(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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Conflict over caste fate in \ social insects: a tragedy of the commons examined. Submitted. \ \>", "Text", FontSize->16], Cell[CellGroupData[{ Cell["Biological question :", "Section", FontSize->25], Cell[TextData[{ "Bourke & Ratnieks (1999) pointed out that in social insects any female is \ related more to own offspring than to a sister's offspring, so that a female \ should benefit from becoming a queen rather than a worker. The predicted \ consequence of selfish manipulation of caste fate would be excess queen \ production. But how excessive should queen prediction be? Clearly, if all \ females would become queens the colony would perish as a result of a dearth \ of workers. So what is the best thing for a female to do? And what are the \ interests of the adult workers and the mother queen? \nI remember that upon \ first reading of Bourke & Ratnieks' paper it occurred to me that there were \ striking similarities between caste fate conflict and Frank's ", StyleBox["tragedy of the commons", FontSlant->"Italic"], " model (1995, 1996), the commons in this case being a common workforce. \ But it was only when Francis showed me an unpublished manuscript (now in \ press: Ratnieks 2001) where he derived the ESS probability for females to \ become queens using a genetic model, that I took renewed interest in this \ problem and arrived at a more general solution using Frank's methods. This \ model (Wenseleers et al. submitted) is explained below. Overall, the model \ can explain why ", StyleBox["Melipona", FontSlant->"Italic"], " stingless bees rear very many queens (most of which are killed upon \ eclosure by the workers), whereas ", StyleBox["Trigona", FontSlant->"Italic"], " stingless bees and honey bees produce very few queens. The reason, as we \ will see, is connected to individual control of caste fate in ", StyleBox["Melipona", FontSlant->"Italic"], ", and social control of caste development in nearly all other social \ insects." }], "Text", FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell["Assumptions :", "Section", FontSize->25], Cell["\<\ We model caste fate conflict for swarming social insects, because here the \ colony optimum is known: just a few queens need to be reared since mainly \ workers are needed for swarming. This matches Frank's assumption (1995, 1996) \ that the optimum for the group is to have very few selfish (queen destined) \ individuals. As in Frank (1995, 1996), we also assume that colony \ productivity is a linear decreasing function of the proportion of selfish \ (queen destined) individuals in the group. Below we model two cases. The \ first model derives the optimum probability for a female to develop as a \ queen assuming caste development is under individual control. This scenario \ is shown to cause excess queen production (i.e. production of more queens \ than needed for swarming). The second model derives the adult worker's \ interests, and shows when they are selected to gain social control over the \ caste allocation process. \ \>", "Text", FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["3.1 Individual control of caste development", "Section", FontSize->25], Cell["\<\ Parameters: y = probability with which a focal female larva develops as a queen z = average probability with which females develop as queens in the colony gf = genotype of a focal female (which does express the caste development \ gene) gm = genotype of males (males do not express the caste development gene, but \ their survival is affected by the caste allocation decisions of females) cf = relative reproductive value of females cm = relative reproductive value of males rf and rm = regression relatedness to females and males Rf and Rm = life-for-life relatedness to females and males\ \>", "Text", FontSize->18], Cell["\<\ The main difference compared to the basic tragedy of the commons model of \ section 2 is that we now have a passive class of relatives, namely males, \ that are affected by the caste allocation decisions of female larvae. Males \ do not themselves express the caste development gene but may transmit copies \ of it to subsequent generations. Therefore, we need to keep track both of the \ effects of correlated behaviour within the active class (female larvae; \ \"correlated selection\", Frank 1997), and the