EFTA01070997.pdf

Journal of Theoretical Biology 399 (2016) 103-116 Contents lists available at ScienceDirect Jam. 1or Theoretical lhoOgy Journal of Theoretical Biology journal homopago::::vw.elsevier.com/locate/yjtbi Evolution of worker policing CrossMatk Jason W. Olejarza, Benjamin Allen b•a'c, Carl Veller aA, Raghavendra Gadagkai Martin A. Nowaka•d-gs ▪ Program for Evolutionary Dynamic. Harvard University, Cambridge. MA 02738, USA b Department of Mathematics. Emmanuel Cortege, Boston, MA 02115. USA Center for Mathematical Sciences and Applications. Harvard University, Cambridge. MA 02738, USA ▪ Department of Organismic and Evolutionary Biology. Harvard University, Cambridge, MA 0213B, USA ' Centre for Ecological Sciences and Centre for Contemporary Studies. Indian Institute of Science, Bangalore 560 012. India Indian National Science Academy, New Delhi 170 002. India 2 Department of Mathematics, Harvard University, Cambridge. AM 02138, 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 II March 2016 policing evolves if the queen mates with more than two mates, 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 find 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 strategics and determine when they are Natural selection evolutionarily stable. We use a population genetics approach that applies to dominant or recessive Evolutionary dynamics Models/simulations mutations of any effect size. e. 2016 Elsevier Ltd. All rights reserved. 1. Introduction (Hughes et al., 2008: Cornwallis et al.. 2010: Queller and Strassmann, 1998: Foster et al.. 2006: Boomsma, 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/(E7_,J1), where n is the number of matings of each queen. and 2001: Wenseleers and Ratnieks, 2006a). Worker policing is .h 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 twol and should prefer that the queen lay all the male Worker policing (Ratnieks, 1988: Rah-licks et al.. 2006: Gadagkar, eggs. Workers would therefore be expected to interfere with each 2001: Wenseleers and Ratnieks, 2006a) and worker sterility (Wilson, others reproduction.' Thus, Starr (1984) was the first to suggest that workers should raise their nephews (sons of other workers) if 1971: Hamilton, 1972: Olejarz et al.. 2015) are two distinct phe- 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 address: martin_nowaloPharvard.edu (MA Nowak). should be absent if queens are singly mated. http:fidx.dokorg/10.1016/mtbi.2016.03.001 0022-51930 2016 Elsevier Ltd. All rights reserved. EFTA01070997 104 OkJorz et al /Journal of theatrical Biology 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 rub (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 crabro (Foster et al., 2002), Camponotusfloridanus (Endler et Ratnieks (1988) considers the invasion of a dominant allele for al.. 2004). Aphaenogasrer smythiesi (Wenseleers and Ratnieks. policing. Using population genetics, he arrives at essentially the 2006b), and Diacamma (Wenseleers and Ratnieks, 2006b). 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 Ratnieks 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, Ratnieks 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. Dotichovespula scuronica, 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 Bonckaert 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, worker 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 (Hammond 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. germanico 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 et 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 Vespula policing allele to be the more realistic possibility because the vulgaris (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 EFTA01070998  J.W. OleJeri et al. /Journal of theoretical Biology 399 (2016) 103-116 105 efficiency function determines whether or not policing is more a likely to evolve for single or multiple matings. In Section 8. we Virgin n Males Fertilized Queens Queens analyze our results for the case where the phenotypic mutation induced by the mutant allele is weak (or. equivalently in our AA +(n —m) A +m a AA,m formalism, the case of weak penetrance). In this setting, the 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 +m a aa,m 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 b OM Ousts MO•Onellos MO inclusive fitness theory, together with the limitations of the Oman Gomm Oen mina Man On VIManelms MO AA A A relatedness-based argument. Section 10 concludes. M 12 /1 As A A•• MA AA•Pa An 5A•• AM *aim An Ana ma As A•• M1 la a 2. The model C IsehOINOIMOE • TOM We investigate worker policing in insect colonies with haplo- Omen Omega Omen Oman I Clain- MINIONIfielt• I la na AnmPa oil A ft...inns diploid genetics. Each queen mates n times. We derive conditions Mon 05-MM•Mann A • Ilp fan-r••ta•h•Ael.