EFTA02702876.pdf

Journal of lhearrtical Biology 399(2016) 103-116 Contents lists available at ScienceDirect 1O4 aRt` SN Journal of Theoretical Biology ELSEVIER journal homepage: Evolution of worker policing Conshlark Jason W. Olejarz a, Benjamin Allen l'"', Carl yeller • Raghavendra Gadagkar -1. Martin A. Nowak a•d•g.s Program for twoltalonary Dynamks. Hanard University, Cambridge, MA 02(38. USA °Department of Mathematics. Emmanuel College, Bosron. MA 02115, USA `Center for Mathematical Sciences and Applicanons. Harvard Unhersky. Cambridge, MA 02138. USA Depaytmcnr of Organismic and Emlutionary Biology. Hamad University. Cambridge. MA 02178. USA • Centre for Ecological Sciences and Centre for Contemporary Studies. Indian Institute of Science. Bangalore 560 072. India 'Indian National Science Academy. New Delhi 110 002. India ▪ Deponent of Mathematics. Harvard University. Cambridge. MA OM& USA ARTICLE INFO ABSTRACT Ankle history. Workers in insect societies are sometimes observed to kill male eggs of other workers, a phenomenon Received 2 February 2015 known as worker policing. We perform a mathematical analysis of the evolutionary dynamics of policing. Received in revised form We investigate the selective forces behind policing for both dominant and recessive mutations for dif- 23 January 2016 ferent numbers of matings of the queen. The traditional, relatedness-based argument suggests that Accepted 2 March 2016 Available online 11 March 2016 policing evolves if the queen mates with more than two males, but does not evolve if the queen mates with a single male. We derive precise conditions for the invasion and stability of policing alleles. We And Keywords: that the relatedness-based argument is not robust with respect to small changes in colony efficiency Sociobiology caused by policing. We also calculate evolutionarily singular strategies and determine when they are Natural selection evolutionarily stable. We use a population genetics approach that applies to dominant or recessive Evolutionary dynamics Modelsisimulations mutations of any effect size. 2016 Elsevier Ltd. All rights reserved. 1. Introduction (Hughes et al.. 2008; Cornwallis et al.. 2010; Queller and Strassmann. 1998; Foster et al.. 2006; Rootnsma, 2007, 2009). In contrast, it is In populations with haplodiploid genetics, unfertilized female believed that polygamy—not monogamy—is important for the evo- workers are capable of laying male eggs. Thus, in a haplodiploid lution of police workers. colony, male eggs can in principle originate from the queen or Several papers have studied the evolution of policing. Starr from the workers. Worker policing is a phenomenon where female (1984) explores various topics in the reproductive biology and workers kill the male eggs of unmated female workers (Ratnieks. sociobiology of eusocial Hymenoptera. He defines promiscuity as 1988; Ratnieks and Visscher. 1989; Ratnieks et al., 2006; Gadagkar. 1 firl_ ill ). where n is the number of matings of each queen, and 2001; Wenseleers and Ratnieks, 20064. Worker policing is J; is the fractional contribution to daughters by the i-th male mate. observed in many social insects, including ants, bees, and wasps. He writes, regarding workers, that "They are on average less Yet the precise conditions for the evolution of worker policing are related to nephews than brothers whenever (promiscuity is still unclear. greater than two) and should prefer that the queen lay all the male Worker policing (Ratnieks. 1988; Ratnieks et al.. 2006; Gadagkar, eggs. Workers would therefore be expected to interfere with each 2001; Wenseleers and Ratnieks. 2006a) and worker sterility (Wilson. other's reproduction." Thus, Starr (1984) was the first to suggest 1971; Hamilton. 1972; Olejarz et al., 2015) are two distinct phe- that workers should raise their nephews (sons of other workers) if the queen mates once, but should only raise their brothers (sons of nomena that are widespread in the eusocial Hymenoptera. In addi- the queen) if the queen mates more than twice. Starr (1984) uses a tion to worker policing, a subset of workers in a colony may forego relatedness-based argument. but he does not provide any calcu- their own reproductive potential to aid in raising their siblings. Prior lation of evolutionary dynamics in support of his argument; he relatedness-based arguments have suggested that queen monogamy uses neither population genetics nor inclusive fitness theory. In a is important for