti-social'(note, that by taking neutral animals as a reference, animals that actively avoid each other would be labeled as anti-social, rather than just being asocial; on the other hand, animals that tolerate each other, yet at levels comparable with the neutral ones, would still be considered as asocial). This operational definition of sociality may not necessarily correspond directly to customary notions of sociality.
The context for our experiments is den occupation patterns. The statistical significance of pot occupancy patterns is established hypothesis-testing methods, with the neutral model serving as a null hypothesis, and by testing statistical contrasts under different experimental treatments. We also develop a statistical model of den occupation based on the maximum entropy principle. The proposed max-ent model transcends hypothesis testing: (i) It treats different experimental setups on the same footing, incorporating together all the measurements and hence increase their statistical power, irrespective of whether these setups are 'un-balanced' or 'un-factorized.' (ii) It in principle enables compilation of meta-data from diverse labs, future measurements, even from different species or eco-systems into a common framework, in which the statistical uncertainties are explicitly stated. (iii) It offers a natural phenomenological explanation for the social behavior in terms of animal (pairwise) interactions. 'Natural' here means a straightforward and simple explanation-using the least structured model, i.e., the one with a minimal set of assumptions which is still consistent with the measurements. (iv) It facilitates verifiable quantitative predictions that may apply outside the measured regime. It is therefore useful for designing new experiments, identifying outliers, and exploring potentially interesting experimental regimes. We can project the trade-off between denser populations and mixed sexes for arbitrary numbers of animals and dens.

Ethical considerations
The research adhered to ASAB/ABS Guidelines for the Use of Animals in Research, in addition to legal and institutional requirements in Japan and the United States. Collection, care, and export of many small non-commercial octopus species, including O. laqueus, are not regulated in Japan, and permits or licenses from a granting authority were not required. Import of O. laqueus from Okinawa to the Marine Biological Laboratory (MBL) in the United States was done in accordance with all applicable US Customs and US Fish and Wildlife regulations. Care of invertebrates like O. laqueus does not fall under United States Animal Welfare Act regulation, and is omitted from the PHS-NIH Guide for the Care and Use of Laboratory Animals. Thus, an Institutional Animal Care and Use Committee, a Committee on Ethics for Animal Experiments, or other granting authority does not formally review and approve experimental procedures on and care of invertebrate species O. laqueus at the MBL. However, in accordance with MBL Institutional Animal Care and Use Committee guidelines for invertebrates, our care and use of O. laqueus in Japan and in the United States generally followed tenets prescribed by the Animal Welfare Act, including the three "R's" (refining, replacing, and reducing unnecessary animal research), and also generally adhered to recent EU regulations and guidelines on the care and use of cephalopods in research [31].

Collection
O. laqueus were collected at night on low tides close to shore in water five to fifty centimeters deep in Okinawa, Japan from Octobrer 2014 to February 2015 and in November 2015. The animals were commonly seen in holes or dens in sand and reef rubble (Fig 1a) and were often observed within a few meters or less of one another, suggesting that each individual is likely to encounter other individuals on a night of foraging and hunting. On three occasions while diving or intertidal walking, sets of two octopus were observed in dens or holes sufficiently nearby for the animals to touch one another, and it was possible that they were sharing a single den with multiple entrances (S1 Video). As a practical matter, animals can be collected only in the winter, by reef-walking at lowest tides that occur at most once or twice a month, over short intervals of time wherein many participating collectors must act simultaneously without the chance to coordinate their efforts until they return to shore when the tide begins to rise. Subject to these conditions, O. laqueus were easily caught when found outside the den. Typically 5-10 animals were placed in a single bucket with seawater during collection (Fig 1a). To allow acclimation to the laboratory environment, behavioral experiments in tanks did not begin until several days after collection and the onset of feeding.

Tolerance in buckets
To obtain a rough indication of whether O. laqueus might be socially tolerant, five replicates of around ten octopuses were placed in ten-liter buckets with several liters of seawater and without lids over the course of collection in the field. Octopuses were left in buckets for one to four hours and observed periodically. In our experience, many species of octopus immediately start trying-often successfully-to climb out of the buckets.

General culturing
O. laqueus was cultured at the Okinawa Institute of Science and Technology (OIST), where clay pot experiments described below were performed, and at the MBL, where a long-term culturing experiment described below was performed. In contrast to most octopus species that in our experience must be singly cultured to prevent fighting or cannibalism, O. laqueus were easy to care for in group cultures in the lab (Fig 1). Animals in OIST were kept at densities up to one animal per 15 liters with four to fifteen animals in 250 liter tanks with filtration, air, and closed circulation. 10-100% of seawater was refreshed every 1-3 days and water quality was checked periodically (pH, nitrates, nitrites, ammonia). Prime (Seachem) was periodically used to help stabilize conditions for short-term cultures (days to weeks). For a longer-term group culture of several months, three young juveniles (one male and two females, each around ten grams) and freshly collected were shipped from OIST in Okinawa, Japan to the MBL in Woods Hole, MA, United States. At the MBL, animals were maintained together in a 75-liter aquarium and a sand-filtered flow-thru seawater system. Animals appeared surprisingly relaxed, and were kept in open tanks without lids or deterrents at both OIST and the MBL (Fig  1b and 1e). O. laqueus brought into the lab typically began eating within one to two days after collection and accepted freshly killed or store-bought frozen shrimp and crabs without training, in addition to live prey. Seawater at OIST was at room temperature, 21˚C; at the MBL temperature was maintained at 23˚C, as room temperature was much lower.

