(*********************************************************************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 406059, 13104]*) (*NotebookOutlinePosition[ 406810, 13130]*) (* CellTagsIndexPosition[ 406766, 13126]*) (*WindowFrame->Normal*) Notebook[{ Cell["4. Assignments", "Title", TextAlignment->Left, FontFamily->"Arial", FontSize->40, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.500008, 0, 0.500008], Background->GrayLevel[1]], Cell[CellGroupData[{ Cell["Purpose :", "Section", FontSize->25], Cell[TextData[{ "- Over the next few days you will be given a chance to carefully go over \ the enclosed ", StyleBox["Mathematica", FontSlant->"Italic"], " sheets and the Questions & Answers sections to better understand all \ this. Enthusiasts can even attempt to solve some problems for themselves - \ below are some suggestions. \n- Depending on how much previous experience you \ have in modelling, you can either select an easy or a difficult problem. Each \ problem below has an associated difficulty score ranging from * (easy), over \ ** (intermediate) to *** (difficult) and **** (masterful). For people with no \ previous experience in modelling it may be best to just go through the ", StyleBox["Mathematica", FontSlant->"Italic"], " sheets and attempt to solve the very easy problems mentioned under ", StyleBox["Questions & Answers", FontSlant->"Italic"], ". \n- Below there is also a section of ", StyleBox["'Sociobiology Classics'", FontSlant->"Italic"], " - classical problems that have featured prominently in the social \ evolution literature: Trivers & Hare's (1976) sex ratio theory, sex ratios \ under local mate competition (Hamilton 1967, 1979), split sex ratio theory \ (Boomsma & Grafen 1991), Hamilton & May's (1977) dispersal model and skew \ theory (e.g. Reeve & Ratnieks 1993). The enclosed ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook contains model solutions to each of these problems (and some of \ the other problems too).\n- Finally, I have prepared a list of discussion \ topics. These are more general thoughts about where our field might be \ heading. Ideal to discuss over beer. " }], "Text", FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell["4.1 Problems related to Section 1 on 2 Player Games", "Section", FontSize->25], Cell[CellGroupData[{ Cell["\<\ 1. (*) Derive the mixed ESS level of cooperation for the 2-player game with \ general payoffs E(C,C), E(C,D), E(D,C) and E(D,D) using the method outlined \ in section 1.1. \ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], Cell["Mixed ESS :", "Text", FontSlant->"Italic"], Cell[BoxData[ \(\(w1 = y1*y2*ECC + \((1 - y1)\)*y2*EDC + y1*\((1 - y2)\)*ECD + \((1 - y1)\)*\((1 - y2)\)*EDD;\)\)], "Input"], Cell[BoxData[ \(\(selection = \((D[w1, y1] + D[w1, y2]*r)\) /. {y2 \[Rule] y1};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(mESS = Solve[selection \[Equal] 0, y1]\)], "Input"], Cell[BoxData[ \({{y1 \[Rule] \(-\(\(\(-ECD\) + EDD - EDC\ r + EDD\ r\)\/\(\(-ECC\) + ECD + EDC - EDD - ECC\ r + ECD\ r + EDC\ r - EDD\ r\)\)\)}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 2. (*) Check the equilibrium frequency of driving genes in a population as \ derived by Wenseleers & Ratnieks (submitted) using the method as outlined in \ section 1.2.\ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], Cell["Pure ESS :", "Text", FontSlant->"Italic"], Cell[BoxData[ \(\(w1 = y1*y2*GDD/2 + \((1 - y1)\)*y2*GDC*\((1 - k)\) + y1*\((1 - y2)\)*GDC*k + \((1 - y1)\)*\((1 - y2)\)*1/2;\)\)], "Input"], Cell[BoxData[ \(\(wDRIVING = w1 /. {y1 \[Rule] 1, y2 \[Rule] p + r*\((1 - p)\)};\)\)], "Input"], Cell[BoxData[ \(\(wCOOPERATIVE = w1 /. {y1 \[Rule] 0, y2 \[Rule] p - r*p};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(pESS = Solve[wDRIVING \[Equal] wCOOPERATIVE, p]\)], "Input"], Cell[BoxData[ \({{p \[Rule] \(-\(\(\(-1\) + 2\ GDC\ k + GDD\ r - 2\ GDC\ k\ r\)\/\(1 - 2\ GDC + GDD - r + 2\ GDC\ r - GDD\ r\)\)\)}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 3. (**) Investigate the evolution of reciprocal altruism. For this, replace \ the 'Cooperate' strategy in the 2-player mixed strategy game by 'TIT-FOR-TAT' \ (TFT, cooperate on the first move and then do what your opponent did on the \ previous move); let defect stand for unconditional defection ('always \ defect', AD). Assume that the game is played an unknown number of times, and \ that after each play of Prisoner's dilemma, the next play will occur with \ probability p, with 0 < p < 1 (this game is known as the iterated Prisoner's \ dilemma). Calculate the payoffs E(AD,AD), E(AD,TFT), E(TFT, AD) and E(TFT, \ TFT), and check (1) when TFT can invade an all-defecting population and (2) \ when AD can invade an all-TFT population. Assume for simplicity that the \ players are unrelated to each other.\ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ StyleBox["The payoff of TFT playing against itself (E(TFT,TFT)) is\n\ (b-c)+p(b-c)+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^2\)]], StyleBox["(b-c)+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^3\)]], StyleBox["(b-c)+\[Ellipsis]=(b-c)/(1-p) (see proof below)\nThe other \ payoffs remain unaffected: E(AD,AD)=0, E(TFT,AD)=-c and E(AD,TFT)=b. \nWith \ these payoffs it is easy to check that (1) TFT can never invade an \ all-defecting population (because you need to be two to gain any advantage) \ and (2) that AD invades an all-TFT population when b.p < c. Note the \ similarity of the latter condition to Hamilton's rule, with p taking over the \ role of relatedness ", FontSlant->"Italic"], StyleBox["[in fact p is a nongenetic type of relatedness (Skyrms 1996); as \ Hamilton (1971, p. 65) put it: \[OpenCurlyDoubleQuote]Rather than continue in \ the jangling partnership, the disillusioned cooperator can part quietly from \ the selfish companion at the first clear sign of unfairness and try his luck \ in another union. The result would be some degree of assortative pairing.\ \[CloseCurlyDoubleQuote]]. ", FontSlant->"Italic"], StyleBox["Putting (genetic) relatedness into the equation one would find \ (1) that TFT can only invade an all-defecting population when b.r > c and (2) \ that AD invades an all-TFT population when b.(p+r)/(1+p.r) < c. All this \ means that if non-cooperation is the primitive condition, it is difficult to \ see how reciprocal altruism could evolve from it. Axelrod & Hamilton (1981) \ suggest that cooperative behaviour might originate as altruism between \ relatives selected by kin selection and then spread to encompass \ nonrelatives, or it might spread from a small cluster of cooperative \ individuals. But clearly, the transition from nonnoncooperation to reciprocal \ altruism is a difficult one.