Eusociality outcompetes egalitarian and solitary strategies when resources are limited and reproduction is costly

Abstract Explaining the evolution and maintenance of animal groups remains a challenge. Surprisingly, fundamental ecological factors, such as resource variance and competition for limited resources, tend to be ignored in models of cooperation. We use a mathematical model previously developed to quantify the influence of different group sizes on resource use efficiency in egalitarian groups and extend its scope to groups with severe reproductive skew (eusocial groups). Accounting for resource limitation, the model allows calculation of optimal group sizes (highest resource use efficiency) and equilibrium population sizes in egalitarian as well as eusocial groups for a broad spectrum of environmental conditions (variance of resource supply). We show that, in contrast to egalitarian groups, eusocial groups may not only reduce variance in resource supply for survival, thus reducing the risk of starvation, they may also increase variance in resource supply for reproduction. The latter effect allows reproduction even in situations when resources are scarce. These two facets of eusocial groups, resource sharing for survival and resource pooling for reproduction, constitute two beneficial mechanisms of group formation. In a majority of environmental situations, these two benefits of eusociality increase resource use efficiency and lead to supersaturation—a strong increase in carrying capacity. The increase in resource use efficiency provides indirect benefits to group members even for low intra‐group relatedness and may represent one potential explanation for the evolution and especially the maintenance of eusociality and cooperative breeding.