- under which a mutation that effects worker policing can spread in 16 -nIsa•maa In-MA•ln•mla a population. We make the simplifying assumption, as do Flg. I (a) The possible mating events with haplodiploid genetics are shown. Each Woyciechowski and Lomnicki (1987) and Ratnieks (1988), that the queen mates with n males. in denotes the number of times hat a queen mates with mutant type a males and can take values between 0 and n Thus, there are 3(n +1) colony's sex ratio is not affected by the intensity of worker types of colonies. (b)Ifeach queen mates with only a single male, then there are six policing. types of colonies. The female and male offspring (right hree columns) of each First we consider the case of a dominant mutant allele. Because colony (leftmost column) are shown. For example. AA. I co onies arise from a type AA female mating with a single mutant type a male. M.1 queens produce female the policing allele confers a gain of function on its bearer, the 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 M. I colonies are 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, Ac. and aa. M. I colonies are of type a. (c) The female and male offspring (right three columns) If the mutant allele is dominant, then Aa and aa workers kill the of each colony (leftmost column) when each queen mates it times are shown. male eggs of other workers, while AA workers do not. (Alter- natively. AA workers police with intensity 44 while Aa and aa Possible Forms for the Function p : workers police with intensity ZA0 = Z 2a = 44+w. We consider this case in Section 8.) For n matings, there are 3(n + 1) types of mated 0 queens. We use the notation AA. m: Aa,m: and aa,m to denote the 0 2 0.75 E genome of the queen and the number. m. of her matings that were with mutant males, a. The parameter m can assume values rn 0,1...., n. A schematic of the possible mating events is shown in gi 0.50 Fig. 1(a). There are three types of females, AA. Act, and ca. and there are n+1 possible combinations of males that each queen can mate wo 0.25 with. (For example, a queen that mates three times (n=3) can O 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 LI- 0 0.25 0.50 0.75 males.) Fig. 1(b) shows the different colony types and the offspring Fraction of police workers, z of each type of colony when each queen is singly mated. Fig. 1 Fig 2. The queen's production of male eggs, p., increases with the fraction of (c) shows the different colony types and the offspring of each type workers that are policing, 2. 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 of removed. mutant allele only depends on a subset of colony types. The cal- Three possibilities for a monotonically increasing function pt are shown. culations of invasion conditions are presented in detail in 22. Colony efficiency as a function of policing Appendix A. re 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 Pr represents the fraction of males that are queen-derived if the of generality, we can set ro = 1. 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 re 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 pr 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 EFTA01070999 106 OkJarz et al. 1puma, of theoretical Biology 399 (2016)193-116 Possible Forms for the Function r the entire worker population is unclear. It is possible that a frac- tion ze 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 0 correspond to colony efficiency rz reaching a maximum value for 1.1 some 0 < z < 1. As another possibility, suppose that police workers, when their number is rare, directly decrease colony efficiency by the act of z 0.9 killing male eggs. It is possible that for some z < 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 rz may reach a minimum value for some Hg. 3. The functional dependence of colony efficiency. rz. on the fraction of 0 ez e 1. workers that are policing. z, may take any one of many possibilities. 2.3. 