\nproof: \nfor any p\[NotEqual]1, 1+p+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^2\)]], StyleBox["+\[Ellipsis]+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^\(m - 1\)\)]], StyleBox["=(1-", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^m\)]], StyleBox[")/(1-p) since (1-p)(1+p+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^2\)]], StyleBox["+\[Ellipsis]+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^\(m - 1\)\)]], StyleBox[")=1-", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^m\)]], "\n", StyleBox["now, as m goes to infinity, ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^m\)]], StyleBox[" goes to zero so that 1+p+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^2\)]], StyleBox["+", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`p\^3\)]], StyleBox["+\[Ellipsis]=1/(1-p)", FontSlant->"Italic"] }], "Text"], Cell[TextData[StyleBox["Or you can just count on Mathematica's intelligence \ :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((b - c)\)*\(\[Sum]\+\(i = 0\)\%\[Infinity] p\^i\)\)], "Input"], Cell[BoxData[ \(\(b - c\)\/\(1 - p\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Smarter than I thought!", FontSlant->"Italic"]], "Text"], Cell[BoxData[ \(\(w1 = y1*y2*\((B - C)\)/\((1 - p)\) + \((1 - y1)\)*y2*B + y1*\((1 - y2)\)*\((\(-C\))\) + \((1 - y1)\)*\((1 - y2)\)*0;\)\)], "Input"], Cell[BoxData[ \(\(selection = \((D[w1, y1] + D[w1, y2]*r)\) /. {y2 \[Rule] y1};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(mESS = Solve[selection \[Equal] 0, y1]\)], "Input"], Cell[BoxData[ \({{y1 \[Rule] \(\((\(-1\) + p)\)\ \((C - B\ r)\)\)\/\(\(-B\)\ p + C\ p - \ B\ p\ r + C\ p\ r\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(selection = \((D[w1, y1] + D[w1, y2]*r)\) /. {y2 \[Rule] 0, y1 \[Rule] 0}\)], "Input"], Cell[BoxData[ \(\(-C\) + B\ r\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "4a. (***) Godfray (1987) uses an explicit genetic model to derive when \ parasitoid wasp larvae should either tolerate or eat each other within their \ host. Show that Godfray's results can be derived in a much easier way using \ Steve Frank's methods. For simplicity, model only the case with a clutch size \ ", StyleBox["c", FontSlant->"Italic"], " of 2. To parallel Godfray's notation, assume the following payoffs to \ larva 1: E(fighter,fighter)=1/2 (it survives half of the time), \ E(fighter,tolerant)=1 (the fighter larva gets the host for itself), \ E(tolerant,fighter)=0 (the tolerant larva gets killed) and E(tolerant, \ tolerant)=f where f is the ratio of the reproduction as a pair over the \ reproduction alone. Furthermore, assume that competing larvae are of a random \ sex and that their mother is singly mated so that the average relatedness \ among them is \.bd. Calculate (1) when tolerance can invade in an all-fighter \ population and (2) when fighting will invade in an all-tolerant population \ assuming either discrete or mixed strategies. Also calculate the mixed and \ pure strategy ESSs, not calculated in Godfray's paper. \n4b. Assume the \ parasitoid wasp mother mates twice. How would this affect the likely \ evolution of tolerance? If the level of siblicide among larvae were \ conditional on relatedness structure, how would this affect the op-timal \ mating strategy of the mother?\n4c. In parasitoid wasps, after egg \ deposition, larvae are left on their own, i.e. there is no parental care. How \ could parental care affect the possible outcome? " }], "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], Cell[TextData[StyleBox["4a.\nBulmer (1994, p. 193) mentions that this problem \ represents \"an extreme example of a situation in which inclusive-fitness \ arguments cannot be used, because the costs and benefits of altruism are not \ constant\". But let's see how we can tackle it using the methods above.", FontSlant->"Italic"]], "Text"], Cell[TextData[StyleBox["First, write the fitness of larva 1 as a function of \ its own (y1) and its partner (y2) phenotype (whether or not to behave \ tolerant), as in ", FontSlant->"Italic"]], "Text"], Cell[BoxData[ \(\(w1 = \((1 - y1)\)*\((1 - y2)\)*\((1/2)\) + \((1 - y1)\)*y2*1 + y1*y2*f;\)\)], "Input"], Cell[TextData[StyleBox["MIXED STRATEGY CASE :\nUnder 'relaxing assumptions', \ point 7 'Incomplete penetrance' (p. 229) Godfray mentions this case.\nAn \ increase in the probability of playing 'tolerant' is favoured when (eqn. 1.1) \ :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(selectiontol = \((D[w1, y1]*1 + D[w1, y2]*r)\) /. {y1 \[Rule] y, y2 \[Rule] y}\)], "Input"], Cell[BoxData[ \(1\/2\ \((\(-1\) + y)\) - y + f\ y + r\ \((1 + 1\/2\ \((\(-1\) + y)\) - y + f\ y)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Where the cost of playing tolerant = D[w1,y1]= \ -(1/2)(1-y2)-(1-f).y2 and the benefit of having a tolerant partner = \ D[w1,y2]=(1/2)(1-y1)+f.y1. Again, as in the hawk-dove game, one can see that \ the costs and benefit depend on what the opponent does so that Hamilton's \ assumption of constant costs & benefits is violated. But a Hamilton's rule \ defined as in eqn. 1.1 is always valid for the mixed or continuous strategy \ case (for a derivation of a Hamilton's rule that is also correct for the \ discontinuous strategy case, see Wenseleers & Ratnieks submitted). \nNear the \ ESS, y1=y2=y, so that the equation above can be solved to yield the ESS \ probability with which any larva should behave tolerantly : \ y*=(1-r)/[(2f-1)(1+r)]", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(mESS[r_, f_] = \(\(\(Solve[selectiontol \[Equal] 0, y]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)], "Input"], Cell[BoxData[ \(\(-\(\(\(-1\) + r\)\/\(\(-1\) + 2\ f - r + 2\ f\ r\)\)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Tolerance will invade in an all-fighting population \ when", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((selectiontol /. {y \[Rule] 0})\) > 0\)], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\) + r\/2 > 0\)], "Output"] }, Open ]], Cell[TextData[StyleBox["which can never be the case, even not when two larvae \ would be identical twins ! (cf. Bulmer 1994, p. 194 and Godfray 1987 p. 229 \ 1st paragraph with penetrance z approaching zero)\nFighting will invade in an \ all-tolerant population when", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((selectiontol /. {y \[Rule] 1})\) < 0\)], "Input"], Cell[BoxData[ \(\(-1\) + f + f\ r < 0\)], "Output"] }, Open ]], Cell[TextData[StyleBox["i.e. when f < 2/3 for r = \.bd. (cf. Godfray 1987 p. \ 229 eqn. 12 with clutch size c=2)", FontSlant->"Italic"]], "Text"], Cell[TextData[{ StyleBox["DISCRETE (PURE) STRATEGY CASE :\nTolerance is favoured over \ fighting when the expected reproduction of larva 1 is higher as a tolerant \ than as a fighter (eqn. 1.2). ", FontSlant->"Italic"], "\n", StyleBox["The fitness of larva 1 when it is of the tolerant type is ", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Wtolerant = w1 /. {y1 \[Rule] 1, y2 \[Rule] p + r*\((1 - p)\)}\)], "Input"], Cell[BoxData[ \(f\ \((p + \((1 - p)\)\ r)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["The fitness of larva 1 when it is of the fighter type \ is ", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Wfighter = w1 /. {y1 \[Rule] 0, y2 \[Rule] p + r*\((0 - p)\)}\)], "Input"], Cell[BoxData[ \(p - p\ r + 1\/2\ \((1 - p + p\ r)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Tolerance will invade in an all-fighting population \ (p=0) when", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\((Wtolerant - Wfighter)\) /. {p \[Rule] 0})\) > 0\)], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\) + f\ r > 0\)], "Output"] }, Open ]], Cell[TextData[StyleBox["i.e. when f > 1 for r =\.bd. This requires that an \ individual should have a higher fitness as one of a pair than when alone \ (implying some unlikely Allee effect). This result is identical to the \ population genetic result of Godfray 1987 (eqn. 2 p. 223), and shows how \ difficult it is to evolve tolerance once fighting has evolved. Bull & Charnov \ (1985) call this irreversible evolution.