Identification
Animals were visually identified to species [29] and weighed. Sex identifications were also made based on male curling of the right third arm while moving, and on the presence of two large suckers at the proximal end of the arms in males but not females. For identification, octopuses were tagged with silicone-based fluorescent elastomer (Northwest Marine Technology) that was injected into a small area in the dorsal mantle [2,32] (S2 Video). Because of its potential adverse effect on behavior in days after treatment and due to the risk of mortality, anesthesia was not used. Injected octopuses seemed lethargic immediately after injection but recovered within a few hours or by the next morning. Experiments were not begun until several days after injection to ensure all animals had recovered and were behaving normally.

Social behavior experiments
For social behavior experiments, tagged animals were sorted into four groups of five or six, and each group was cultured in one of four identical circular tanks, with a balance of mixed sizes and sex across tanks of the same treatment when possible (S1-S4 Tables). Animals were maintained on an 11:13 hour light-dark cycle that roughly matched the local light cycle in Okinawa in late November and early December. The tanks were loosely covered with light-proof lids at the onset of the dark cycle to keep out most indoor light, but very dim light was admitted by the mostly but not fully opaque plastic sides of the tank, roughly approximating nocturnal natural light (Fig 1b). Small clay pots (15 cm tall) were used as dens in the tanks. Each pot had 4 large slits along the sides and a hole on top, allowing animals to readily enter and leave a pot and monitor activity outside it (Fig 1c). Clay pots and tanks were scored for animals three hours after the start of the light cycle (Fig 1d). Individuals were identified based on their elastomer tags and their health was generally assessed at this time. To minimize stress from repeated handling, animals were transferred to small individual feeding containers immediately after assessment but prior to feeding. The containers (Critter Keepers) included very small clay pots (5 cm tall) as dens. The containers were returned to the larger tanks after the animals were added. One hour prior to the dark cycle, the feeding containers with animals were moved to the bench top and two live or frozen shrimp or crab were added to each container. At the start of the dark cycle, the small containers were covered to block room lighting. A few hours after adding food to the containers, animals were returned to their main tanks to roam freely. This procedure ensured that each animal was equally and adequately fed and prevented fouling of the main tanks from left-over food, which rotted quickly in the warm conditions.
To quantify social tolerance versus social repulsion through pot occupancy in communal tanks, two social behavior treatments were performed with the experimental setup described above: "Pots Equal" (PE) and "Pots Limited" (PL). To balance sizes, each tank included one to two large, three medium, and one small animal. Ranges for the three sizes classes were determined based on the distribution of animal weights across sexes. The PE treatment was preformed in three configurations: mixed sexes (FM), all-female (FF) and all-male (MM). (i) In the FM case, an equal number of octopuses and pots were placed in a tank, and pot and tank occupancy was scored daily for five or six days, with two replicates, twelve octopus, and eleven tank assessments in total. The male:female ratio was 1:1 in all FM replicates (Table 1). (ii) In the PL treatment, two or three octopus per pot were placed in a tank, and pot and tank occupancy was scored daily for seven days, with two replicates, twelve octopus (one octopus was replaced after Day 1 of the second replicate), and fourteen tank assessments in total ( Table 2). The male:female ratio was 1:1 in the first replicate, but 1:2 in the second replicate because only a limited number of animals was available. (iii) Using the same animals, potential sex-based differences in social tolerance of shared den occupancy were subsequently tested in a second round of PE treatment, with two all-female (FF) replicates scored daily for five days (twelve octopus and ten tank assessments in total (Table 3) and two all-male (MM) replicates scored daily for five days (eleven octopus and ten tank assessments in total (Table 4). Re-sampling of individuals for testing sex-based differences brought some novel animals together within a Table 1. Occupation numbers for mixed sexes, K = 6 animals in N = 6 pots (K f = K m = 3). (a) and (b) are two replicas with equal numbers of females and males. S d = ∑ i n i I[n i − 2] is the daily sharing level and σ ff , σ mm , σ fm are, respectively, the number of female-female, male-male and female-male pairs [Eq (14)]. Bottom lines-the mean values.

Experimental design and statistical power
To estimate statistical power, we defined a daily sharing-level S d (N, K) for K identical animals distributed in N pots. Thus, denoting the number of animals in pot i by n i (i = 1, 2, . . ., N), the Table 3. Occupation numbers for equal number of pots and animals-all females in two replicas (a) and (b). n 0 indicates the number of outsiders that stay out of the pots. S d = ∑ i n i I[n i − 2] is the daily sharing level and σ ff = ∑ i n i (n i − 1)/2 is the number of female-female pairs. bottom lines-the mean values.