. In this context, group formation has traditionally been seen as a risk-averse mechanism reducing the variance in resource supply (Caraco, 1981;Caraco, Uetz, Gillespie, & Giraldeau, 1995;Clark & Mangel, 1986;Uetz, 1996;Uetz & Hieber, 1997;Wenzel & Pickering, 1991). The simple idea behind these models is that foraging success may vary: individuals may find more resources than what they are able to fully utilize, or alternatively, they may not find any resources at all, which leads to certain death (Figure 1b). Foraging with subsequent egalitarian resource sharing in groups allows animals to dampen such environmental variance (Figure 1a), as all group members will receive an intermediate amount of resources which may guarantee survival and reproduction (see also ).
Yet, as Poethke and Liebig (2008) point out, group formation is not necessarily a variance-reducing mechanism. It may be seen as an important means of variance manipulation in general: whether variance in resource availability is reduced or increased depends on the degree of reproductive division of labor. While egalitarian resource allocation decreases intra-group variance as explained above (Figure 1a), skewed resource allocation, by contrast, increases variance ( Figure 1c). If resource availability and variance are low, solitary foragers may collect more resources than needed for survival, but not enough to reproduce in one reproductive period. If individuals forage, subsequently pool the surplus of resources not needed for survival and then redirect this surplus toward one (or a few) individual(s), individuals in such groups will survive and specific group members have a chance to reproduce. Throughout our analysis, we will use the term "eusocial" for highly skewed reproduction with one reproductive individual only. Depending on the shape of the fertility function such skewed reproduction clearly may increase fitness, either through direct fitness benefits (for the reproductive individual) or through indirect fitness benefits (via intra-group relatedness).
Thus, Poethke and Liebig (2008) suggest that egalitarian groups, as a risk-reducing foraging strategy, should be favored in environments with high resource variance and eusocial animal groups should be favored in habitats with low resource variance, since this group structure increases inter-individual variance.
Yet, in nature, egalitarian animal groups are only rarely found (Packer, Pusey, & Eberly, 2001). We assume that this discrepancy between model predictions and empirical observations stems from the fact that previous theoretical work on eusocial group formation as a risk-sensitive foraging strategy accounts for a limited individual foraging rate (including Poethke & Liebig, 2008) but ignores the feedback of competition for limited resources, that is, the interaction between population size and resource availability. More concretely, while the relevant models may assume foraging success to be a function of forager strategies, foraging success is usually not modeled as being a function of the emerging number of individuals, that is, competition for limited resources which leads to density dependence at the population level. However, competition for resources has been shown to be of high relevance in the context of risk-sensitive foraging in general (Fronhofer, Pasurka, Poitrineau, Mitesser, & Poethke, 2011) and for egalitarian resource sharing in particular . The latter work clearly demonstrates that including resource limitation into models of egalitarian resource sharing yields a more complex evolutionary pattern than the simple dichotomy of risk-prone or risk-averse behavior.
In the following we will extend the work of , which focuses on egalitarian resource sharing, to groups with reproductive skew. We will thus compare two types of cooperative animal groups: On the one hand, individuals forming egalitarian groups forage and subsequently share the pooled resources so that every group member receives roughly the same amount of resources (examples include lions and social spiders: Packer et al., 2001;Whitehouse & Lubin, 2005). On the other hand, one can find animal groups in which just one individual receives all the resources for reproduction while the other members F I G U R E 1 Schematic comparison of different modes of resource sharing in groups of animals. Individuals collect resources individually and vary in their success over one reproductive period (i.e., a season). Solitary individuals (b) thus differ in the amount of resources individuals may use for survival (light gray) and reproduction (dark gray). Some individuals (3, 5, and N) will die of starvation and not reproduce because they are not able to cross the survival threshold (dashed line). In egalitarian groups (a), resources are evenly shared after solitary foraging (b). All individuals can survive and receive a small amount for reproduction. In eusocial groups (c), individuals receive sufficient resources for survival and channel all remaining resources to the reproductive dominant individual (here individual 1). Of course, the evolutionary advantages of these resource sharing strategies will be impacted by how resources are exactly translated into survival and offspring, that is, the mortality and fertility functions, which is beyond the scope of this schematic representation. Note that the term "eusocial" group in this context only implies that only one individual will reproduce (a) (b) (c) E of the group only obtain a share necessary for their survival (Wilson, 1971;Clutton-Brock, West, Ratnieks, & Foley, 2009, for example, eusocial insects or mole-rats). Such groups are henceforth termed "eusocial." The term "despotic," which one may also find in the literature, is equivalent here. Of course, egalitarian and eusocial groups are rare situations at the ends of a continuum of different degrees of reproductive division of labor (that is, skew, for a review see Reeve & Keller, 2001). Evidently, as Sherman, Lacey, Reeve, and Keller (1995) point out, other degrees of reproductive division of labor in between these two extremes are possible and often encountered (for numerous examples from insect societies alone see Wilson, 1971;Hölldobler & Wilson, 1990;Costa, 2006;Hölldobler & Wilson, 2009). Nevertheless, theory suggests that these extremes may be favored over intermediate strategies (Cooper & West, 2018). We will here focus on the extremes for simplicity.
We will calculate the amount of resources required by groups across different group sizes in population equilibrium. This allows us to compare the competitive ability of the different strategies (type T ∊ {"egalitarian", "eusocial"} and size of group N ∊ {1, 2, 3, …}) and to determine optimum group sizes for egalitarian and eusocial groups by identifying the group sizes that are most competitive, that is, minimize the amount of resources required at equilibrium. We find that, in contrast to egalitarian groups, eusocial groups may not only reduce variance in resource supply for survival, thus reducing the risk of starvation, they may also increase variance in resource supply for reproduction within the group. The latter effect allows reproduction even in situations when resources are scarce, which gives eusocial groups a competitive advantage over egalitarian groups and solitary strategies, specifically when resources are limiting and reproduction is costly. At the population level, this competitive advantage leads to increased carrying capacities, a phenomenon which has been termed "supersaturation" in cooperatively breeding birds (Dickinson & Hatchwell, 2004).

| Resource availability
We assume stochastic foraging, that is, individual foraging success follows a random distribution and the per capita probability of collecting an amount x of resources during one reproductive period is given by a probability density function P (x,x, ). For the sake of simplicity, we assume that individuals collect resource items of limited size. Thus variance in foraging success is determined by a mean resource item size θ and a distribution of resources can easily be described by a Gamma distribution: with mean x, scale parameter θ, and the gamma function Γ (Andrews, Askey, & Roy, 2001).
For integer ratios =x −1 , this Gamma distribution results from summing up κ independent, identical and exponentially distributed random variables with scale parameter θ (and thus mean θ). Such a random number can be interpreted as the size of an item collected during one of κ foraging trips of a single individual. The variance of acquired resources is 2 =x and the coefficient of variation . For a constant mean amount of resources x collected by an individual, an increase in item size will necessarily be accompanied by an increase in the variance of the amount of resources collected ( Figure 2a). In the following, we will therefore use mean resource item size θ as a proxy for environmental variance.