\nFighting will invade in an \ all-tolerant population (p=1) when ", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\((Wfighter - Wtolerant)\) /. {p \[Rule] 1})\) > 0\)], "Input"], Cell[BoxData[ \(1 - f - r\/2 > 0\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["For r =\.bd this is when f < 3/4. By setting c=2 in eqn. 5 of \ Godfray 1987, it can again be checked that this is identical to the \ population genetic result. ", FontSlant->"Italic"], "\n", StyleBox["A pure strategy ESS is reached when the fitness of a fighting \ larva equals the fitness of a tolerant larva, which is when", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[Wtolerant \[Equal] Wfighter, p]\)], "Input"], Cell[BoxData[ \({{p \[Rule] \(-\(\(1 - 2\ f\ r\)\/\(\((\(-1\) + 2\ f)\)\ \((\(-1\) + r)\)\)\)\)}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Overall it can be seen that this method is much easier, and \ vastly more economical and general than the corresponding population genetic \ model, e.g. you don't need to compile massive mating tables, it does not \ require any complex matrix algebra, is valid under any relatedness structure, \ etc\[Ellipsis] Nevertheless, if you want to get your papers into Am. Nat. you \ may be forced to give your models more of an air of mathematical sophistry \ (and pointing out that siblicide in parasitoid wasps is just like the \ hawk-dove game won't help).", FontSlant->"Italic"], "\n\n4", StyleBox["b. The average relatedness among the mother's offspring of random \ sex is given by r= \.bc.rMM+\.bc.rMF+\.bc.rFM +\.bc.rFF where rMM, rMF, rFM \ and rFF stand for relatedness between 2 male offspring, between a male and a \ female offspring, between a female and a male offspring and between 2 female \ offspring (we assume a 1:1 sex ratio in the brood). Under single mating rMM =\ \.bd , rMF =\.bd , rFM =\.bc and rFF =\.be , giving r= \.bd. But under \ double mating, rFF drops from \.be to \.bd so that r= 7/16, which is lower \ than \.bd. This lower relatedness will make it harder for tolerance to evolve \ (substitute in the above equations). \nIf larvae adjust their probability of \ being siblicidal based on the average relatedness within the brood, it might \ pay for mothers not to mate twice (or not to superparasitise a host, which \ may be another cause of low relatedness).", FontSlant->"Italic"], "\n\n4", StyleBox["c. If the parent rears her offspring to adulthood he/she might \ prevent offspring from behaving aggressively, or punish them if they behave \ aggressively. Punishment, in turn, may make it unprofitable for any offspring \ to even try to behave aggressively. This may be why so many parasitoid wasps \ have larvae with huge mandibles, adapted for fighting, something that is \ rarely seen in species with parental care. Hamilton (1964) suggested that \ bees and wasps might have evolved combs to prevent larvae from eating each \ other. ", FontSlant->"Italic"], "\n" }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 4.2 Problems related to Sections 2 and 3 on Tragedies of the Commons\ \>", "Section", FontSize->25], Cell[CellGroupData[{ Cell["\<\ 1.(*) Section 2: When a social insect colony loses its queen it seems likely \ that not all workers are totipotent and can switch to a reproductive role \ (this should be true especially for foragers with regressed ovaries). Also, \ the workers that do become reproductive may be able to both lay eggs and do \ some foraging. How could this be accomodated in the worker reproduction \ model?\ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["If t is the fraction of totipotent workers then a \ fraction (1-t) might keep foraging. This tends to reduce the costs to colony \ function, making a higher level of breeding favourable from the perspective \ of the totipotent workers. To take this into account in Model 1, let colony \ success be (1-z(1-t)) instead of just (1-z). By the same methods as above, \ this yields an ESS probability of becoming a reproductive worker of \ (1-r)/(1-t) :", FontSlant->"Italic"]], "Text", FontSize->16], Cell[CellGroupData[{ Cell[BoxData[{ \(\(w = \((y/z)\)*\((1 - z*\((1 - t)\))\);\)\), "\[IndentingNewLine]", \(\(dwdg = Dt[w, g] /. {Dt[y, g] \[Rule] 1, Dt[z, g] \[Rule] r, Dt[t, g] \[Rule] 0, y \[Rule] z};\)\), "\[IndentingNewLine]", \(mESS[r_, t_] = \(\(\(Solve[dwdg \[Equal] 0, z]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)}], "Input"], Cell[BoxData[ \(\(\(-1\) + r\)\/\(\(-1\) + t\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["For example, if half of the workers are totipotent at \ the time the queen is lost (t=0.5), then 50% of them should become \ reproductives instead of just 25% if all are totipotent (this assumes single \ mating). \nIf reproductive workers can spend a fraction f of their time \ foraging, then, again costs to colony function would not be as severe. \ Formally, colony success would become (1-z.(1-f)) rather than just (1-z). So \ this affects the ESS in the same way as above. ", FontSlant->"Italic"]], "Text", FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 2. (*) Section 2: Suppose that colony success is not a decreasing linear but \ a decreasing concave or a convex function of the proportion of reproductive \ workers within the group. How would this affect the predictions of the model?\ \ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ StyleBox["With a concave colony productivity function unrestrained breeding \ would reduce group output more than with a linear cost function. The \ consequence would be that in Model 1, the ESS probability of breeding would \ be lower than with a linear cost function. The inverse is true for a convex \ productivity function. To formalise this, let group success be G=", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\((1 - z)\)\^k\)], FontSlant->"Italic"], StyleBox[" instead of just (1-z) in Model 1 (k=1, >1 and <1 then \ correspond to a linear, a concave and a convex cost function). By the same \ methods as above, this yields an ESS probability of becoming a reproductive \ worker of z*=(1-r)/(1+r.(k-1)) :", FontSlant->"Italic"] }], "Text", FontSize->16], Cell[CellGroupData[{ Cell[BoxData[{ \(\(w = \((y/z)\)*\((1 - z)\)\^k;\)\), "\[IndentingNewLine]", \(\(dwdg = FullSimplify[ Dt[w, g] /. {Dt[y, g] \[Rule] 1, Dt[z, g] \[Rule] r, Dt[k, g] \[Rule] 0, y \[Rule] z}];\)\), "\[IndentingNewLine]", \(mESS[r_, k_] = \(\(\(Solve[dwdg \[Equal] 0, z]\)[\([2]\)]\)[\([1]\)]\)[\([2]\)]\)}], "Input"], Cell[BoxData[ \(\(-\(\(\(-1\) + r\)\/\(1 - r + k\ r\)\)\)\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 3. (**) Section 3: A slight artificiality of the caste conflict 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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"Input"], Cell[BoxData[ \({z \[Rule] \(1 + \((\(-1\) + k)\)\ Rf + k\ Rm\)\/\(1 + Rm\)}\)], \ "Output"] }, Open ]], Cell[TextData[StyleBox["e.g. for k=0.05 z*=24% rather than 20% if k=0. But \ more realistically, k is probably in the range of 0.1%, in which case the \ deviation from the simple model becomes negligibly small.", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ESS /. {Rf \[Rule] 0.75, Rm \[Rule] 0.25, k \[Rule] 0.05}\)], "Input"], Cell[BoxData[ \({z \[Rule] 0.2400000000000001`}\)], "Output"] }, Open ]] }, Closed]], Cell["\<\ 4. (***) Calculate the ESS probability with which a slime mould slug cell \ should become a spore rather than a stalk cell as a function of the number of \ constituent clones (assume that the different clones are unrelated to each \ other). Assume that the global optimum, i.e. the one that yields the most \ succesful spores, is 20% stalk cells, and that when fewer than 20% stalk \ cells are formed, total net spore production will be reduced (cf. Q&A section \ 3.3 question 4). Initially assume that there are no benefits of being larger \ when composed of 2 or several clones, then relax this assumption. If it were \ found that the ratio of spore to stalk cells is independent of the number of \ constituent clones, how could this be explained? What would be the result if \ cell fate were under social rather than individual cellular control? \ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], Cell[CellGroupData[{ Cell["\<\ 5. (****) Section 2: How could the worker reproduction be extended so that it \ also applies to the evolution of worker laying under queenright condition \ (hint: first take a look at Section 3 on caste conflict)? Should the \ predicted proportion of reproductive workers be higher or lower? And if \ worker policing is allowed for, could it be that it is favoured at any queen \ mating frequency, rather than just at queen mating frequencies higher than \ two, as predicted by Ratnieks (1988)?\ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["Haven't done this yet, but the predicted level of \ worker laying should be lower than under queenless condition, because worker \ reproduction would also reduce the number of queens reared, i.e. it would \ have additional inclusive fitness costs. The idea that worker reproduction \ would reduce colony productivity would still apply, and therefore it seems \ likely that if policing can be performed at low cost, it would be favoured at \ any queen mating frequency (although for a different reason than envisaged by \ Ratnieks (1988), see Question 7). ", FontSlant->"Italic"]], "Text", FontSize->16] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "6. (****) Tragedies of the commons often result from an excessive breeding \ or exploitation ", StyleBox["rate", FontSlant->"Italic"], ". How could one make a more detailed model that derives the optimal \ breeding or exploitation ", StyleBox["rate", FontSlant->"Italic"], " as a function of relatedness?" }], "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], Cell[TextData[StyleBox["No idea. If you know how to do this, let me know. ", FontSlant->"Italic"]], "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["4.3 Sociobiology Classics", "Section", FontSize->25], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["1a. (*) ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["SKEW THEORY", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\nConsider the basic reproductive skew model of Reeve & Ratnieks \ (1993; for an accessible review see Keller & Reeve 1994). Two paper wasps may \ either nest on their own, or share their reproduction on a single nest. Using \ Frank's approach (terminology cf. section 3), derive when (1) a subordinate \ wasp is selected to join a dominant? and (2) how much reproduction the \ subordinate should be able to obtain to make staying worthwile ('the staying \ incentive')? Let r be the relatedness between two foundresses, p the fraction \ of a nest' direct reproduction that is the subordinate's, k the total \ reproductive output of a group with two females, relative to a value of 1 for \ an already established nest with one female (k will usually be >1) and x the \ expected reproductive success of a subordinate if she nests elsewhere, \ relative to a value of 1 for an already established nest (usually x will be \ <1). Let yS stand for the probability with which a subordinate stays.\nTo \ make things easy I have already calculated the payoffs to subordinate and \ dominant as a function of these parameters\n(first payoff is to subordinate, \ second to dominant):\n\n\t\t\t\t\t\t\tDOMINANT\nSUBORDINATE\tLEAVE (prob. \ 1-yS)\t\t(x,1)\n\t\t\tSTAY (prob. yS)\t\t(p.k,(1-p).k)", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"] }], "Subsection"], Cell[TextData[StyleBox["The direct fitness of subordinate and dominant can be \ written as", FontSlant->"Italic"]], "Text"], Cell[BoxData[{ \(\(Wsub = \((1 - yS)\)*x + yS*p*k;\)\), "\[IndentingNewLine]", \(\(Wdom = \((1 - yS)\)*1 + yS*\((1 - p)\)*k;\)\)}], "Input"], Cell[TextData[StyleBox["The subordinate is selected to stay when this eqn. > \ 0", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(selsubstaying = D[Wsub, yS]*1 + D[Wdom, yS]*r\)], "Input"], Cell[BoxData[ \(k\ p + \((\(-1\) + k\ \((1 - p)\))\)\ r - x\)], "Output"] }, Open ]], Cell[TextData[StyleBox["That is when p>(x-r.(k-1))/k(1-r). This is called the \ staying incentive :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(stayingincentive = \(\(\(FullSimplify[ Solve[selsubstaying \[Equal] 0, p]]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)], "Input"], Cell[BoxData[ \(\(r - k\ r + x\)\/\(k - k\ r\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Usually the staying incentive is interpreted as the \ fraction that the dominant should allow to the subordinate to make her stay. \ Actually I think it would be more properly described as the reproduction that \ the subordinate should be able to obtain to make staying worthwile (this is \ not the same, as we will see below).", FontSlant->"Italic"]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["1b. (**) In 1a we derived what fraction of the reproduction the \ subordinate would like to obtain to make staying worthwile ('staying \ incentive'). Usually the staying incentive is interpreted a bit differently \ as the fraction of the reproduction that the ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["dominant", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"Underline"->True}], StyleBox[" would like to allow to the subordinate to make her stay. But is \ the dominant really selected to share any reproduction with the subordinate? \ To investigate this, try to derive what fraction of the reproduction that the \ dominant is selected to give. To do this, substitute ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["p", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" in the previous fitness equations by a continuous character ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["yD ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox["(the fraction of the reproduction that the dominant allows the \ subordinate to have), and make a few field plots of the joint evolution of \ subordinate and dominant behaviour. ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"] }], "Subsection"], Cell[TextData[StyleBox["The direct fitness of subordinate and dominant is the \ same as above, except that we have substituted p by yD", FontSlant->"Italic"]], "Text"], Cell[BoxData[{ \(\(wSUB = \((1 - yS)\)*x + yS*yD*k;\)\), "\[IndentingNewLine]", \(\(wDOM = \((1 - yS)\)*1 + yS*\((1 - yD)\)*k;\)\)}], "Input"], Cell[TextData[StyleBox["The subordinate is selected to stay when this eqn. > \ 0 (cf. eqns. 3.1-3.2 )", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Selsubstaying[k_, x_, r_, yD_] = D[wSUB, yS]*1 + D[wDOM, yS]*r\)], "Input"], Cell[BoxData[ \(\(-x\) + r\ \((\(-1\) + k\ \((1 - yD)\))\) + k\ yD\)], "Output"] }, Open ]], Cell[TextData[StyleBox["The dominant is selected to allow the subordinate a \ greater share of the reproduction when this eqn. > 0 (cf. eqns. 3.1-3.2 )", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Seldomallowsubrepr[k_, r_, yS_] = D[wDOM, yD]*1 + D[wSUB, yD]*r\)], "Input"], Cell[BoxData[ \(\(-k\)\ yS + k\ r\ yS\)], "Output"] }, Open ]], Cell[TextData[StyleBox["From the last eqn. you can see that when subordinates \ are selected to stay (yS=1), dominants are always selected to give the \ subordinate a small as possible share of the reproduction (and not the \ fraction p derived above). This is because -k+k.r=k(r-1) is always <0.", FontSlant->"Italic"]], "Text"], Cell[TextData[StyleBox["But when the dominant does not allow the subordinate \ to reproduce, staying only remains favourable when r > x/(k-1) :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[Selsubstaying[k, x, r, 0] \[Equal] 0, r]\)], "Input"], Cell[BoxData[ \({{r \[Rule] x\/\(\(-1\) + k\)}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["So let's see what happens by drawing a few field \ plots.", FontSlant->"Italic"]], "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["\<\ Case 1 : r > x/(k-1) -> subordinate staying favoured, but dominant is not \ selected to give any reproduction to subordinate (r=0.5, k=1.5, x=0.1)\ \>", "Subsubsection"], Cell[BoxData[ \(\(\(rex = 0.5\) \); \(\(kex = 1.5\) \); xex = 0.1;\)], "Input"], Cell[BoxData[{ \(\(fieldplot = PlotVectorField[{Selsubstaying[kex, xex, rex, yD], \ Seldomallowsubrepr[kex, rex, yS]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ {yS, \ 0, \ 1}, \ {yD, \ 0, \ 1}, PlotPoints -> 17];\)\), "\[IndentingNewLine]", \(\(equillines = ImplicitPlot[{Selsubstaying[kex, xex, rex, yD] \[Equal] 0, \ Seldomallowsubrepr[kex, rex, yS] \[Equal] 0}, {yS, 0, 1}, {yD, 0, 1}, PlotStyle \[Rule] {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(ESS = Graphics[{\[IndentingNewLine]{RGBColor[0, 0, 1], Disk[{1, 0}, 0.02]}, \[IndentingNewLine]{RGBColor[0, 0, 1], Text["\", {0.95, 0.05}]}, \[IndentingNewLine]}];\)\ \ \ \ \ \)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Show[{fieldplot, equillines, ESS}, AspectRatio -> 1, Frame -> True, PlotRange -> {{0, 1}, {0, 1}}, FrameLabel -> {"\", "\"}]\)], 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Of course, one \ could make the model more complex and assume that when the dominant wants all \ reproduction for herself she has to engage in costly aggression. In that \ case, the dominant might be selected to give a small share of the \ reproduction to the subordinate. But then, of course, the subordinate might \ be selected to fight back. We then arrive at the hawk-dove dynamics that we \ discussed under section 1. But note that for such a model one has to specify \ whether one is modelling the mixed or pure strategy case, as one will obtain \ different results for each.", FontSlant->"Italic"]], "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["2a. (**) ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["BASIC SEX RATIO THEORY", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\nRederive using Frank's approach the classical result of Trivers \ & Hare (1976) that social hymenopteran workers are selected to allocate 3 \ times more to sisters than to brothers (assuming single mating) (use the \ terminology of section 3). First assume an infinite population size then try \ to derive the ESS for a population of size N. Let y and z stand for the \ allocation to queens in a focal colony and the population at large (focal \ colony included). ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"] }], "Subsection"], Cell[TextData[StyleBox["The fitness of males and females (queens) are given \ by", FontSlant->"Italic"]], "Text"], Cell[BoxData[{ \(\(Wm = \((1 - y)\)*RMSm;\)\), "\[IndentingNewLine]", \(\(Wf = y*RMSf;\)\)}], "Input"], Cell[TextData[StyleBox["where RMSm and RMSf are the relative mating success \ of males and females, which depends on the average sex allocation in the \ population at large:", FontSlant->"Italic"]], "Text"], Cell[BoxData[ \(RMSm = z; RMSf = \((1 - z)\);\)], "Input"], Cell[TextData[StyleBox["First we assume an infinite population size. The \ equation that says when workers are selected to allocate more resources to \ sisters is :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(selWallocF = \((Dt[Wm, gm] /. {Dt[y, gm] \[Rule] Rm, Dt[z, gm] \[Rule] 0, y \[Rule] z})\)\[IndentingNewLine] + \ \((Dt[Wf, gf] /. {Dt[y, gf] \[Rule] Rf, Dt[z, gf] \[Rule] 0, y \[Rule] z})\)\)], "Input"], Cell[BoxData[ \(Rf\ \((1 - z)\) - Rm\ z\)], "Output"] }, Open ]], Cell[TextData[StyleBox["This is a Hamilton's rule with the cost of reducing \ allocation to males and the benefit to queens equal to male and female mating \ success. ", FontSlant->"Italic"]], "Text"], Cell[TextData[StyleBox["This is the worker's sex allocation optimum :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[selWallocF \[Equal] 0, z]\)], "Input"], Cell[BoxData[ \({{z \[Rule] Rf\/\(Rf + Rm\)}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Under single mating, this is 3/4 allocation to \ sisters, as was to be shown :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[selWallocF \[Equal] 0, z] /. {Rf \[Rule] 0.75, Rm \[Rule] 0.25}\)], "Input"], Cell[BoxData[ \({{z \[Rule] 0.75`}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Actually this is also the sex ratio optimum of an X \ chromosome in diploids, because from the perspective of an X chromosome the \ relatedness to sisters is 3/4 and the relatedness to brothers is 1/4 (under \ male heterogamy).", FontSlant->"Italic"]], "Text"], Cell[TextData[StyleBox["How does small population size influence the optimal \ sex ratio? Assume e.g. that the population consisted of a single colony \ (stranded on a remote Finnish island). In that case the population sex ratio \ z would equal the colony sex ratio y, and Dt[z,gf]=Rf and Dt[z,gm]=Rm. One \ can see that the optimum strategy for workers is then to raise an even sex \ ratio of males and queens, irrespective of relatedness to males or queens:", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(selWallocF = \((Dt[Wm, gm] /. {Dt[y, gm] \[Rule] Rm, Dt[z, gm] \[Rule] Rm, y \[Rule] z})\)\[IndentingNewLine] + \ \((Dt[Wf, gf] /. {Dt[y, gf] \[Rule] Rf, Dt[z, gf] \[Rule] Rf, y \[Rule] z})\);\)\), "\[IndentingNewLine]", \(Solve[selWallocF \[Equal] 0, z]\)}], "Input"], Cell[BoxData[ \({{z \[Rule] 1\/2}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Any selfish biasing of the sex ratio would only \ translate in a reduced mating success of the queens produced. More generally, \ with population size N, and assuming random mating, Dt[z,gf]=(1/N).Rf and \ Dt[z,gm]=(1/N).Rm. The optimum sex ratio (investment in queens) expressed as \ a function of the population size is then", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(selWallocF = \((Dt[Wm, gm] /. {Dt[y, gm] \[Rule] Rm, Dt[z, gm] \[Rule] \((1/N)\)*Rm, y \[Rule] z})\)\[IndentingNewLine] + \ \((Dt[Wf, gf] /. {Dt[y, gf] \[Rule] Rf, Dt[z, gf] \[Rule] \((1/N)\)*Rf, y \[Rule] z})\);\)\), "\[IndentingNewLine]", \(FullSimplify[Solve[selWallocF \[Equal] 0, z]]\)}], "Input"], Cell[BoxData[ \({{z \[Rule] \(N\ Rf + Rm\)\/\(Rf + N\ Rf + Rm + N\ Rm\)}}\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Conclusion: a small population size selects workers \ to be less selfish because the costs of a biased sex ratio would be \ reciprocated directly on the sexuals that are produced. The effect is the \ same as that discussed in section 2 and 3, and is what D.S. Wilson terms weak \ altruism - altruism caused by reciprocated costs of selfishness. ", FontSlant->"Italic"]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["2b. (***) ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["SPLIT SEX RATIOS", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\nAssume we have variation in mating structure with a proportion \ p of all queens doubly mated and a proportion 1-p singly mated. Derive the \ joint sex allocation optimum of workers in single and double mated colonies. \ Derive this graphically, using a field plot. ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"] }], "Subsection"], Cell[TextData[StyleBox["The relative mating success of males and females \ depends on the average sex allocation in the population at large, now given \ as a weighted average of the sex allocation of single and double mated \ colonies (zS and zD are the mean population investment in females by single \ and double mated colonies):", FontSlant->"Italic"]], "Text"], Cell[BoxData[ \(RMSm = zS*\((1 - p)\) + zD*p; RMSf = \((1 - zS)\)*\((1 - p)\) + \((1 - zD)\)*p;\)], "Input"], Cell[TextData[StyleBox["The fitness of males and females (queens) in single \ mated colonies are given by", FontSlant->"Italic"]], "Text"], Cell[BoxData[{ \(\(WmS = \((1 - yS)\)*RMSm;\)\), "\[IndentingNewLine]", \(\(WfS = yS*RMSf;\)\)}], "Input"], Cell[TextData[StyleBox["The fitness of males and females (queens) in double \ mated colonies are given by", FontSlant->"Italic"]], "Text"], Cell[BoxData[{ \(\(WmD = \((1 - yD)\)*RMSm;\)\), "\[IndentingNewLine]", \(\(WfD = yD*RMSf;\)\)}], "Input"], Cell[TextData[StyleBox["Workers in single mated colonies are selected to \ allocate more resources to sisters when :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(selSMColonies[RfS_, RmS_, p_, zS_, zD_] = \((D[WmS, yS]*RmS + D[WfS, yS]*RfS)\) /. yS \[Rule] zS\)], "Input"], Cell[BoxData[ \(RfS\ \((p\ \((1 - zD)\) + \((1 - p)\)\ \((1 - zS)\))\) + RmS\ \((\(-p\)\ zD - \((1 - p)\)\ zS)\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["Workers in double mated colonies are selected to \ allocate more resources to sisters when :", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(selDMColonies[RfD_, RmD_, p_, zS_, zD_] = \((D[WmD, yD]*RmD + D[WfD, yD]*RfD)\) /. yD \[Rule] zD\)], "Input"], Cell[BoxData[ \(RfD\ \((p\ \((1 - zD)\) + \((1 - p)\)\ \((1 - zS)\))\) + RmD\ \((\(-p\)\ zD - \((1 - p)\)\ zS)\)\)], "Output"] }, Open ]], 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[TextData[StyleBox["Some example values : half of all colonies with a \ singly, half with a doubly mated queen :", FontSlant->"Italic"]], "Text"], Cell[BoxData[ \(pex = 0.5; RfSex = 0.75; RmSex = 0.25; RfDex = 0.5; RmDex = 0.25;\)], "Input"], Cell[TextData[StyleBox["Now draw the field plot and equilibrium lines :", FontSlant->"Italic"]], "Text"], Cell[BoxData[{ \(\(fieldplot = PlotVectorField[{selSMColonies[RfSex, RmSex, pex, zS, zD], \ selDMColonies[RfDex, RmDex, pex, zS, zD]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ {zS, \ 0, \ 1}, \ {zD, \ 0, \ 1}, PlotPoints -> 17];\)\), "\[IndentingNewLine]", \(\(equillines = ImplicitPlot[{selSMColonies[RfSex, RmSex, pex, zS, zD] \[Equal] 0, \ selDMColonies[RfDex, RmDex, pex, zS, zD] \[Equal] 0}, {zS, 0, 1}, {zD, 0, 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ratio: workers in single mated colonies are selected to specialise \ on female production, workers in double mated colonies on male production \ (with our parameters, single mated colonies should rear only females; double \ mated colonies should rear 1/3 females and 2/3 males). This is the classical \ idea of split sex ratios, first derived by Boomsma & Grafen (1991), and now \ documented in quite a few species. As a good exercise, you could try to \ extend the analysis to three classes of colonies (e.g. one single mated, one \ double mated and one triple mated) and see what happens using a 3D field plot \ (use the commond ", FontSlant->"Italic"], "PlotField3D[]", StyleBox["). Or you could take the two classes to be monogynous and \ polygynous colonies and see what the optima are.", FontSlant->"Italic"] }], "Text"], Cell[TextData[StyleBox["The average population sex ratio in our case will \ tend to (1/2).1+(1/2).(1/3)=2/3=67%, actually slightly less female-biased \ than when workers would just respond to average relatedness structure (the \ optimum is then 5/7=71% allocation to females). ", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Rf\/\(Rf + Rm\) /. {Rf \[Rule] \((1/2)\)*\((3/4)\) + \((1/2)\)*\((1/ 2)\), Rm \[Rule] 1/4}\)], "Input"], Cell[BoxData[ \(5\/7\)], "Output"] }, Open ]], Cell[TextData[StyleBox["You can also plot the optimal allocation in single \ and double mated colonies as a function of the proportion of doubly mated \ colonies in the population (p) (cf. 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(***) ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["SEX RATIOS UNDER LOCAL MATE COMPETITION", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\nConsider the local mate competition scenario as envisaged by \ Hamilton (1967, 1979) with a life cycle as in fig wasps: (1) N foundresses \ lay eggs within a fig, (2) sexuals emerge, (3) males compete locally for \ mating with undispersed females, (4) mated females disperse, .... What \ optimal sex ratio should a foundress produce? Use the following parameters: N \ = foundress number, y = a single focal foundresses' allocation to males, z = \ the average allocation to males within a fig (average of all foundresses; the \ population average sex allocation is not needed, since mating is independent \ of this; mating occurs locally), rm and rf = the relatedness to a son and \ daughter and Rm = the relatedness between a foundress and an average male \ within a fig (includes sons with probability 1/N and sons of other \ foundresses with a probability of (N-1)/N.\n(hint: allocating more resources \ to sons is a tragedy of the commons: it increases your competitiveness \ relative to other foundresses (more of your own sons will succeed in finding \ a mate), but if all foundresses do this there will be no females left to mate \ with, hence the tragedy). ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"] }], "Subsection"], Cell["\<\ The relative mating success of males and females within a fig is given by\ \>", "Text", FontSlant->"Italic"], Cell[BoxData[ \(RMSm = \((1 - z)\)/z; RMSf = 1;\)], "Input"], Cell["\<\ So that we can write the direct fitness of males and females within a fig \ as\ \>", "Text", FontSlant->"Italic"], Cell[BoxData[{ \(\(Wm = y*RMSm;\)\), "\[IndentingNewLine]", \(\(Wf = \((1 - y)\)*RMSf;\)\)}], "Input"], Cell["\<\ Note that the equation for male fitness is identical to the equation for \ female fitness in the caste conflict model of section 3: (y/z).(1-z). Here, \ allocating more resources to sons is a tragedy of the commons: it increases \ your competitiveness relative to other foundresses (more of your own sons \ will succeed in finding a mate), but if all foundresses do this there will be \ no females left to mate with, hence the tragedy.\ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(Wm\)], "Input"], Cell[BoxData[ \(\(y\ \((1 - z)\)\)\/z\)], "Output"] }, Open ]], Cell["\<\ The selection equation that says when a mother is selected to allocate more \ to males is\ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(SelincrMalloc = \[IndentingNewLine]\((Dt[Wm, gm] /. {Dt[y, gm] -> rm, Dt[z, gm] -> Rm, y \[Rule] z})\)\[IndentingNewLine] + \[IndentingNewLine]\((Dt[ Wf, gf] /. {Dt[y, gf] -> rf, y \[Rule] z})\)\)], "Input"], Cell[BoxData[ \(\(-rf\) - Rm + \(rm\ \((1 - z)\)\)\/z - \(Rm\ \((1 - z)\)\)\/z\)], "Output"] }, Open ]], Cell["So that the ESS allocation to males is given by", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(ESS = \(\(\(Solve[SelincrMalloc == 0, z]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)], "Input"], Cell[BoxData[ \(\(rm - Rm\)\/\(rf + rm\)\)], "Output"] }, Open ]], Cell["\<\ If in each fig there are N foundresses and all foundresses are unrelated to \ each other then Rm=rm/N :\ \>", "Text", FontSlant->"Italic"], Cell[BoxData[ \(\(Rm = rm/N;\)\)], "Input"], Cell["And the ESS allocation to males becomes", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(ESSallocM[rf_, rm_] = FullSimplify[ESS]\)], "Input"], Cell[BoxData[ \(\(\((\(-1\) + N)\)\ rm\)\/\(N\ \((rf + rm)\)\)\)], "Output"] }, Open ]], Cell["\<\ And since the relatedness to sons and daughters is both 1/2, we get an \ unbeatable sex ratio of \ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(ESSfoundress[N_] = ESSallocM[1/2, 1/2]\)], "Input"], Cell[BoxData[ \(\(\(-1\) + N\)\/\(2\ N\)\)], "Output"] }, Open ]], Cell["\<\ This is female biased 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there is LMC. Here are a few examples:\ \>", "Text", FontSlant->"Italic"], Cell["\<\ \tY in XY male: rf=0, rm=1 -> ESS = (N-1)/N (cf. Hamilton 1967) \tcytoplasmic element or Y in XY female: rf=1, rm=0 -> ESS = 0 (cf. \ Hamilton 1967)\ \>", "Text", FontSlant->"Italic"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["3. (****) ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["EVOLUTION OF DISPERSAL", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\nConsider a population of dandelions (", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox["Taraxacum", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox[") that has completely taken in its habitat.", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"], StyleBox[" ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"], StyleBox["Assume that the population is divided into patches and that each \ patch can support only one plant. The plant is annual, i.e. dies at the end \ of the year, and is replaced the next spring. What proportion of dispersing \ seeds should the plant produce at the end of the season ('winged' seeds; in \ proper terminology seeds with a 'pappus', i.e. 