the evolution of a non-reproductive worker caste book on honeybee ecology, Seeley (1985) also proposed, using a relatedness-based argument, that worker policing should occur in • Corresponding author. colonies with multiply mated queens, but that worker policing E-mail oddmss: martin novakPhaivard.edu (MA. Novak). should be absent if queens are singly mated. latp:ndx.clorcag/1010I6Ultbi.20I6.03.091 0022-5193)&2016 Elsevier Ltd. All rights reserved. EFTA_R1_02076880 EFTA02702876 104 1W. OhIan et al /journal of Theoretkal Bletrov 399 (2016)103-116 Woyciechowski and Lomnicki (1987) perform a calculation studied species to date that are multiply mated.) Worker removal based on population genetics and conclude that workers should of worker-laid eggs is much less prevalent in colonies of the raise their nephews (sons of other workers) if the queen mates bumblebee (Velthuis et al., 2002), the stingless bee, (Peters et al.. once, but should only raise their brothers (sons of the queen) if the 1999), and the wasp, Vespula rata (Wenseleers et al.. 2005), which queen mates more than twice—the case of double mating is neu- are predominantly singly mated. (As mentioned above, worker tral with respect to preference. From this result, they claim that, policing has been found only in about 20% of the studied species to under multiple mating of the queen, natural selection should favor date that are singly mated.) There are some studies based on non-reproductive workers. Woyciechowski and Lomnicki (1987) observational evidence that find policing in singly mated species; consider both dominant and recessive alleles affecting worker examples of species with single mating and worker policing are behavior, but they do not consider colony efficiency effects. Vespa crobro (Foster et al.. 2002), Camponotusjlortdanus (Endler et Ratnieks (1988, considers the invasion of a dominant allele for al., 2004), Aphaenogaster smythiesi (Wenseleers and Ratnieks, policing. Using population genetics, he arrives at essentially the 200bh). and DitIC0MM0 (Wenseleers and Ratnieks, 20061), same conclusion as Woyciechowski and Lomnicki (1987): In the Interspecies comparisons are somewhat problematic, because absence of efficiency effects, policing evolves with triple mating even though phylogeny can be controlled for, there are many but not with single mating. But Ramieks also considers colony (known and unknown) ways in which species differ in addition to efficiency effects, focusing mainly on the case where policing mating frequency that may also affect the absence or presence of improves colony efficiency. Since policing occurs alongside other worker policing. Furthermore. many empirical studies are based maintenance tasks (such as cleaning of cells. removal of patho- on genetic analyses of male parentage. (Though studies of some gens, incubation of brood), and since eating worker-laid eggs species are based on actual observational evidence: see. e.g., might allow workers to recycle some of the energy lost from laying Wenseleers and Ratnieks, 2006b.) Regarding species for which the eggs. Ramieks supposes that policing improves colony efficiency. study of policing is based on genetic analyses, policing is often He finds that worker policing with singly mated queens may inferred if males are found to originate predominantly from the evolve if policing improves colony reproductive efficiency. He also queen. But such an inference, in cases where it is made, pre- finds that worker policing with triply mated queens may not supposes that workers actively try to lay male eggs in the first evolve if policing reduces colony reproductive efficiency, but he place. It is therefore not clear how reliably genetic investigations considered this case to be unlikely on empirical grounds. Ratnieks can measure policing. does not study recessive policing alleles. He also does not calculate The small number of attempts at measuring the prevalence of evolutionary stability conditions. worker policing in intraspecific experiments have also returned Both papers (Woyciechowski and Lomnicki. 1987; Ratnieks, conflicting results. Foster and Ratnieks (2000) report that facul- 1988) offer calculations based on population genetics without tative worker policing in the saxon wasp. Dolichovespula saxonica. mentioning or calculating inclusive fitness. These early studies is more common in colonies headed by multiply mated queens. (Starr. 1984; Seeley, 1985; Woyciechowski and Lomnicki. 