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. Le, queen-laid eggs may accidentally be removed by workers. Recognitional errors could result in modifications to the sex ratio, which is an important extension of Ptin +Pia Cm) (rig) >2 _ ro ro CI/2) _ ro _ Pito Cin) ro (1) 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. r im. r la, pun. and colony efficiency. It has been hypothesized that the eggs which are pia. 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 (Velthuis et al.. 2002: Pirk et al., 1999: Cadagkat 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. AA, has mated with n wild- Oldroyd. 2005: Helantera et al., 2006: Zanette et al., 2012). If less- type males. A, and 0 mutant males, a. In addition, the colonies Act 0 viable worker-laid eggs are competing with more-viable queen-laid and AA, 1 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 Ac. 0 colony, half of all workers are Ratnieks, 2001a: Wenseleers et al.. 2004a,b: Wenseleers and policing, and therefore the parameters r112 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 un and pim in Eq. (1). reproduce due to policing would only arise on a short time scale if Next. we ask the converse 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 +n)(2 +Pi )+Pan- oita,o(n —2) (2) likely to lay eggs (which are high in nutritional content. of course) If the average police worker is of condition below the average egg- roan-mart 2(2+n+npi ) 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, ri, ran _ wan). Pt. and Pan-102M? 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 aa.n: aa,n — 1: Aa,n: n-2: Aa,n-1: and AA, n. The Although monotonically increasing or monotonically decreas- colonies aa.n: mai —1: Aa,n: aa, n —2: and AA, n do not contain ing functions rz are the simplest possibilities, these cases are not non-police workers: the efficiency of those colonies is r 1, 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 At n —1 produce a fraction of 1/(2n) non-police workers. EFTA01071000  J.W. Oleforz er al. /Journal of theoretical Biology 399 (2016) M3-116 107 Single mating. n=1 Double mating, n=2 0 10 10 1 To 104 1104 ' .6 10 4 4 10 1. !1.1. 104 .7 2 6 Time (104) Fig. a Numerical simulations of the evolutionary dynamics of worker policing confirm the condition given by Eq. (1). The policing allele is dominant. For numerically probing invasion, we use the initial condition XA40 —1-10 s and Kai — 10 a. We set ra —1 without loss of generality. Other parameters are: (a) pi,2 —0.75, pi —0.9, and rig -1A1: (b)pl a —0.6, p,—0.8. — 1.005, and r,-1.01. Police allele can invade and is evolutionarily stable can invade, but is not stable cannot invade. but is stable cannot invade and is not stable 0 1 kequency of police allele S. There are four possibilities for the dynamical behavior in the proximity of two pure equilibria a n= 1 mating P1/2 = 0.75. p1 =1 b n = 1 mating pv2 = 0.99. Pi = I 1.2 1.2 Invades Stable Colony efficiency. r1 Does Not 1.1 Stable Invade Invades Stable Does Not Unstable Invade 1 Invades Stable Unstable Does Not a Invade 8 0.9 Unstable 0.9 Does Not Invade 0.8 0.8 08 0.9 1 1.1 12 08 0.9 1 1.1 12 Colony efficiency. r„2 Colony efficiency. r112 Mg. 6. If queens are singly mated (n-1). then a plot of ri versus rin clearly shows all four possibilities for the behavior around the two pure equilibria. For (a). we set pi,2 —0.75 and p, — I. For (b), we set pc.2 —0.99 and pi — I. which explains the occurrence of rota _ I va," and pan _ i)!2,0 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. EFTA01071001 108 Okjarz er aL /journal of theoretical Biology 399 (2016)103-n6 a The police allele invades and is stable. b The non-police allele invades and is stable. p„.2=0.75. 1.0344. r1=1.0767 p.„2=0.75, p1=1, r„e1.0344, r,=1.0567 rue Frequency of policing allele, a to 0.8 co :9. • 0.6 1 o 0.4 8. 0.2 1 2 3 4 82 84 86 88 Time (101) Time (101) The police and non-police alleles are bistable. d The police and non-police alleles coexist. pw=0.75, p1=1. r„2=1.0244. r1.1.0667 pm l3.75. p1=1, re 1.0444. r1=1.0667 1 1 1 0.8 0.8 2' 0.8 0.8 0.4 8 0.4 0.2 0.2 2 LL LL OB 10 15 2 4 8 10 Time(10) Time (104) Fig. 7. Numerical simulations of the evolutionary dynamics of worker policing that show the four behaviors in fig. 6(a). The policing allele is dominant. For each of the four panels. we use the initial conditions: la) Xm,0 and443 -10-1; (b)X„,2 — 1 —10"1 and X„0 (C)440 0.02 and X2A.I — 0.98 (lower cum), and XAA0 —0.01 and X,,,„, —0.99 (upper cum): (d)X.„,„ — 1-10-2 and X.,," — 10-2 (lower curve), and X„,,, —1-10 2 and Xmo —10 2 (upper curie) We set re —1 without loss of generality. The stability con