| Fertility and mortality
We assume that individual mortality M is a function of the amount of resources x s allocated to survival and, as a simplification, we use a step function. We therefore assume that an animal dies if it receives Effect of group size (N) Resources, x Probability density, P with the resource independent baseline mortality M b , resulting from predation or disease, for instance. We previously analyzed the influence of including a sigmoid function for mortality and could show that this does not change our results qualitatively . where F max determines fecundity, that is, the maximal value the reproduction function can take. For low values of x r , the steepness of the fertility function is determined by 1/c 0 . Therefore, c 0 can be interpreted as the cost of reproduction. For an overview of parameter combinations under consideration see Table 1.   yields an implicit relation that allows us to determine the influence of group size N and resource item size θ on the minimal mean amount of resources x N per individual needed to balance reproduction and mortality (Figure 4a). The carrying capacity is then determined by the total amount of resources available (X) and the minimal amount of resources required per individual x N : If current total population size N tot is less than K, the mean per capita amount of resources available for individuals (x) is greater than x N resulting in an increase in population size as natality is greater than mortality. The opposite is the case if N tot > K.
We assume that the evolution of an optimal grouping strategy (group size and type) will increase resource use efficiency, thus minimizing resource requirement in equilibrium and maximizing carrying capacity (see among others MacArthur, 1962;MacArthur & Wilson, 1967;Boyce, 1984;Lande, Engen, & Saether, 2009). A minimization of x N (Figure 4) thus allows us to determine the optimal group size N opt Fronhofer, Pasurka, Poitrineau, et al., 2011). Clearly, it is well known that evolution does not generally maximize carrying capacity (e.g., Fronhofer & Altermatt, 2015;Fronhofer, Nitsche, & Altermatt, 2017;Matessi & Gatto, 1984;Reznick, Bryant, & Bashey, 2002). In order to show that, under the model assumptions outlined above, optimal strategies that maximize carrying capacity are indeed continuously stable strategies, we compare the results of our optimality approach with an invasibility analysis in the Appendix S1.
Strictly speaking, our reasoning only holds if all group members have the possibility to reproduce, which, of course, is given in egalitarian groups and holds for eusocial groups if the reproductive individual is determined by a lottery. However, in eusocial groups, subordinates may never be able to reproduce. In the latter case, the optimal group size derived as described above may not be evolutionarily stable, as subordinates will mainly benefit from indirect fitness gains via relatedness. This implies that we have to take into account the degree of intra-group relatedness. For simplicity, we will first present results that hold true if the reproductive individual of eusocial groups is defined by a lottery or if intra-group relatedness equals 1. We will then relax this assumption, introduce intra-group relatedness <1 into our model and explore its robustness in the section "Joining or leaving a group: evolutionary stability of eusocial groups" below.

| Optimal group sizes and minimum resource requirements
As Equation 8 cannot be solved analytically, we approximated the results numerically. Figure 5 gives the resulting mean amount of resources needed at population equilibrium and the correspondent optimal group sizes for a broad range of environmental vari-  Clearly, these three phases result from our choice of a sigmoid fertility function (Equation 5) which exhibits both convex and concave parts. Our fundamental reasoning based on Jensen's inequality (Ruel & Ayres, 1999) of course also applies for other shapes of fertility functions. However, specifically functions that are purely convex or concave will modulate results: for instance, increased convexity will increase the potential benefit of eusocial groups, as pointed out above.

| The effect of fecundity and mortality
The success of eusocial groups depends on the ability of the reproductive individual to effectively use the resources it receives from members of the group. This ability is, however, critically limited by the maximum reproductive capacity F max which limits the amount of baseline mortality that can be compensated by reproduction. Thus, at population equilibrium, the group size of eusocial groups is limited . It increases with increasing fecundity F max (compare