effects of the active class on \ a passive class of relatives (males; \"kin selection\", Frank 1997). Frank's \ (1997) direct fitness method, that allows to analyse such complex problems, \ is described below. \ \>", "Text", FontSize->14], Cell["\<\ Step 1: Write individual reproductive success W of a random recipient as a \ function of the behaviour of the active class\ \>", "Definition", FontSize->16], Cell["\<\ The first step is to write the direct fitness of members of the active \ (females) and passive class as a function of the behaviour of members of the \ active class. As before, we assume that the colony's total productivity G, measured over \ the colony's lifetime, is given as (1-z), where z is the average probability \ with which larvae become queens. Thus the direct fitness of any male Wm is given by\ \>", "Text", FontSize->14], Cell[BoxData[ \(\(Wm = 1 - z;\)\)], "Input"], Cell["\<\ The direct fitness of female larvae is directly proportional to the \ probability with which it becomes a queen (y), inversely proportional to the \ probability with which other larvae become queens (z, they compete for \ inheriting the swarm or colony) and proportional to the number of workers \ available (1-z), because these are needed for swarming. Thus we get \ \>", "Text", FontSize->14], Cell[BoxData[ \(\(Wf = \((1 - z)\)*\((y/z)\);\)\)], "Input"], Cell[TextData[{ "y/z can also be thought of as the individual success of any larvae \ relative to others (", StyleBox["I", FontSlant->"Italic"], "), and 1-z as the success of any daughter swarm or colony (G, proportional \ to the number of workers available, and thus equal to 1-z)." }], "Text", FontSize->14], Cell["\<\ Step 2: Analyse when the slope of a recipient's fitness on its genotype is \ positive\ \>", "Definition", FontSize->16], Cell["\<\ The second step is to analyse when the slope of a random recipient's fitness \ on its genotype is positive. The only difference compared to the models of \ Section 1 and 2 are that a random recipient is now taken to be either a \ female (a member of the active class) or a male (a member of the passive \ class). Therefore, a gene causing a larva to develop as a queen with a \ slighly higher probability is selected for when\ \>", "Text", FontSize->14], Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@0002`d0@0006`00002000000000:L0000N0000 00000000000F6`006@D00215CDH00040=2`0080000070000000000000000000080<005P2001:0@00 l000000000000000000001091@20Z@<0AP000;`:002^2P00AdA9@`4008000`00X1CU1`00002F2P00 0@090003B`D000P01`400000500002H63`0N0?ooool401@0001GKg9T3P1=JF=bKg=_IW@PEfmbI0D0 000;0P000005000030:J03<370000?/240070000002l0P0000010P8RDgUcM6E]000000X000040000 003ooooo0@000000<0040000;@4000D000020@40000500002@80000270000?/2c_l00000002@0@00 0004@00BE6U]IGIGLP DVm]HFh03EGeMaIEmGL10000000`00@0000]0@D03000038:000000<0000X;2T0HP5:0@031@0001@2 X09j31`0003k0X3n00000000T0400000108045AYKFEc84iUMb1BKfeQKP0=EOEg5UGeM`4000000300 100002d11P040000l04500`0000b2P0000030000;Rh`7RH6m0@00`D0000D0^01<@lL0000n`8Ro`00 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0P0002P0000<00000`0002P0000<0000100002P0000<00001@0002P0000<00001P0000h0000D0000 00000100000D0000 \>"], "Text", ImageSize->{218, 39.5625}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["(eqn.3.1)", FontSize->14], "\nwhere Wf and Wm are the direct fitnesses of females and males and gf and \ gm are their genotypes. Note that the two components of selection have been \ weighted by class (sex)-specific reproductive values cf and cm. This is \ necessary because in haplodiploids twice as many genes are present in females \ than in males (if there is no worker reproduction), so that effects on female \ fitness change gene frequency twice as much as effects on male fitness. (in \ cases where interactions occur between different age classes, the c's stand \ for age class specific reproductive value)\nExpanding the total derivatives \ of eqn. (3.1) we get :" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ regression\ of\ recip . \ fitness\ on\ actor\ breeding\ value\ *) \)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\(dWdg = \(\((D[Wf, y] + Simplify[D[Wf, z]]*Rf + D[Wm, z]\ *Rm)\)\(/.\)\({y \[Rule] z}\)\(\ \ \ \)\( (*\ note : \ in\ Mathematica, \ \(\(/.\)\(\ \)\(means\ Replace\ by\)\)\ *) \ \)\)\)\)\)], "Input"], Cell[BoxData[ \(\(-Rm\) - Rf\/z + \(1 - z\)\/z\)], "Output"] }, Open ]], Cell["Step 3: Solve to yield the ESS", "Definition", FontSize->16], Cell["\<\ Near the ESS, we can set y=z=z*, and eqn. 3.1 should equal 0. 