1987; But their sample size is only nine colonies. The phenomenon was Ratnieks. 1988) were instrumental in establishing the field of reinvestigated by Ronckaert et al. (2011) who report no evidence worker policing. of facultative worker policing depending on queen mating fre- Testing theoretical predictions on the evolution of worker quencies, and argue that the previous result may have been flawed policing in the field or in the lab is difficult Due to the complex- or that there were interpopulational variations. ities inherent in insect sociality, published empirical results are not Many empirical studies have emphasized that factors besides always easy to interpret. While, so far, winter policing has been intracolony relatedness—including the effects of policing on a found in all species with multiple mating that have been studied, it colony's rate of production of offspring—may play a role in has also been found in about 20% of species with singly mated explaining evolution of worker policing (Foster and Ratnieks queens (I lammond and Keller. 2004; Wenseleers and 2001a,c: Hartmann et al_ 2003; Hammond and Keller. 2004; Ratnieks. 2006b; Bonckaert et al.. 2008). Herein lies the difficulty: Wenseleers and Ratnieks. 2006b; Helantera and Sundstrom. 2007; When worker policing is found in multiply mated species and Khila and Abouheif. 2008; Zanette et al. 2012). Yet reliable pub- found to be absent in singly mated species, this is taken as evi- lished data on the effect that policing has on colony reproductive dence supporting the relatedness argument, and when worker efficiency are often hard to find. (For some exceptions, see policing is found in singly mated species, it is explained away as Wenseleers et al.. 2013 and references therein.) not being evidence against the theory, but as having evolved for In this paper, we derive precise conditions for the evolutionary other reasons (such as colony efficiency). See, for example, the invasion and evolutionary stability of police alleles. We consider following quotation by Bonckaert et al. (2008): 'Nevertheless. our any number of matings, changes in the proportion of queen- results are important in that they show that V. germanica forms no derived males, changes in colony efficiency, and both dominant exception to the rule that worker reproduction should be effec- and recessive mutations that affect the intensity of policing, tively policed in a species where queens mate multiple times Our paper is based on an analysis of evolutionary dynamics and (Ratnieks, 1988). Indeed, any exception to this pattern would be a population genetics of haplodiploid species (Nowak et al., 2010; much bigger challenge to the theory than the occurrence of Olejarz et al., 2015). It does not use inclusive fitness theory. Spe- worker policing in species with single mating, which can be cifically, we adapt the mathematical approach that was developed readily explained (Ratnieks, 1988; Foster and Ratnieks. 2001b)." by Olejarz in al. (2015) for the evolution of non-reproductive This is precisely why a careful simultaneous consideration of workers. We derive evolutionary invasion and stability conditions relatedness, male parentage, and colony efficiency is important for for police alleles. Mathematical details are given in Appendix A. understanding worker policing. In Section 2, we present the basic model and state the general We do not aim to provide an exhaustive catalog of all species in result for any number of matings for dominant policing alleles. In which worker policing has been studied. We merely cite some Sections 3-5, we specifically discuss single, double. and triple specific examples to add context Policing is rampant in colonies of mating for dominant policing alleles. We take dominance of the the honeybee (Ratnieks and Visscher. 