| Joining or leaving a group: evolutionary stability of eusocial groups
For all results presented above, we assume that, whenever a large population of groups of size N pop > 1 utilizes resources more efficiently (that is, reaches a higher carrying capacity) than a population of solitary individuals it can, in principle, not be invaded by individuals following a solitary strategy. This phenomenon is known from cooperatively breeding birds as "supersaturation" (Dickinson & Hatchwell, 2004). The group strategy of size N pop > 1 evolves because, at population equilibrium, the groups would drive mean resource availability below the critical value that allows the growth of a solitary strategy.
As pointed out above, our results hold true for egalitarian groups in general and for eusocial groups if the reproductive individual is determined by a lottery or if intra-group relatedness equals 1.
However, in eusocial groups, the reproductive individual is often not determined by a lottery and intra-group relatedness is likely <1.
Therefore, direct fitness of subordinates is zero and the advantage of subordinates living in a eusocial group is solely determined by indirect fitness benefits, that is, by its relatedness to the offspring of the dominant individual (Hamilton, 1964a,b), if we ignore other direct benefits such as queuing for a dominant position (see, for example, Kokko & Johnstone, 1999 4) of individuals as Ψ = (N,x, ) (N,x, ) . Ψ is a function of the size N of the group an individual is a member of, the mean size of resource items θ collected by individuals and the mean amount of resources collected x. As the latter is itself an emergent property resulting from intra-specific competition in an equilibrium population of groups of size N pop , we may denote it as Ψ (N, N pop , θ). Note that, strictly speaking, the inclusive fitness approach only holds as long as a strategy is not more likely to interact with itself than with unrelated strategies (for a detailed discussion see, for example, Hines & Maynard Smith, 1979). Subordinates should leave the group whenever leaving would result in a net increase in inclusive fitness, that is, when Equation 12 allows to derive the minimum relatedness r min preventing individuals from leaving a group, that is, the minimum relatedness that allows the evolutionary stability of eusocial groups (see Figure 6). Figure 6) show that, particularly for low environmental variance θ, the benefit of eusocial groups is sufficient to make the role of subordinate group members attractive even for individuals only modestly related to (N,N, ).

| D ISCUSS I ON
In contrast to the work of Poethke and Liebig (2008), the present model explicitly quantifies birth and death rates as functions of resource availability. This allows us to take into account competition for resources between individuals (see also Pen & Weissing, 2000) which reduces resource availability and ultimately results in selection for resource-use efficiency. Our results show that, for a broad spectrum of model parameters, cooperative breeding with resource sharing, and in particular, the formation of eusocial groups with extreme reproductive skew, may substantially increase carrying capacity (Figure 5a-d). This will lead to the competitive exclusion of solitary foragers and breeders, a phenomenon known from cooperatively breeding birds as "supersaturation" (Dickinson & Hatchwell, 2004) which also makes a reversion to solitary breeding less likely.
Our results demonstrate the potential of variance manipulation as a driving force for the evolution of cooperative animal groups.
It may thus have contributed to the evolution of eusocial animal groups. More importantly, the demonstrated ecological benefit of group formation may have been important for the stabilization of cooperative breeding or eusociality after the transition from solitary life had already occurred, as our model does not explicitly consider the initial mechanism of group formation. Our model provides an ecological explanation for the benefit of group formation which sets it apart from previous models of reproductive skew (Johnstone, 2000;Reeve & Keller, 2001;Vehrencamp, 1983) that are often based on a predefined arbitrary benefit of group formation. In our simple consumer-resource model such an assumption is not required, as group formation evolves because of the emergent advantages of variance manipulation.
As mentioned earlier, Poethke and Liebig (2008) demonstrate that egalitarian group formation, a variance reducing foraging strategy, is favored at high resource variances and that, by contrast, eusocial groups or cooperative breeding is advantageous when resource variance is low, because this strategy increases inter-individual variance in resource supply. However, when competition for resources is taken into account, as in the present study as a result of our population equilibrium assumption (Equation 8), these predictions change.
Eusocial groups remain at a clear advantage for low resource variances but become advantageous even for intermediate and rather high variance in resource availability (see Figure 5). This is due to the beneficial effects of eusocial groups on resource variance: (a) inter-individual variance is indeed increased for reproduction, which makes reproduction possible even when solitary individuals do not collect sufficient resources for survival and reproduction. (b) At the same time, for survival the opposite is true, individuals that do not collect sufficient resources for survival as solitaries may survive in the group because they profit from resource sharing. The combined effect of these two mechanisms may explain the dominance of eusociality over egalitarian group.