Females occupation numbers Sharing #links
Tank Day n 0 n 1 n 2 n 3 n 4 n 5 n 6 S d σ ff (a) https://doi.org/10.1371/journal.pone.0233834.t003 Table 4. Occupation numbers for equal number of pots and animals-all males in two replicas (a) and (b). n 0 indicates the number of outsiders, S d = ∑ i n i I[n i − 2] is the daily sharing level and σ mm = ∑ i n i (n i − 1)/2 is the number of male-male pairs. Bottom lines-the mean values.

Males occupation numbers Sharing #links
Tank Day n 0 n 1 n 2 n 3 n 4 n 5 n 6 S d σ mm (a)

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number of animals is K ¼ P N i¼0 n i where n 0 is the number of 'outsiders', i.e., animals that remain inside the tank but outside all pots. The sharing-level is defined as where the indicator I vanishes for n i = 0, 1 (for a list of symbols cf. Table 5). We estimate the average sharing-level � S d ðN; KÞ and compare this quantity to a model of neutral animals (the null hypothesis) as a function of the number of independent samples M [33].
The neutral model assumes that (i) animals completely ignore each other and therefore can be treated as independent non-interacting particles-volume-fraction of octopus is neglected; and (ii) no animal remains outside a pot, n 0 � 0. One can then think of distributing the animals into pots as rolling an N-sided dice K times. For identical but 'distinguishable'(i.e., labeled) animals, the probability of obtaining a specified configuration {n 1 , n 2 , . . ., n N } of such neutral animals is given by the multinomial distribution Denoting the type-II error by β, we set the probability of type-I error α to 0.05 and calculate the expected power (1 − β) as a function of the ratio � S d =S 0 , where S 0 is the average sharinglevel of the neutral model. The distribution of the sample-mean of M observations, The results for N = K = 6 (S 0 = 3.59), are shown in Fig 2. It then follows, that M = 5 samples are Table 5. List of symbols.

Symbol Description
M # independent observations. n 0 # 'outsiders' that stay in the tank, outside of all pots. O total # of configurations.
S d daily sharing level, defined in Eq (1).
S 0 mean sharing level of identical neutral animals, Eq (2). σ linkage = total # of pairs as defined in Eq (14)

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sufficient to detect a deviation from neutrality in the range � S d � 0:55S 0 (anti-social) and � S d > 1:35S 0 (social) at a power of more than 80%.

Modeling pot occupancies
In the following we obtain a statistical description of pot occupancies that extends beyond neutrality. We employ a maximum entropy principle [34] wherein which distributions and correlations, such as the probability distribution of sharing, are derived from a Hamiltonian. Maximum entropy yields the least structured model consistent with the empirical observations, while quantitatively recapitulating the hypothesis-testing values. Our model might be called the "housemate model" you will let me share your home only if I get along with everyone else in the house; similarly, everyone already living there must get along with one another. Such pairwise propensities, affinities, or proclivities are taken to be independent of and uncorrelated with one another. The probability that n i agents get along with each other decays exponentially with the total number of pairwise interactions between, or distinct pairs of, agents within pot i, n i (n i − 1)/2. Of course the model is simplistic, but with readily achievable values of n i , it could be misleading to try to fit a model with more parameters or degrees of freedom.
We first study the max-ent model with single-sex distribution. Assuming K identical animals distributed among N pots, a configuration of animals up to labeling is completely determined by specifying a set of occupation numbers:ñ ¼ fn 0 ; n 1 ; � � � ; n N g. The mean number of outsiders averaged over M days, and the mean number of pair-interactions, is the Hamiltonian, Z is the partition function, obtained by summing over all configurations, (4) is similar to the bosonic Hubbard model which is well known in condensed matter physics [35]. The parameters introduced in (4) are the on-site (or "contact") interaction U, which can be attractive (U < 0) or repulsive (U > 0), and the chemical potential μ which penalizes the outsiders (μ > 0). The chemical potential is the simplest one-body (linear) contribution to a model and the interaction term is quadratic. Positive values of μ and U describe together a balance between staying inside a den because the open environment is unfavorable, and staying out of a den in order to avoid repulsive den-mates. When U = 0 and μ ! 1, one recovers the neutral model as a special case of Eq (4).
The parameters in Eq (4) are found by imposing the conditions so that empirical-averaging coincides with ensemble-averaging with respect to PðñÞ. Namely [here and in the sequel � x stands for the sample-averaging over a quantity x whereas hxi is the ensemble-averaging], Alternatively, the parameters can be obtained by the maximum likelihood condition, When n 0 � 0, μ is traced out of Eq (4) (i.e., μ ! 1) so that the partition function is independent of the chemical potential, Z = Z(U). As a result, Eq (8) reduces to The linear-response term on the right hand side of (9) is related to the variance of σ by the fluctuation-dissipation theorem [37]. Thus, The model of Eq (4) readily extends to experiments with mixed sex/species, so long as distinct species can share a pot without harming one another: Here n f i ðn m i Þ is the number of females (males) occupying pot i out of N and n f 0 ðn m 0 Þ is the number of outsiders; similarly, the subscripts of U denote sex and take the values m or f accordingly. This model allows different interactions between sexes. For example, U fm � 0 � U ff � U mm would describe animals having attractive between-sex interactions and repulsive withinsex interactions, with the females being more social than the males.
The interaction parameters in Eq (10) are determined by maximum likelihood estimation. In full analogy with Eq (7) wherex � ðm; U ff ; U mm ; U fm Þ. The errors in estimating these parameters are given by the 4 × 4 inverse of the Hessian matrix, evaluated at the maximum likelihood solutionx 0 [compare to Eq (8)]: As a result, the uncertainty levels of the observablesỹ � ðn 0 ; s ff ; s mm ; s fm Þ have two sources: one, due to the intrinsic fluctuations which occur as the animals keep moving between different occupancy configurations, and the other, due to errors in estimating the interaction parameters. Expanding the error-matrix hdŷ i dŷ j i to leading order in (1/M) one finds: where hdx i dx j i is the error-matrix given by Eq (12) and h� � �i c are the 2 and 4-point connected correlations for a set of known parameters. The last term on the right hand side of (13) vanishes as the number of experiments M ! 1. The first term, however, is controlled by the size of the system (decreases as K, N ! 1, while the density ρ = K/N is kept finite) and, therefore, remains relatively large for small systems. After calculating the partition function Z and estimating the interaction parameters and their errors [as is given in Eqs (11) and (12)], one obtains the distribution function of pot occupancies for any numbers of pots and animals (N, K) and for an arbitrary mix of sexes (K = K f + K m ). We follow this procedure in the Results section.