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larvae ever have control over \ their own caste fate? Although certainly unusual, individual control of caste \ development is thought to be present in ", StyleBox["Melipona", FontSlant->"Italic"], " stingless bees (Bourke & Ratnieks 1999, Ratnieks 2001), because ", StyleBox["Melipona ", FontSlant->"Italic"], "queens and workers are of the same size, yet are morphologically distinct, \ and are reared in mass provisioned cells that are all of the same type. \ Empirical evidence also provides good support for the predictions of the \ caste conflict model: under optimal conditions, levels of queen production \ range from around 14% in ", StyleBox["Melipona favosa", FontSlant->"Italic"], ", where most of the males are produced by the workers, to 20% in ", StyleBox["Melipona beecheii", FontSlant->"Italic"], ", where all males are produced by the queen (Wenseleers et al. submitted). \ In addition, most of the virgin queens are killed by the workers soon after \ eclosion, which shows that individual control of caste fate causes excess \ queen production from the perspective of the adult workers.\nBut what about \ other swarming social insects such as ", StyleBox["Apis", FontSlant->"Italic"], " honey bees or army ants? Below it is shown that the typically low level \ of queen production is caused by a transition from individual to social \ control over caste development. " }], "Text", FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell["3.2 Social control of caste development", "Section", FontSize->25], Cell[CellGroupData[{ Cell["3.2.1. ANALYTICAL SOLUTION", "Subsection", FontSize->19], Cell["\<\ Parameters: y = probability with which a focal female larva develops as a queen z = average probability with which female larvae develop as queens in the \ colony P = average worker investment in caste policing b = genotype at the breeding locus (bf and bm=genotypes of females and males) p = genotype at the caste policing locus (pf and pm=genotypes of females and \ males) c = colony level cost of policing cf = relative reproductive value of females cm = relative reproductive value of males rf and rm = regression relatedness to females and males Rf and Rm = life-for-life relatedness to females and males\ \>", "Text", FontSize->18], Cell["\<\ We model a transition from individual to social control by assuming that the \ ancestral condition is one where each larva may decide over its own caste \ fate, and then checking when adult workers are selected to take over the \ caste allocation process. Thus we must study the joint evolution of two \ traits, one controlling the individual probability of becoming a queen, and \ the other controlling whether workers should inhibit larvae from becoming \ queens (worker caste policing). \ \>", "Text"], Cell["\<\ Step 1: Write individual reproductive success W of a random recipient as a \ function of the behaviour of the active classes\ \>", "Definition", FontSize->16], Cell["\<\ We now have to write male and female fitness as a function of the behaviour \ of two active classes: (1) females, who may adjust their probability of \ becoming a queen, and (2) workers, who may or may not inhibit immature \ females from developing as a queen. An adequate equation for the average \ success of a colony or swarm would be \ \>", "Text"], Cell[BoxData[ \(\(G = 1 - z*\((1 - P)\) - c*P;\)\)], "Input"], Cell["\<\ because G=1-z in the absence of caste policing (P=0), but 1-c when workers \ can prevent (P=1) any excess queen production (where such prevention is \ assumed to come at a colony level productivity cost c). Similarly, we can \ write the relative individual success of a female larva as \ \>", "Text"], Cell[BoxData[ \(\(Ind = P + \((1 - P)\)*\((y/z)\);\)\)], "Input"], Cell["\<\ because in the absence of worker policing (P=0) the relative success of any \ female larva is given as y/z (the above-average probability of developing as \ a queen), whereas with worker policing (P=1), it becomes impossible for any \ female to gain an advantage over any other (Ind=1). Consequently, the fitness through male and female function are given as\ \>", "Text"], Cell[BoxData[{ \(\(Wm = G;\)\), "\[IndentingNewLine]", \(\(Wf = Ind*G;\)\)}], "Input"], Cell["\<\ Step 2: Analyse