| Model limitations
Throughout this work, we have analyzed the formation of eusocial groups under equilibrium conditions. However, in a temporally and spatially heterogeneous landscape, and particularly in a metapopulation (Fronhofer, Kubisch, Hilker, Hovestadt, & Poethke, 2012), one will always find local populations that have not reached equilibrium density, yet. In newly colonized local habitat patches, for example, resources will usually be rather abundant and competition will be weak. This will necessarily favor solitary strategies with their high potential offspring numbers. Thus, landscape fragmentation and temporal heterogeneity in resource availability may lead to the coexistence of eusocial and solitary strategies. While we do analyze the consequences of relatedness, and show that the ecological benefits of eusociality may be very large, which makes eusocial groups evolutionarily stable even at low levels of relatedness, our modeling procedure implicitly assumes that groups have already been formed and ignores the process of group formation. Group formation strategies are diverse and include the establishment of entire colonies after nest foundation by a single or few individuals as observed in halictid bees or wasps, for example, but also establishment after colony fission as in highly eusocial insect species like honeybees or ants. Our omission of the group formation process necessarily limits the scope of our analyses and highlights that our model may be best thought of as showcasing ecological benefits that are relevant for the maintenance and increase in size of already existing eusocial groups. Note that these restrictions do not apply to eusocial groups in which all members initially have a chance to become the dominant individual. Such groups can evolve by mutualism and indirect fitness benefits via relatedness are not necessary (see e.g., Rissing, Pollock, Higgins, Hagen, & Smith, 1989).
Furthermore, it is important to note that our inclusive fitness analysis is heuristic in the sense that we do not use an explicit model of evolutionary competition between different strategies. As Olejarz, Allen, Veller, and Nowak (2015) have shown recently, invasibility and stability of an altruistic allele need not be linear in any relatedness parameter and our analysis must therefore be seen as a conceptual extension of our model and not as an in-depth analysis.
In addition to the points discussed with regards to relatedness, in both the egalitarian and eusocial case resource redistribution rules according to group type might be violated by cheating individuals which try to increase their reproductive share. However, this additional level of complexity is out of the scope of our approach and but has been analyzed elsewhere, for instance by Hamilton (2004), Wenseleers, Helanterä, Hart, and Ratnieks (2004), or Schneider and Bilde (2008).
A further limitation of our model is its comparison of only the two extreme cases of group formation (egalitarian vs. eusocial groups), while in nature one will observe a continuum of cooperative strategies (see, e.g., Sherman et al., 1995, but see Cooper & West, 2018. While this may impact our results quantitatively, the two beneficial mechanisms of eusocial group formation discussed above remain potentially important ecological mechanisms responsible for the evolution and maintenance of eusocial groups. Of course, other factors (e.g., reviewed in Krause & Ruxton, 2002;Nowak, 2006;Lehmann & Keller, 2006) will also play a role for the evolution of eusociality and the relative importance of the different mechanisms may vary. Nevertheless, our model is general in the sense that dealing with limited resources and variance in resource supply are challenges likely faced by a majority of organisms.
Clearly, the ecological conditions we consider exclusively relate to the distribution and especially the variance in resource supply.
While our model shows the relevance of intraspecific competition for resources, we do not consider interspecific competition or predation, for instance (see Rankin, López-Sepulcre, Foster, & Kokko, 2007;Tsuji, 2013).
Finally, the assumption that evolution will minimize resource requirement and therefore maximize carrying capacity is valid for our model (see Appendix S1 for an invasibility analysis). However, this hinges upon our description of the resource distribution (Equation 1) and the implicit assumption that the environmental resource distribution itself does not change over time (see Appendix S1). Therefore, our results are valid for consumers that feed on abiotic, renewing resources or for other consumer-resource systems in which assimilation efficiency is maximized (see also Fronhofer & Altermatt, 2015;Matessi & Gatto, 1984;Reznick et al., 2002).