Bucket and tank observations
We did not observe obviously aggressive interactions between O. laqueus individuals in our bucket field experiments, occasional color flashes because of disturbances from researchers with flashlights notwithstanding. Instead, O. laqueus would often sit in buckets with arms or bodies in contact with one another, and at times partially atop one another. In contrast to other species such as Abdopus aculeatus and Octopus incella that were also present in the field but encountered much less frequently, O. laqueus rarely attempted to escape from the open buckets, despite the potential stresses of collection, dense conditions, and limited seawater within the buckets.
Our bucket observations suggested that O. laqueus might thrive in communal aquaria and we explored maintaining them in shared tanks without lids in the lab. We found that for over 100 animals of mixed sexes and a range of sizes, only a few O. laqueus ever escaped or disappeared from open tanks housing 3-15 animals at a time. At least one incident of escape appeared to be related to poor water conditions that arose unexpectedly, while another involved a very young juvenile that was much smaller than any other O. laqueus brought in from the field. Three young juveniles that were shipped from Japan to the United States and raised in an open communal tank for almost five months matured and mated, with hatchlings appearing healthy and with the mother dying naturally from senescence after hatching. These results demonstrate that it is possible to collect, ship, and culture wild-caught O. laqueus for up to several months in open communal tanks with little risk of escape, and that the animals appear to thrive and complete their life cycle, including sexual maturation, mating, and hatching of the next generation.
As expected for a nocturnal octopus, we observed that O. laqueus roamed their open communal tanks at night, hunting and eating prey, and interacting with brief arm or sucker contact that did not appear to be aggressive. Each morning, octopus would select a pot to occupy for the day, rarely remaining outside all pots within the tank. Surprisingly, even in tanks with at least as many pots as octopus, multiple individuals would share a single pot for the day, often within arm's reach or in non-aggressive contact with one another inside the pot. As in our bucket experiments, these observations of non-aggressive co-occupancy of a communal tank over periods of days to months suggest that O. laqueus is much more socially tolerant then we expected based on studies of other octopus species, where octopus are housed in isolation or must be size-matched and well fed. That two or more O. laqueus will share not only a tank but even a pot serving as a den is remarkable, as den sharing in aquaria or in the field was until now unreported for octopus, so far as we know, outside the exceptional occurrence of matepair bonding in Octopus LPSO, wherein a mating male and female will share a den for several days [14].

Replicates and balance in the experimental design
An advantage of laboratory experiments on behavior over those done in the field is that the degree of control often allows the number of replicates and balance in treatments to match the ideal experimental design. However, because animals could be obtained only with considerable difficulty, our control was limited. In this context, the experiments above have a relatively low number of replicates (2 for each of the treatments) and are at times unbalanced with respect to sex or size. Specifically, there are only two replicates of each of the FF, MM, and FM Pots Equal (PE) treatments and of the Pots Limited treatment (PL). Further, size and sex representations are not balanced across a given treatment to a varying degrees for all treatments. The low number of replicates and unbalance in the data versus an ideally balanced experimental design is due largely to limitations encountered in collecting O. laqueus within a limited window to do experiments. Importantly, despite these limitations, we are able to identify with statistical significance that O. laqueus share dens yet they are far from being neutral independent animals.