1989), the wasp Vesputa policing allele to be the more realistic possibility because the vutgaris (Foster and Ratnieks, 2001c), and the wasp Vespula ger- policing phenotype is a gained function. Nonetheless. for com- manica (Bonckaert et al., 2008), which are all multiply mated. (As pleteness, we give the general result for recessive policing alleles mentioned above, worker policing has been found in all of the in Section 6. In Section 7, we discuss how the shape of the EFTA_R1_02076881 EFTA02702877  Olefore et at /Journal of Theorrtkol Biology 399 (20:6) I03-II6 105 efficiency function determines whether or not policing is more a likely to evolve for single or multiple matings. In Section 8. we vimin Males Fertiliad Queens Queens analyze our results for the case where the phenotypic mutation induced by the mutant allele is weak (or, equivalently in our formalism, the case of weak penetrance). In this setting, the PA + (n —m) A +m a p AA,m quantity of interest is the intensity of policing. We locate the evolutionarily singular strategies. These are the values of intensity Aa + (n —m) A +m a Aa.m of policing for which mutant workers with slightly different policing behavior are, to first order in the mutant phenotype. aa + (n —m ) A+ma aafri neither advantageous nor disadvantageous. We then determine if a singular strategy is an evolutionarily stable strategy (ESS). In Section 9. we discuss the relationship between policing and San Cbiab ninisrb• Tem Clusll inclusive fitness theory. together with the limitations of the O•WIIPO Ors•if• INI•oss• Os" Ns Maw' Moo AAA A. • A relatedness-based argument. Section 10 concludes. Ms Aa A •• • ••• A•••• • •• SA0a AAA M•a• A••• Awl* A• A •.0 rs •• 2. The model C ra Come likes a Usua We investigate worker policing in insect colonies with haplo- Oman timain• Porn% ClougAlm• 0.••••• scos- C•i•Ase Sam -ANAA•mAs A (2.•••1••••• diploid genetics. Each queen mates n times. We derive conditions Awn \10-1•0•A•nAs•maa A•• 1.0 A • In•2al• la-m)M•air In-elA•01•••• under which a mutation that effects worker policing can spread in ran a population. We make the simplifying assumption, as do nip I (a) The possible fluting events with haplodiploid genetics are shown. Each Wowiechowski and Loninicki (1987) and Rat nicks (1988). that the queen mates with n malts. m denotes the number of times that a queen mates with mutant type a males and can take values between 0 and n. Thus. there are 3Ot I I) colony's sex ratio is not affected by the intensity of worker types of colonies. (b) If each queen mates with only a single male, then there are six policing. types of colonies. The female and male offspring (right three columns) of each First we consider the case of a dominant mutant allele. Because colony (leftmost column) are shown. For example. M, I colonies arise from a type the policing allele confers a gain of function on its bearer, the M female fluting with a single mutant type a male. AA. I queens produce female offspring of type Aa and male offspring of type A. 50% of the offspring of workers in assumption that it is dominant is reasonable. There are two types AA. I colonies ate of type A. while the remaining 50% of the offspring of workers in of males. A and a. There are three types of females, AA. Aa, and aa. M. I colonies are of type o. (c) The female and male offspring fright three columns) If the mutant allele is dominant then Aa and aa workers kill the of each colony (leftmost column) when each queen mates n times are shown. male eggs of other workers, while M workers do not. (Alter- natively, M workers police with intensity ZAA, while Aa and ao Possible Forms for the Function p, workers police with intensity Zaa = =Zm+w. We consider this case in Section 8.) For n matings, there are 3m+ 1) types of mated queens. We use the notation AA, in; Ac. in; and art m to denote the a genome of the queen and the number, m. of her matings that were 5ei 0.75 with mutant males. a. The parameter m can assume values 0.1....,n. A schematic of the possible mating events is shown in ei, 0.50 Fig. 1(a). 0 There are three types of females, AA. Aa. and aft and there are n+1 possible combinations of males that each queen can mate 0.25 with. (For example, a queen that mates three times (n=3) can mate with three type A males. two type A males and one type a 0_ male, one type A male and two type a males, or three type a 0.25 0.50 0.75 1 males.) Fig. I (b) shows the different colony types and the offspring Fraction of police workers, of each type of colony when each queen is singly mated. Fig. I Ng. 2. The queen's production of male eggs. pt. increases with the fraction of (c) shows the different colony types and the offspring of each type waiters that are policing. Z. This is intuitive, since having a larger worker police of colony when each queen mates n times. The invasion of the force means that a greater amount of worker-laid eggs can be eaten or removed. mutant allele only depends on a subset of colony types. The cal- Three possibilities for a monotonically increasing function p, are shown. culations of invasion conditions are presented in detail in Appendix A. 22 Colony efficiency as a function of policing r, represents the rate at which a colony produces offspring 2.1. Fraction of male offspring produced by the queen (virgin queens and males) if the fraction of police workers is z. (This quantity was also employed by Ratnieks. 