| Empirical examples
It is interesting to note that, in our model, the increase in carrying capacity is generally more pronounced in eusocial than in egalitarian groups. Our model thus suggests that eusocial groups should dominate for a majority of environmental settings and life-history strategies. Although our model is very simple and compares only the extreme cases of egalitarian and eusocial groups, the dominance of eusocial groups in nature can be observed empirically: most cooperative societies are eusocial while truly egalitarian groups seem to be rare (Packer et al., 2001).
Typical eusocial groups are found among insects. In accordance with our model, the ubiquitously present and very successful ants alone show a fascinating array of different life-history strategies and feed on resources with typically low but also high variance (Hölldobler & Wilson, 1990). Interestingly, egalitarian societies have been reported from two ant species, Ooceraea (formerly Cerapachys) biroi (Tsuji & Yamauchi, 1995) and Pristomyrmex punctatus (Tsuji & Dobata, 2011), in which all workers reproduce and help others. While we can only speculate with regards to the evolutionary forces responsible for this secondarily evolved egalitarian behavior, Pristomyrmex punctatus shows rather low fecundities (Tsuji, 1988) and their nomadic life history may suggest important variance in resources, which is in line with our model predictions.
While these examples come from highly derived insect societies, our model may be more appropriate for primitively eusocial insects where subordinates are not sterile, for instance. An additional example are polistine wasps (reviewed in the context of skew theory in Reeve & Keller, 2001): While in the founding phase of a wasp nest the chance of becoming the reproductively dominant will make joining another female an attractive strategy, the probability to stay and accept the role of a "worker" will ultimately depend on the relatedness with the reproductively dominant individual. However, when an expensive nest is a prerequisite of successful reproduction this will change the shape of the fertility function. Such primary investments may be modeled as an offset that shifts the fertility function toward higher amounts of resources needed .
Additional investments make reproduction more costly and will thus severely reduce the relatedness r min (see Figure 6c) necessary to stabilize eusocial groups.
Cooperative systems with non-reproductive helpers can also be found in cooperatively breeding birds (Dickinson & Hatchwell, 2004) and the phenomenon of "supersaturation" has been well described in his context. In line with our results that predict an advantage of eusocial groups at low baseline mortalities, Arnold and Owens (1998) report that cooperatively breeding birds that demonstrate some reproductive skew are generally characterized by low mortality rates. Furthermore, cooperative breeding seems to be consistently associated with low environmental variance in nature (Arnold & Owens, 1998Ford, Bell, Nias, & Noske, 1988;Gonzalez, Sheldon, & Tobias, 2013), although Jetz and Rubenstein (2011) find evidence for the opposite pattern. Our model corroborates these findings as it predicts an advantage for cooperative breeding and eusocial groups for both low and high resource variance.
By contrast, eusocial societies are rare in mammals (Clutton-Brock et al., 2009). Cooperative breeding with high reproductive skew or eusociality has only evolved in four taxa: marmosets and tamarins, dogs, diurnal mongooses and African mole-rats. Typically, females in these groups show unusually high levels of fecundity.
Of course, also some examples of egalitarian groups are known.
Social spiders have been discussed at length elsewhere (e.g., . Our model predicts that egalitarian animal societies evolve when resource variance is high and offspring are few. These life-history traits are typically found in large mammals like lions (Packer et al., 2001) which do form egalitarian groups.
All these examples show that global patterns of the occurrence of eusocial and cooperatively breeding groups in natural arthropod and vertebrate systems can, at least tentatively, be explained by the above presented model, specifically by the influence of resource variance and life-history parameters (offspring cost and number), despite its great simplicity and caveats.

| CON CLUS IONS
In egalitarian as well as in eusocial groups, pooling of resources reduces the risk of starvation. In eusocial groups, it has the additional effect that it may increase intra-group variance in the amount of resources individuals may invest in reproduction. For upward convex fertility functions, eusocial groups thus out-compete solitary individuals as well as egalitarian groups. Whenever population growth is limited by resource availability, resources will necessarily be scarce and reproductive output will be dominantly determined by the convex part of the fertility function.
We show that in situations of limited food supply risk-sensitive group formation has the potential to lead to the evolution of cooperative breeding and eusociality ( Figure 5). More importantly, risksensitivity is likely important for the maintenance of eusocial groups and in the transition from small to larger groups that had previously formed due to other mechanisms. In our model, selection for increased resource-use efficiency leads to supersaturation (Dickinson & Hatchwell, 2004) of the environment, that is, an increase in equilibrium population density (Figure 4). Finally, our model yields some clear and testable predictions.
In summary, these are (