Observations of pots occupancy
Daily pots occupancies, for both Pots Equal and Pots Limited experiments and observed over 45 days, are shown in Tables 1-4. Each table specifies the number of available pots N, the number of females K f or males K m in the tank (K m + K f = K), the number of animals that were found each day inside pot i, n i (i = 1, . . ., N), and the number animals that stayed in the open space outside the clay pots, n 0 , The tables also specify the daily sharing levels S d [Eq (1)] and the pairwise linkage, i.e., the total number of pairs formed by female-female, male-male and female-male, respectively: As opposed to S d , the pairwise linkage is sensitive both to the sex and to the density of animals in a pot.

Occupancy in Pots Equal experiments.
For Pots Equal experiments, pot occupancy across 31 days of observation having an equal number of octopus and pots (N = K) ranged from zero to three animals in a pot. The total number of sharing animals per day, S d , ranged from zero to five. Specifically, we found that S d � 2 i.e., at least two animals were sharing a pot, in 19 out of 31 days (61%). Then, looking at the subset of N = K = 6 (omitting the all-male N = 5 replicates in Table 4a, and a single incident with n 0 6 ¼ 0 in Table 1b), we found that S d � 2 in 16 out of 25 days (64%). These numbers are sufficiently high to demonstrate that O. laqueus are not totally solitary and can be tolerant of sharing a clay pot or den with one or more individuals. At the same time, it's clear that the animals are far from being neutral (indifferent to the presence of others) because, for K independent animals distributed among N jars, the probability of non-sharing would be [see Eq (2)] P(S d = 0) = K!N −k = 1.5%. Averaging the occupation numbers of all the pots over 25 days, we also found that verifying thereby that all pots are statistically identical (the deviations compared to the expected mean values of � n i ¼ 1 were tested using a 6-level 1-way-ANOVA and turned out to be insignificant with a p-value = 0.23). Thus, clay pot or den selection is not an entirely random process, and there is an anti-social behavioral component at play, keeping pot sharing at levels lower than predicted by a neutral random model. Sex analysis. Sex analysis of the pot occupancy data suggests that the anti-social behavior component is coming primarily from male-male interactions but occurs at statistically significant levels even in all-female tanks. Indeed, the average sharing number of all-females configurations is � S f d ¼ 2:7 whereas the all-males average sharing is � S m d ¼ ð0:8 þ 0:4Þ=2 ¼ 0:6 [for estimation of the error-bars, see Eqs (20a) and (20b)]. Therefore, females are much friendlier than males, however, both sexes are less friendly than neutral animals. For comparison, the average sharing levels of independent animals are: Furthermore, considering the case of mixed sexes (Table 1), one can verify that most of the sharing events (8/10) were by female-male pairs. This tendency persists even in the case of limited number of dens, N < K ( Table 2), so that in all experiments � s fm > � s ff . Occupancy in Pots Limited experiments. For the Pots Limited experiments, pot occupancies across all the experiments having more octopus than pots (N < K) ranged from two to four animals in a pot (Table 2), and the daily sharing numbers ranged accordingly from four to six sharing animals per day. We found that, out of fourteen tank examinations, S d = 4, 6 each occurred twice and S d = 5 occurred 10 times (71%). However, limiting dens also increased the number of 'outsiders'(namely those animals, either females or males, that stay in the open environment outside the pots) so that a solitary octopus was found outside the pots on most days. Specifically, we found that n 0 = 0 occurred only twice (14.3%), n 0 = 1 occurred 11 times (78.6%), and n 0 = 2 happened once (7.1%). Thus, overall, limiting dens increased the amount of social sharing but, at the same time, forced some fraction of the animals to stay out of the dens.

Tests against neutrality
The sample-mean and sample-variance of S d for all the experimental treatments are shown in Table 6. Also shown the mean values of the neutral model, that serves as a null hypothesis, and the corresponding p-values of the one-sample t-tests. The deviations from neutrality are significant for all experimental setups, except FF (all females) with a p-value = 0.07. Obviously, the significance is further increased by pooling two replicates together. As is readily verified by looking at the main lobe of the temporal correlation function CðtÞ ¼ dS d ðmÞdS d ðm À tÞ with dS d � S d À � S d , the correlation time is less than a day in all the experiments. Furthermore, the power ratio of main-lobe to the side-lobes of C(τ) is 8dB and the relaxation time [38] is t 0 ¼ 2 R 1 0 dtCðtÞ ¼ 0:82 days. Thus, practically, the daily measurements are statistically independent.