'parachute' seeds)? Assume \ that a dispersing seed is equally likely to arrive at any patch and let the \ relative chances of survival of dispersing vs. stay at home seeds be 1-c, \ where c is the cost of dispersal. Calculte the optimum both from the mother \ plant's and an individual seed's perspective and check that (1) the optimal \ dispersal rate is very high even when the chances of surviving are very low \ (high c) (Hamilton & May 1977) and (2) that the mother plant favours higher \ dispersal than any individual seed (Motro 1983). If flowers are pollinated by \ many different bees or bees that have visited many flowers, would this \ influence the optimal dispersal rate? ", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain"] }], "Subsection"], Cell["\<\ Parameters: D = proportion of dispersing seeds produced in the population at large d = proportion of dispersing seeds produced by a given plant \[Delta] = individual seed's probability of becoming a dispersing seed c = cost of dispersal r = relatedness between seeds on a given plant\ \>", "Text", FontSize->18], Cell["\<\ The expected reproductive success of any seed is given as a weighted average \ of the reproductive success of dispersing and stay-at-home seeds (with the \ weighting given by their relative frequencies) :\ \>", "Text", FontSlant->"Italic"], Cell[BoxData[ \(\(w = \[Delta]*Wdisperser + \((1 - \[Delta])\)* Wstayathome;\)\)], "Input"], Cell["\<\ If a seed just inherits the mother's breeding spot, its success will be \ inversely proportional to the proportion of seeds on the same plant that also \ decide to stay at home (1-d) plus the number of seeds that arrive there by \ wind (D of which a fraction 1-c makes it=D.(1-c)).\ \>", "Text", FontSlant->"Italic"], Cell[BoxData[ \(\(Wstayathome = 1/\((\((1 - d)\) + D*\((1 - c)\))\);\)\)], "Input"], Cell["\<\ Similarly, the success of a dispersing seed is proportional to the \ probability of surviving the dispersal act (1-c), but inversely proportional \ to the number of seeds that are already present on the patch where it lands \ (the proportion of nondispersing seeds of the plant in that patch, 1-D) plus \ the number of seeds that arrive there by wind (D of which a fraction 1-c \ makes it=D.(1-c)).\ \>", "Text", FontSlant->"Italic"], Cell[BoxData[ \(\(Wdisperser = \((1 - c)\)*1/\((\((1 - D)\) + D*\((1 - c)\))\);\)\)], "Input"], Cell["\<\ The personal cost of dispersal is then = c / (1-c.D), i.e. the higher the \ cost of dispersal (high c) and the less dispersing seeds produced in the \ population at large (low D), the lower the cost of dispersal\ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(CostDispersal = FullSimplify[\(-D[w, \[Delta]]\) /. {d \[Rule] D}]\)], "Input"], Cell[BoxData[ \(c\/\(1 - c\ D\)\)], "Output"] }, Open ]], Cell[TextData[{ "The benefit of dispersal in terms of reducing competition with related \ seeds on the other hand = (1-D) / ", Cell[BoxData[ \(TraditionalForm\`\((1 - c . D)\)\^2\)]], "), i.e. the more that seeds disperse (high D), and the more that this \ costs (high c), the lower the benefit" }], "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(BenefitDispersal = FullSimplify[D[w, d] /. {\[Delta] \[Rule] D, d \[Rule] D}]\)], "Input"], Cell[BoxData[ \(\(1 - D\)\/\((\(-1\) + c\ D)\)\^2\)], "Output"] }, Open ]], Cell["\<\ Consequently, there will be positive selection for more dispersal when this \ eqn. >0\ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(SelDispersal = \(-CostDispersal\) + BenefitDispersal*r\)], "Input"], Cell[BoxData[ \(\(-\(c\/\(1 - c\ D\)\)\) + \(\((1 - D)\)\ r\)\/\((\(-1\) + c\ D)\)\^2\)], \ "Output"] }, Open ]], Cell["\<\ So that the optimal dispersal rate (probability of becoming a dispersing \ seed) from the perspective of an individual seed is given by\ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(ESSd[c_, r_] = \(\(\(FullSimplify[ Solve[SelDispersal \[Equal] 0, D]]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]\)], "Input"], Cell[BoxData[ \(\(c - r\)\/\(c\^2 - r\)\)], "Output"] }, Open ]], Cell["\<\ This ESS dispersal rate is lower the lower the relatedness among the seeds \ produced by a given plant. This is not surprising: dispersal is favoured \ because it reduces competition with related seeds, and the higher \ relatedness, the more that altruistic dispersal pays off. More surprising is \ how high the predicted level of dispersal is. Even when c approaches 1 (so \ that virtually no dispersing seeds survive) and r=1 (as under \ self-pollination), half of all seeds are selected to disperse. Such a high \ level of altruistic dispersal may actually be deleterious at the population \ level since with a high cost of dispersal the population optimum would be not \ to disperse. This is a truly unusual aspect of this model: in most cases \ selfishness causes a suboptimal outcome at the population or species level, \ but here altruism has a cost at the population level. It can be seen that the ESS dispersal rate is 0 when r < c, i.e. when \ seed-seed relatedness < the cost of dispersal (but if r>c there is a conflict \ between the mother plant and an individual seed over the dispersal rate, see \ below). If flowers are pollinated by pollen from many different flowers, then \ this will tend to reduce relatedness, and hence the optimal dispersal rate. \ Note that r, as in the tragedy of the commons model of section 2, is the \ relatedness to an average seed produced by a given plant, which includes self \ (r=1) with a probability n (if n is the number of seeds produced by a plant) \ and other seeds with a probability of (n-1)/n. \ \>", "Text", FontSlant->"Italic"], Cell["\<\ From the mother's perspective the relatedness to any seed is the same, so \ that from her perspective r =1 (maternal control of seed dispersal makes the \ dispersal probabilities of all seeds on a plant completely correlated; Frank \ 1998, p. 119). Consequently, her optimum is to produce a proportion of \ dispersing seeds = 1/(1+c) (cf. eqn. 1 of Hamilton 1977 with p=1-c):\ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[ESSd[c, 1]]\)], "Input"], Cell[BoxData[ \(1\/\(1 + c\)\)], "Output"] }, Open ]], Cell["\<\ This dispersal rate tends to be higher than that favoured by any individual \ seed, i.e. there is a parent-offspring conflict over the dispersal rate \ (Motro 1983). \ \>", "Text", FontSlant->"Italic"], Cell["\<\ Here the ESS dispersal rate is plotted as a function of the cost of dispersal \ and seed-seed relatedness (upper curve: r=1=self-pollination or mother's \ optimum, middle curve: r=0.5, lower curve: r=0.1).