1988.) Without loss pa represents the fraction of males that are queen-derived if the of generality, we can set rc, = I. For a given mutation that affects fraction of police workers is z. (This quantity was already the intensity of policing, and for a given biological setting, the employed by Ratnieks. 1988.) The parameter z can vary between efficiency function ra may take any one of a variety of forms 0 and 1. For z=0. there are no police workers in the colony, and for (Fig. 3). z= 1, all workers in the colony are policing. We expect that pa is an Colony efficiency depends on interactions among police work- increasing function of z. Increasing the fraction of police workers ers and other colony members. It also depends on the interactions increases the fraction of surviving male eggs that come from the of colonies and their environment. There are some obvious nega- queen (Fig. 2). tive effects that policing can have on colony efficiency. By the act EFTA_R1_02076882 EFTA02702878 1013 ale/an et at /journal of Theoretical Biology 399 (2016) 103-116 Possible Forms for the Function r, the entire worker population is unclear. It is possible that a frac- tion z < 1 of police workers can effectively police the entire 1.3 population, and adding additional police workers beyond a certain point could result in wasted energy, inefficient use of colony 1.2 resources, additional recognitional errors. etc. These effects may correspond to colony efficiency r, reaching a maximum value for j 1.1 73 some 0 < z <1. I 1 As another possibility, suppose that police workers, when their number is rare, directly decrease colony efficiency by the act of I 0.9 killing male eggs. It is possible that for some z c 1, police workers are sufficiently abundant that their presence can be detected by 0.8 other workers. Assuming the possibility of some type of facultative response, the potentially reproductive workers may behaviorally 0 0.25 0.50 0.75 adapt by reducing their propensity to lay male eggs, instead Fraction of police workers, z directing their energy at raising the queen's offspring. In this scenario, colony efficiency r, may reach a minimum value for some Fig. 3. The functional dependence of colony efficiency. r,, on the fraction of 0<z<1. workers that are policing. z. may take any one of many possibilities. 23. Main results for dominant police alleles of killing eggs, police workers are directly diminishing the number of potential offspring. In the process of identifying and killing We derive the following main results for dominant police nephews, police workers may also be expending energy that could alleles. If the queen mates with n males, then the a allele for otherwise be spent on important colony maintenance tasks (Cole, policing can invade an A resident population provided the fol- 1986; Naeger et al.. 2013). Policing can also be costly if there are lowing "evolutionary invasion condition" holds: recognitional mistakes. i.e.. queen-laid eggs may accidentally pun +pia (rim\ ma) > 2 _ I'l_ _„ .1 trim\ be removed by workers. Recognitional errors could result in ( (1) modifications to the sex ratio. which is an important extension of 2 ro ro 1'0 l ring ro our model but is beyond the scope of this paper. When considering only one mutation. ro can be set as 1 without We can also identify positive effects that policing may have on loss of generality. Why are the four parameters. rim. ruz, p", and colony efficiency. It has been hypothesized that the eggs which are p12, sufficient to quantify the condition for invasion of the mutant killed by police workers may be less viable than other male eggs allele, a? Since we consider invasion of a. the frequency of the (Velthuts et al., 2002: Pirk et al.. 1999: Gadagkar. 2004; Nonacs, mutant allele is low. Therefore, almost all colonies are of type 2006), although this possibility has been disputed (Beekman and AA 0. which means a wild-type queen.M. has mated with n wild- Oldmyd. 