The effect of treatments
To test for significant fixed effects among daily sharing levels S d , we performed a 4-level-1-way unbalanced ANOVA test using Matlab © . A 2-way-ANOVA is not applicable in our case, due to insufficient degrees of freedom (dof) which leads to singular cross terms (we only have 3 dof at our disposal, whereas a full 2-way analysis would require 5 dof). The ANOVA levels correspond to the above mentioned 4 groups: MM, FF, FM, and PL. As shown in Table 7, the pvalue is extremely low, of the order Oð10 À 9 Þ. This allows us to proceed in trying to identify the significant treatments among the 4 groups using six post-hoc t-tests. To account for varying

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sample sizes, we used the Welch unbalanced two-sample t-tests. As the mean of the mixed FM group lies in between the means of all-females and all-males groups, the differences |FF − FM| and |MM − FM| are insignificant. Other contrasts, including the |FM − PL| between mix sexes at different densities, are statistically significant (see Table 7). These contrasts remain significant after correcting by a factor of 12 (a factor of 6 is the Bonferroni correction and another factor of 2 comes from doing double-sided tests).

Estimation of the maximum entropy interaction parameters
The S d statistic is not sensitive to sex-mixtures and local densities and therefore cannot resolve, for example, the difference between the two replicates of Table 2. A possible way to overcome such limitations, as well as un-balanced and un-factorised experimental designs, is by using the max-ent model. As explained in the Methods section, an estimation of the interaction parametersx ¼ ðm; U ff ; U mm ; U fm Þ involves a computation of the partition function ZðxÞ, and depends on the empirical dataỹ ¼ ð� n 0 ; � s ff ; � s mm ; � s fm Þ given in Table 8. Interaction parameters of single-sex populations. Let's start with the simpler estimation for a single-sex populations, described by Eq (4). In this case, the total number of configurations for K identical animals occupying N pots is where δ � 1 if � n 0 ¼ 0 and zero otherwise (� n 0 6 ¼ 0 means that the animals can dwell somewhere in the tank outside the pots. Combinatorially, this amounts to having an additional available

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'slot'). Referring to the first two rows in Table 8, with N = K = 6 and δ = 1, the number of con- The calculation of the partition function is, therefore, amendable to numerical computation. The partition function (more precisely, the free energy F � −log Z) as a function of U is plotted in Fig 3. Therefore, with F(U) given, and by solving Eq (6) � s ¼ @F=@U for U, we find that As expected, in a non-mixed environment females are friendlier than males. However, compared to neutral animals (U = 0) both sexes exhibit significant repulsive interaction. The t-statistic for the difference between sexes is t ¼ j3:45 À 0:82j= ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 0:36 2 þ 0:66 2 p ¼ 3:50 with a pvalue = 0.002. These results are based on combining two replicates consisting of a total of 10 measurements for each sex (see Tables 3 and 4). For males, since N 1 = K 1 = 5 and N 2 = K 2 = 6, the combined free energy for two replicates is given by a weighted average, and Eq (6) takes the form: is the effective number of links. Eq (19) demonstrates how independent data sets (in this example, unbalanced male replicates 1&2) are compiled together into a single set with proper averaging.
We computed the following quantities as a function of the interaction parameter U (Fig 4): (i) the average number of links hσi = @F/@U (ii) the canonical distribution PðñÞ ¼ gðñÞexp½À HðñÞ�=ZðUÞ and the log-likelihood function W � P M m¼1 log Pðñ m Þ ¼ MðF À U � sÞ and (iii) the fluctuation dÛ according to Eq (9). As a consistency check of the model, we also calculated the average sharing-number in terms of the canonical distribution PðñÞ. Namely, hS d i ¼ Pñ PðñÞd½S d À P i n i Iðn i À 2Þ�. Note that, by its construction, the canonical distribution PðñÞ always reproduces the average number of links hsi ¼ � s. However, functions

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like P obs (S d ) and P cal (S d ), i.e., the empirical and derived distributions of S d , are more complicated objects and, as such, they don't necessarily need to agree with each other [consistency is nevertheless maintained, because D[P obs (S d )||P cal (S d )] is minimized exactly at the same value of U which solves the maximum likelihood condition � s ¼ @F=@U]. We found that the average sharing numbers, calculated at the corresponding maximum likelihood solutions (18)  In particular, the estimated errors in Eq (20a) are smaller than the empirical ones and, as shown in Fig 5, the empirical values lay well within the estimated confidence levels. Since, � S d ðU ¼ 3Þ ' 1 (Fig 5), it follows that, for U � 3 one typically observes at least one pot with sharing animals, whereas for U > 3 sharing is much suppressed. Also note that both values in (20a) differ significantly from the expected sharing level of neutral animals, Eq (16). The full sharing distribution as a function of the interaction parameter, P S d ðkjUÞ {k = 0, 2, . . ., K}, is shown in Fig 6 (for N = K = 6). We find that the non-sharing probability P 0 � P S d ðk ¼ 0jUÞ, evaluated at the maximum likelihood points Eq (18), is P f 0 ¼ 7% for females and P m 0 ¼ 67% for males. Clearly, both values are larger than the non-sharing probability of neutral animals. More generally, we examined the Kullback-Leibler distance between the empirical sharing distribution, P obs (k) = M −1 ∑ m δ[S d (m) − k], and the probability P S d ðkjUÞ calculated as a function of U by using the distribution function PðñÞ. We found (Fig 7), that the KL-distance D[P obs (S d )||P cal (S d |U)] assumes its minimal value-respectively for females and males, at U = (0.82, 3.45) which is again very close to the maximum likelihood solution Eq (18). Remarkably, this holds even though the number of observations, M = 10, is pretty small.