when the slope of a recipient's fitness on its genotype is \ positive\ \>", "Definition", FontSize->16], Cell["\<\ We now have 2 equations: one specifying selection for female larvae to become \ queens, the other specifying selection for worker caste policing, i.e. we are \ again analysing a 2-locus kin selection problem. As before, these two \ selection equations need to be simultaneously maximised to provide the joint \ ESS. For a female larva, a higher probability of developing as a queen is favoured \ when dW/db >0, with dW/db =\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(dWdb[Rf_, Rm_, c_, z_, P_] = \((D[Wf, y] + D[Wf, z]*Rf + D[Wm, z]*Rm)\) /. {y \[Rule] z}\)], "Input"], Cell[BoxData[ \(\((\(-1\) + P)\)\ Rm + \(\((1 - P)\)\ \((1 - c\ P - \((1 - P)\)\ z)\)\)\/z + Rf\ \((\(-1\) + P - \(\((1 - P)\)\ \((1 - c\ P - \((1 - P)\)\ z)\)\)\/z)\)\)], \ "Output"] }, Open ]], Cell["\<\ where derivatives are evaluated near the ESS where y = z = z* and P = P* and \ terms involving Dt[P,bf] and Dt[P,bm] are dropped, because we assume \ that the caste determining locus of female larvae is not linked to the worker \ caste policing locus. \ \>", "Text"], Cell["\<\ Similarly, an increase in the investment in worker caste policing is favoured \ when\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ equation\ for\ selection\ for\ policing\ \ *) \)\(\[IndentingNewLine]\)\(dWdp[Rf_, Rm_, c_, z_, P_] = Simplify[\((D[Wf, P]*Rf + D[Wm, P]*Rm)\) /. {y \[Rule] z}]\)\)\)], "Input"], Cell[BoxData[ \(\((Rf + Rm)\)\ \((\(-c\) + z)\)\)], "Output"] }, Open ]], Cell["\<\ where derivatives are also evaluated near the ESS where y = z = z* and P = P* \ and terms involving Dt[y,pf], Dt[z,pf], Dt[y,pm] or Dt[z,pm] are dropped, \ because we assume that the caste determining locus of female larvae is not \ linked to the worker caste policing locus. \ \>", "Text"], Cell["Step 3: Joint maximisation to provide the joint ESS", "Definition", FontSize->16], Cell["\<\ Joint maximisation of these 2 equations is difficult, but as a first start \ let us check when worker caste policing can invade in a sitation where all \ larvae are becoming queens with their own optimum probability \ (z*=(1-Rf)/(1+Rm), see Section 3.1 above). For this, set the probability of \ developing as a queen (z*) to (1-Rf)/(1+Rm) and the level of worker caste \ policing P to 0 in the selection equation for policing, and see when an \ increase in policing is favoured.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ equation\ for\ selection\ for\ policing\ with\ z = \(\((1 - Rf)\)/\((1 + Rm)\)\ and\ P = 0\)\ *) \)\(\[IndentingNewLine]\)\(dWdp[Rf, Rm, c, \((1 - Rf)\)/\((1 + Rm)\), 0]\)\)\)], "Input"], Cell[BoxData[ \(\((Rf + Rm)\)\ \((\(-c\) + \(1 - Rf\)\/\(1 + Rm\))\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ for\ an\ increase\ in\ policing\ to\ be\ favoured\ this\ should\ be\ > 0\ *) \)\(\[IndentingNewLine]\)\(Solve[\(-c\) + \(1 - Rf\)\/\(1 + \ Rm\) \[Equal] 0, c]\)\)\)], "Input"], Cell[BoxData[ \({{c \[Rule] \(-\(\(\(-1\) + Rf\)\/\(1 + Rm\)\)\)}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Conclusion :", FontSize->18, FontWeight->"Bold"], "\nWhen the cost of policing c < (1-Rf) / (1+Rm), worker caste policing, \ i.e. social control over caste development, is favoured. Below it is checked \ graphically that whenever worker caste policing invades it will also tend to \ be maximally favoured. This can best be done using field plots.\nThe \ intuitive explanation for this result is that excess queen production will \ reduce colony productivity by a fraction (1-Rf) / (1+Rm), and that worker \ caste policing can prevent this to happen at a productivity cost c (the cost \ of policing). Thus caste policing is favoured when the cost of policing < the \ cost of having a colony with excess queen production. \nSocial control over \ caste development is the reason why ", StyleBox["Apis", FontSlant->"Italic"], " produces few queens, in keeping with colony needs: queens are larger than \ workers, and if females are to become queens, they need to be reared in \ specially built cells and given special food (royal jelly). Note, however, \ that adult workers constantly force individual females to become workers, \ even when this is not in the inclusive fitness interests of any single female \ (check from eqn. 3.5 what their optimum is). This means that the increased \ efficiency of a honeybee relative to a ", StyleBox["Melipona", FontSlant->"Italic"], " colony also comes at a cost: a complete loss of individual freedom. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["3.2.1 GRAPHICAL SOLUTION", "Subsection", FontSize->19], Cell["\<\ As in section 2.2.2, the way the ESS is reached can also be illustrated \ graphically using a field plot.\ \>", "Text"], Cell[BoxData[{ \( (*\ load\ the\ PlotField\ package\ to\ enable\ you\ to\ draw\ field\ plots\ \ and\ the\ ImplicitPlot\ package\ to\ allow\ you\ to\ make\ plots\ of\ the\ \ equilibrium\ lines\ *) \[IndentingNewLine]<< Graphics`PlotField`\), "\[IndentingNewLine]", \(<< Graphics`ImplicitPlot`\)}], "Input"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Case 1 : c > (1-Rf)/(1+Rm)\n(Rf=0.75, Rm=0.25, \ c=0.5)", FontSize->17]], "Subsubsection"], Cell[BoxData[ \(\(\(\(Rfex = 0.75\) \)\(;\)\(\(Rmex = 0.25\) \)\(;\)\(cex = 0.5\)\(;\)\(\ \ \ \ \ \ \)\( (*\ Some\ example\ values\ for\ Rf, \ Rm\ and\ c\ *) \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(fieldplot = PlotVectorField[{dWdp[Rfex, Rmex, cex, z, P], \ dWdq[Rfex, Rmex, cex, z, P]}, \n\ \ \ \ 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With what probability does a female larva in an ", StyleBox["Apis", FontSlant->"Italic"], " honeybee colony prefer to develop as a queen? Why is the observed level \ of queen production lower than this? " }], "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["In honeybee colonies, Rf=0.15 due to multiple mating \ and Rm=0.25. Therefore, any larva is selected to become a queen with a \ probability of (1-0.15)/(1+0.25)=68%. Of course, Apis colonies produce far \ fewer queens than this, usually just a few over a whole season. This is \ because caste is under complete social control in honeybees. Note that caste \ is often said to be non-genetically determined, i.e. set purely by the \ feeding regime imposed on by the workers. But could the way workers feed \ larvae not be heritable? ", FontSlant->"Italic"]], "Text", FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 2. In an unpubished study, Strassmann et al. have produced a caste conflict \ model using Hamilton's constant cost/benefit rule.What is the problem with \ such an approach? \ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["The assumption of constant costs and benefits is \ inappropriate. E.g., the benefit of becoming a queen depends on what other \ larvae do within the colony (the benefit becomes less as more larvae develop \ as queens). With frequency dependent costs and benefits (as derived above) \ one can calculate the ESS probability for a larva to develop as a queen, but \ with a constant cost/benefit rule this is not possible.", FontSlant->"Italic"]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "3. In ", StyleBox["Melipona ", FontSlant->"Italic"], "nearly all virgin queens are killed by the workers soon after eclosion. \ What is there to gain then from becoming a queen? That is, why develop as a \ queen with a probability of 20% if no one is going to survive anyway? " }], "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["No, queen execution by workers does not affect the \ optimum probability for female larvae to develop as queens. The reason is \ that if, say, half of all queens get executed, the remaining queens' fitness \ will be twice as high, since they compete with half as few queens for heading \ a daughter swarm. Probability one half of twice as much yields the same \ payoff. The same argument has been used to show that if many males die before \ mating, this has no influence on optimal sex allocation. ", FontSlant->"Italic"]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 4. A slight artificiality of the model is that it assumes that zero \ allocation to queens is the colony optimum. Actually, it may be that colonies \ that produce 5 queens do better than queens that produce only 1 queen (and \ certainly better than colonies that produce no queens), because they can \ split in more daughter colonies. How could this problem be alleviated?\ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["One way around this would be to assume that the \ optimum allocation to queens from a colony level perspective is k, and to use \ a colony success function G that has a maximum at k and then decreases as \ more larvae become queens. 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