2005; Helantera et al.. 2006; Zanetre et al.. 2012). If less- type males, A. and 0 mutant males, a. In addition, the colonies Aa, 0 viable worker-laid eggs are competing with more-viable queen-laid and M. I are relevant. These are all colony types that include male eggs, then policing may contribute positively to overall colony exactly one mutant allele. Colony types that include more than one efficiency. Moreover, policing decreases the incentive for workers to mutant allele (such as Aa.1 or AA 2) are too rare to contribute to expend their energy laying eggs in the first place (Foster and the invasion dynamics. For an Aa.O colony, half of all workers are Ratnieks, 2001a; Wenseleers et al.. 2004a,b: Wenseleers and policing and therefore the parameters r1j2 and p1, 2 occur in Eq. Ratnieks. 2006a). which could be another positive influence on (1). For an AA.1 colony, 1/n of all workers are policing, which colony efficiency. (However, the decrease in incentive for workers to explains the occurrence of r im and p,,,, in Eq. (1). reproduce due to policing would only arise on a short time scale if Next, we ask the convene question: What happens if a popu- there is a facultative response to policing which is unlikely.) lation in which all workers are policing is perturbed by the As another speculative possibility: Could it be that worker egg- introduction of a rare mutant allele that prevents workers from laying and subsequent policing acts as a form of redistribution policing? If the a allele for worker policing is fully dominant, and if within the colony? That is. suppose that it is better for colony colony efficiency is affected by policing then a resident policing efficiency to have many average-condition workers than to have population is stable against invasion by non-police workers if the some in poor condition and some in good condition. Suppose following "evolutionary stability condition" holds: further, as seems realistic. that good-condition workers are more ri pi) +pa„_ ivamtn-2) likely to lay eggs (which are high in nutritional content, of course). > (2 + 0)(2 +2(2+n+npi ) (2) If the average police worker is of condition below the average egg- ran- mom laying worker, then worker egg-laying and policing serves to What is the intuition behind the occurrence of the four para- redistribute condition among the workers, improving overall col- meters, r1. ran _ wo p pi. and pan _ warn? The condition applies to ony efficiency. a population in which all workers are initially policing. Note that. The special case, where policing has no effect on colony effi- because the allele, a. for policing is fully dominant in our treat- ciency and which has informed the conventional wisdom, is ment. non-policing behavior arises if at least two mutant A alleles ungeneric, because policing certainly has energetic consequences for non-policing are present in the genome of the colony. which is for the colony that cannot be expected to balance out completely. the combination of the queen's genome and the sperm she has An early theoretical investigation of colony efficiency effects stored. To study the invasion of a non-policing mutant allele. we regarding invasion of dominant mutations that effect worker must consider all colony types that have 0,1, or 2 mutant A alleles: policing was performed by Ratnieks (1988). these are Oa. n; aa.n —1: Ao.n: aa.n —2; Ao.n-1; and M.n. The Although monotonically increasing or monotonically decreas- colonies aa,n: aa.n —1: Aa.n: aa.n —2: and Mae do not contain ing functions r, are the simplest possibilities, these cases are not non-police workers: the efficiency of those colonies is r,, and the exhaustive. For example, a small or moderate amount of policing fraction of male eggs that originate from the queen in those may be expected to improve colony efficiency. However, the pre- colonies is pi. Both of these parameters occur in Eq. (2:. Colonies of cise number of police workers that are needed to effectively police type Aa.n - 1 produce a fraction of 1/(2r) non-police workers. EFTA_R1_02076883 EFTA02702879  Olejarz et at /Journal of Theorrtkol Biology 399 (2016)103-N6 107 a Single mating. n=1 100 a 101 — r •052 - rp1.061 11s1.0130 — r et•069 10 10 15 Time (104) Fig. 4. Numerical simulations of the evolutionary dynamics of wetter policing team the condition given by Eq. ' I :. The policing allele is dominant. For numerically probing invasion we use the initial condition XA•o= 1-10- 3 and XAA 1=10 3. We set ro= I without loss of generality. Other parameters are: (a)Pi7 =0.75. pl =0.9. and (b) Ave 0.6, r — 1005. and ri -1.01. Police allele ca ' ade and is evolutionarily stable • ca ade. but is not stable C— to not vale but s stable C— c not d nil is not stable 0 1 Frequency of police allele Fig. 5. There are four possibilities for the dynamical behavior in the proximity of two pure equilibria. a n = 1 mating p1,2 = 0.75, p, = 1 b n = 1 mating p,,, = 0.99. p, = 1 1.2 1.2 Stably Colony efficiency, r1 .