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In addition, the one-parameter model H 1 ðñÞ ¼ ðU=2Þ P i n i ðn i À 1Þ, resulting from Eq (4) by setting μ ! 1, has the smaller AIC as compared other polynomial models (see Table 9 as well as Fig 8).

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The female-female interaction is consistent with the previous result [Eq (18)] obtained in a single-sex environment. The chemical potential μ being of the same order of magnitude as U ff is sufficient to prevent females from staying outside the pots. The female-male interaction U fm is much less repulsive than either U ff or U mm . The error estimates in (22) are obtained, as in (12), by calculating the Gaussian fluctuation of the free-energy at the maximum-likelihood solution.
Next, we considered the case of dense pots N = 2 < K f + K m = 6 For tank #4, containing 4 females and 2 males that are sharing 2 pots, Eq (21) gives O ¼ 6

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configurations. For tank #2, with 3 females and 3 males O ¼ 5 3 À � 2 ¼ 100. Such small number of configurations enables one to obtain the exact partition function and infer the four coupling constants of HðñÞ. In practice however, the number of samples M = 7 is also very small, so the expected accuracy of these parameters is rather low. The results are summarized in Table 10. Tank #4 looks promising: females are as social as males and the f-m interaction is on the verge   This 'anomaly' can be traced back to a high degree of individual variety (it turns out, see S4 Table, that a certain large female, named 2RG, sits most of the time out of the pots and seems to be extremely anti-social). The combined interaction parameters. All the experimental results can be treated on the same footing by combining the interaction parameters, obtained separately under different experimental conditions, into a single set of properly weighed parameters, as is done in Eq (19). Referring to Table 10 and combining together the results of five different setups, (1+2), (3 +4), (5+6), (7) and (8) [see also Eqs (18), (22), (23) and (24)], we find: The t-statistic for the difference between sexes in Eq (25) is t = 2.01. This contrast is lower than the corresponding single-sex statistic [Eq (18)]. However, it's still significant with a p-value = 0.025. Eq (25) specifies the most probable set of interaction parameters that are consistent with the total of 45 available measurements. These values can be used in 2 ways: first, for identifying potential outliers and second, for the prediction of the behavior over a large set of experimental designs. As an example, let's consider K = 6 animals distributed among a varying number of pots N = (1, 2, . . ., 8) with several possible mixtures of sexes, K f = 0, 1, . . ., 6 (K m = K − K f ). In this case, all quantities of interest, such as the number of outsiders n 0 or the female-male linkage σ fm (which may well affect factors like potential mating, rate of cannibalism etc.), are determined by two parameters: the specific volume N/K and the sex mixture K f /K.
In Fig 9a and 9b, hn 0 i, hσ fm i are shown as functions of N and K f . As expected, both hn 0 i and hσ fm i assume their maximal values when the number of pots is limited (N = 2) and the mixture of sexes is balanced (K f = K m ). Fig 9 suggests that the two empirical points (b, c), described by Eqs (23) and (24), lay reasonably close to the respectively calculated curves. On the other hand, the point (a) corresponding to Eq (22), forms an 'outlier'. This discrepancy can be attributed to the unusual total lack of male-male sharing as seen in Table 1. (see the levels of confidence in Fig 10). Fig 10 demonstrates the tradeoff between gain by having a high female-male linkage and loss to a large number of outsiders. Thus, as one increases the density of animals, by reducing the number of pots, hn 0 i and hσ fm i start growing together and keep increasing monotonically, until reaching a turning-point (in our case, that point is specified as N = 2) where further increase of the density causes a decrease of hσ fm i, accompanied by continuing increase of hn 0 i. Fig 10 also presents the expected uncertainties in n 0 and σ fm which are essential for making comparison with experiments, especially for small systems [see Eq (13)]. The opposite case of a large system 1 � N � K is of particular interest. Referring to Eq (4) and setting r � μ/U, we find that for weak interaction the density ρ � K/N and the average linkage per den ξ � hσi/N are smooth functions of r. However, as U increases (U ' 4π), ρ and ξ cross over to staircaselike curves (Fig 11) which resembles the Mott-Hubbard transition [39].

Conclusion
In pioneering studies of "use of space"by the octopus O. joubini, the psychologist J. Mather characterized spatial distributions of octopus in the lab and in the wild [1,7]. Minimal, if any, spatial organization was evident, a striking contrast to other cephalopods such as squid and certain species of cuttlefish that at least before maturity are often observed in schools [40,41]. Mather studied densities of individuals, potential clustering indicative of spatial inhomogeneity, and measures of hierarchy/dominance and territoriality. As she observed, aggregation does not necessarily reflect interaction.
One of her principal conclusions can be recast in standard physics terminology as the finding that over characteristic scales exceeding a few octopus lengths, static spatial octopus distributions-although not homogeneous in the lab, where edge effects asserted themselves-are fully determined by single-particle correlations (one-body densities) and exhibit no order even at short ranges. That is, at least insofar as her measurements were concerned, observed densities of these live animals were indistinguishable from those of classical non-interacting particles in an external field, in this instance the local seafloor shell density that presumably reflected prey abundance.
Mather suggested [15,30] that the term "asocial"is a suitable designation for this behavior wherein the animals ignore one another. Unfortunately, certain pivotal quantities such as mean free path and impact parameter that would pin down how often animals approached one another within a specified distance.
A pivotal, if implicit, contribution of Mather was her elucidation of a link between animals' "use of space"-simplistically, their spatial organization, although temporal components may also be relevant-and "social behavior."Spatial organization is measurable even when attribution of observed use of space specifically to social behavior-or even to animate agency-is uncertain, and may itself not be quantifiable. "Sociality"is an elusive concept; the difficulties that arise in trying to define it crystallize in robotics, for example, wherein inanimate objects  N � K). In red-weak interaction, ρ = r + 1/2 and ξ = r 2 /2. In blue-strong repulsive interaction U ' 12. https://doi.org/10.1371/journal.pone.0233834.g011 can exhibit collective swarm-like behaviors [42]. Once living organisms are viewed as wetware machinery, the arbitrariness inherent to any particular definition of "sociality"is uncontroversial. Nevertheless, once one has in mind a specific purpose, definitions of sociality customized to achieve clearly articulated predictions of behavior on explicitly stated terms may become possible. Thus, as discussed at the 2018 Aspen Center for Physics workshop on 'Physics of behavior', any quantitative measure of 'sociality' is heavily dependent on context.
For these reasons, we revisited the ideas raised by Mather in a slightly different context and with modern quantitative tools. We were motivated by field observations of unanticipated behavior of a native Okinawan octopus that frequents coral reefs, O. laqueus. Individuals were observed anecdotally in our field expeditions to share dens, which cephalopod specialists found surprising for what they customarily regard as an asocial genus. We studied O. laqueus under laboratory configurations wherein we could vary the number of dens and octopus within a tank and directly measure how multiple occupancy depended on those values. We inferred parameters of a minimal model to maintain predictive value, and we painstakingly characterized uncertainty, so that our findings can in principle be invalidated. Our study is of potential importance for humane laboratory and industrial culture of cephalopods under conditions wherein they share a tank.
We aim to develop reproducible laboratory measures that reflect (and eventually predict) field observations that could be relevant for successful commercial culture of the animal. The field observations reported here of octopus O. laqueus engaging in den-sharing, a behavior which is thought to be atypical of most octopus species, could indicate that they are more readily cultured in the lab without cannibalization than are other species of octopus. Anecdotal evidence suggested that O. laqueus individuals tolerate one another: field observations of two animals apparently sharing the same den; the willingness of multiple individuals to cohabit indefinitely within a single tank without a lid, a condition wherein many octopus species would-in our experience-flee the tank to certain death in a dark corner of the lab. The challenge is to move beyond anecdote. As with all biological systems, experiments in the lab and their modeling often come at the cost of artificial or unnatural settings. Octopuses that are not well-fed, for example, may harm one another, but EU guidelines and animal welfare considerations preclude keeping octopus under conditions wherein they may be subject to harm.
Our anecdotal observation of den sharing in the field, first reported here, suggested to us that den sharing could be recast into a laboratory measure that might plausibly reflect certain aspects of sociality. In our hands, O. laqueus in laboratory tanks equipped with clay pots, exhibit distinctive behavior wherein they explore the dens in the morning hours before settling in for the day. Indeed, it is this observation-suggestive of ergodicity-that could account for the apparent validity of the equilibrium theory invoked here. Den sharing provides a readily measurable observable amenable to parameterization by number of dens and number of animals. Because our measure was crude, we were able to establish statistical uncertainty by assessing the independence of measurements with a suitably-defined correlation time without which statistical characterizations often performed in the literature on sociality are rendered meaningless.
We studied the social tolerance of O. laqueus by measuring the den occupancy of dens in the lab for varying densities of animals and several sex-mixtures. We found that O. laqueus tolerate other individuals by sharing tanks and dens, with typically no loss to cannibalism or escape. However, animals also exhibit significant levels of repulsion, and individuals often chose a solitary den when given the option. The patterns of den occupancy were studied with a maximum entropy model that treated animals as particles with on-site pair interaction. The three interaction parameters that determine the amount of social attraction/repulsion between animals according to sex, together with the chemical potential that confines animals to dens, were estimated from the experiment by a standard maximum likelihood calculation. The parameters computed in this way were then used to characterize the social behavior in large set of experimental conditions and to identify potential outliers. This procedure, as well as the general applicability of a maximum entropy model in this context, remain to be verified in future experiments with independently obtained or larger sample statistics.