-- 11 Does Not \ 1.1 Invade Invades co Stable \ Unstable e.= 1 Invades C Unstable / Does Not O O Invade Does Not 0.9 0.9 - Invade Bitable Unstable 0.9 1 1.1 12 10.8 0.9 1 1.1 1.2 Colony efficiency, r v2 Colony efficiency, r1,2 Fig. S. If queens are singly mated (n-1). then a plot of I.' versus r 1,2 dearly shows all four possibilities for the behavior around the two pure equilibria. For (a), we set pi 0.75 and pi 1. For (b). we set pi = 0.99 and p,-1. which explains the occurrence of ri2,102a, and pa,,_ 102„, in Eq. and be stable. The possibilities are shown in Fig. 5. In the cases (2). where policing cannot invade but is stable, or where policing can Numerical simulations of the evolutionary dynamics with a invade but is unstable. Brouwer's fixed-point theorem guarantees dominant police allele are shown in Fig. 4. the existence of at least one mixed equilibrium. In the case where Generally, four scenarios regarding the two pure equilibria are policing can invade but is unstable, police and non-police workers possible: Policing may not be able to invade and be unstable, will coexist indefinitely. policing may not be able to invade but be stable, policing may be We will now discuss the implications of our results for parti- able to invade but be unstable, or policing may be able to invade cular numbers of matings. EF TA_R 1_02076884 EFTA02702880 lea J.W. Otejaa ef al / Journal of Theo:ilk& Swim 399 (2016)103-116 a The police allele invades and is stable. b The non.polico allele invades and is stable. p,2 0.75. r,? 1.0344, re1.0787 pin=0.75. p.=1. rig=1.0344. r,=1.0567 1 1 0.8 0.8 1 0.6 0.8 - a -8 OA I 0.2 - 0 2 3 4 84 88 88 Time (10') Thin (10') The police and non-police alleles are bistable. d The police and non-poise alleles meat p,a 40.75. rflu1.0244. r,a1.0667 pws0.75. pint ri4e1.0444. ?ter 1 0.8 1 0.8 0.6 - •O 0.8 a 0 OA 15 0.4 LL • 0.2 5 10 15 20 5 LL 02 1 00 4 6 8 10 Time (103) Time (10') Flg. 7. Nurnencal simulations of the evolutionary dynamics of worker policing that show the four behaviors in hg (,(a). The policing allele is dominant For each of the four panels. we use the initial conditions: (a) XAto I -10 l and gm - 10 1:(b)X,ai - I -10 s and r eso - 10 3: 01%Am) 0.02 and Xµl— 098 (lower curve). and Xmo = 0 01 and XAA l =0.99 (upper cum): (d)44.0 =1-10 -2 and XAAI =10 -2 (lower curve). and )(mi I —10-l and 4, 10 -2 (upper curve). We set ro • I without loss of sawrality. n = 1 mating The stability condition for a dominant police allele is 1.003 6—pia+3p,I-112 (4) ri > 6+2p, 4.." 1.002 Evolution of policing is highly sensitive to changes in colony effi- 1.001 ciency. For example, let us consider p, ,2 = 0.99 and pi = 1. This means that if half of all workers police then 99% of all males come from the queen. Ifall workers police then all males come from the queen. In this case, efficiency values such as r 1,2 = 1.001 and r, = 1.0031 lead to the Petty Wades bilis unstable evolution of policing. In principle, arbitrarily small increases in colony 0.999 Pollee Insides aid et HAW — Mktg der not Merle and a unstable efficiency can lead to the evolution of policing for single mating. — eons do a Minna, but e ;We A plot of r, versus rif2 for singly mated queens (Fig. 6) illus- 0.998 trates the rich behavior highlighted in Fig. 5. Numerical simula- 0 1/2 1 tions of the evolutionary dynamics are shown in Fig. 7. Fraction of police workers. Another intriguing feature is that increases in colony efficiency Fig. 8. Possible r, efficiency curves for n=1 mating which demonstrate different due to policing do not necessarily result in a higher frequency of behaviors. For this plot, we set pre =099 and m 41. Here. each curve has the police workers at equilibrium Fig. S illustrates this phenomenon. functional form r1 - 1 +ea +922. For exam*. we can have: (blue) policing Invades Four possibilities for the efficiency function r., are shown. Notice that but is unstable, o-0.003. )l -0.0004: (green) policing invades and is stable. (7=0.0026. 4=0: (red) policing does not invade and is unstable. 0=0.0024. p=0: the r2 curve which results in coexistence of police workers and non- (black) policing does not Invade but is stable. 0=0.002. 4=00004. (For Inter-