\ \>", "Text", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{ESSd[c, 1], ESSd[c, 1/2], ESSd[c, 1/10]}, {c, 0, 1}, Frame \[Rule] True, FrameLabel \[Rule] {"\", "\"}, PlotRange \[Rule] {0, 1}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: 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Ool00`00Oomoo`08Ool00`00Oomoo`0"], ImageRangeCache->{{{0, 355}, {218.875, 0}} -> {-0.173197, -0.136166, \ 0.00343929, 0.00529989}}] }, Open ]], Cell["\<\ Limitations of this model are that it assumes that each mother plant dies and \ is replaced in each generation, that dispersal occurs over the entire \ population and equally likely to any site and that the habitat is assumed to \ be completely saturated (i.e. there are no vacant sites). But it does \ illustrate some important points. \ \>", "Text", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["4.4 Discussion Topics", "Section", FontSize->25], Cell[CellGroupData[{ Cell[TextData[{ "1. George Price recognized that his equation was a step towards a more \ general theory of selection. Here is the opening paragraph of Price's \ manuscript", StyleBox[" \"The Nature of Selection\"", FontSlant->"Italic"], " (Price 1995) :\n\n", StyleBox["Selection has been studied mainly in genetics but of course there \ is much more to selection than just genetical selection. In psychology for \ example trial-and-error learning is simply learning by selection. In \ chemistry selection operates in a recrystallisation under equilibrium \ conditions, with impure and irregular crystals dissolving, and pure, \ well-formed crystals growing. In paleontology and archaeology selection \ especially favours stones, pottery, and teeth, and greatly increases the \ frequency of mandibles among the bones of hominid skeletons. In linguistics, \ selection unceasingly shapes and reshapes phonetics, grammar, and vocabulary. \ In history we see political selection in the rise of Macedonia, Rome, and \ Muscovy. Similarly, economic selection in private enterprise systems causes \ the rise and fall of firms and products. And science itself is shaped in part \ by selection, with experimental tests and other criteria selecting among \ rival hypotheses.\n\n", FontSize->12], "Price then remarks that he \"hopes that the concept of selection \ \[Ellipsis] will contribute to the future development of \"selection theory\" \ as helpfully as Hartley's concept of information contributed to Shannon's \ communication theory.\" Indeed, the Price equation is already providing \ deeper insight into cultural evolution (e.g. Frank 1998, p. 55), \ trial-and-error learning (Frank 1997a), developmental selection and \ self-organization (Frank 1997). " }], "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["Read Price (1995) and Frank (1995) and reflect on the \ potential of such a \"general theory of selection\". ", FontSlant->"Italic"]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ 2. Currently, self-organisation models are quite succesful at making \ predictions of dynamic aspects of social organisation, but entirely neglect \ the possibility of conflict (e.g. Bonabeau et al. 1996, 1999 consider the \ dynamics of dominance hierarchy formation, but don't seem to be aware of the \ conflicts that arise in this context). Kin selection models on the other hand \ make useful predictions on the evolution of conflict, but are poor at making \ dynamic predictions, and the methods available for studying dynamic kin \ selection problems have not been developed in great detail (but see McNamara \ et al. 1994; Day & Taylor 1997, 2000; Iwasa 2000). For an example of a \ dynamic problem think about how much resources a social insect colony should \ allocate to reproduction vs. growth at any given time. Macevicz & Oster \ (1976) showed that for annual social insects the colony optimum is to produce \ workers early on in the colony lifecycle, and switch to sexual production \ only towards the very end of the season (also see Bulmer 1994, p. 83-84). But \ suppose that workers can lay eggs, would this not favour an earlier onset of \ the sexual (male) production phase? Or if females could manipulate their \ caste fate and become queens would this not favour an earlier onset of the \ queen production phase?\ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["Discuss the relative merits of \ complexity/self-organisation models and evolutionary models. E.g., can both \ make testable predictions? In what ways do the predictions differ? Is there a \ differents emphasis on proximate vs. ultimate aspects of social organisation, \ etc... Hardcore modellers can discuss possible means by which dynamic aspects \ of conflict evolution could be studied. ", FontSlant->"Italic"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 3. In biology we take as a dogma that all behaviour is genetically \ determined. In the social science there is an almost exclusive focus on \ cultural inheritance. The study of how genetic and cultural evolution might \ interact is remarkably slow-going, but there seems to be a prospect that the \ Price equation might bring some formalism to it all. \ \>", "Subsection", FontFamily->"Times New Roman", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[StyleBox["Read the rebellious child model of Frank (1998, p. \ 55), a model of the cultural evolution of an altruistic trait. Think about \ the following: (1) could nurture ever go against nature? (i.e. could a \ culturally inherited trait that harms ones biological fitness spread through \ a population?), (2) are gene-culture arms-races ever expected?, (3) if so, \ which type of evolution will tend to be ahead?, (3) what are the fundamental \ differences in the inheritance laws and the way phenotypic variance is \ created? (e.g. mutation rate etc...). Take self sacrifice propagated by some \ religious sect as an example of a culturally inherited trait. ", FontSlant->"Italic"]], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["4.5 References", "Section", FontSize->25], Cell[TextData[{ "Bonabeau, E., Theraulaz, G. & Deneubourg, J. L. (1996) Mathematical model \ of self-organizing hierarchies in animal societies. ", StyleBox["Bulletin of Mathematical Biology", FontSlant->"Italic"], " 58: 661-717.\nBonabeau, E., Theraulaz, G. & Deneubourg, J. L. (1999) \ Dominance orders in animal societies: The self-organization hypothesis \ revisited. ", StyleBox["Bulletin of Mathematical Biology", FontSlant->"Italic"], " 61: 727-757.\nBulmer, M. (1994) ", StyleBox["Theoretical Evolutionary Ecology. ", FontSlant->"Italic"], "Sinauer Associates Publishers, Sunderland, Massachusetts.\nDay, T. & \ Taylor, P. D. (1997) Hamilton's rule meets the Hamiltonian: Kin selection on \ dynamic characters. ", StyleBox["Proceedings of the Royal Society of London Series B-Biological \ Sciences", FontSlant->"Italic"], " 264: 639-644.\nDay, T. & Taylor, P. D. (2000) A generalization of \ Pontryagin's maximum principle for dynamic evolutionary games among \ relatives. ", StyleBox["Theoretical Population Biology", FontSlant->"Italic"], " 57: 339-356.\nDawkins, R. (1976) ", StyleBox["The Selfish Gene. ", FontSlant->"Italic"], "Oxford University Press.\nFrank, S.A. (1995) George Price's contributions \ to evolutionary genetics. ", StyleBox["Journal of Theoretical Biology", FontSlant->"Italic"], " 175: 373-388.\nFrank, S. A. (1997a). The design of adaptive systems: \ Optimal parameters for variation and selection in learning and development. \ ", StyleBox["Journal of Theoretical Biology", FontSlant->"Italic"], " 184: 31-39.\nFrank, S. A. (1997b). Developmental selection and \ self-organization. ", StyleBox["Biosystems", FontSlant->"Italic"], " 40: 237-243.\nFrank, S.A. (1998) ", StyleBox["Foundations of Social Evolution. ", FontSlant->"Italic"], "Princeton University Press, Princeton, New Jersey.\nGodfray, H.C.J. (1987) \ The evolution of clutch size in parasitic wasps. ", StyleBox["American Naturalist ", FontSlant->"Italic"], "129: 221-233.\nGodfray, H.C.J. & Parker, G.A. (1991) Clutch size, \ fecundity and parent-offspring conflict. ", StyleBox["Philosophical Transactions of the Royal Society of London Series \ B-Biological Sciences ", FontSlant->"Italic"], "332: 67-79.\n", "Hamilton, W. D. & May, R. M. (1977) Dispersal in stable habitats. ", StyleBox["Nature", FontSlant->"Italic"], " 269: 578-581.", "\nIwasa, Y. (2000) Dynamic optimization of plant growth. ", StyleBox["Evolutionary Ecology Research", FontSlant->"Italic"], " 2: 437-455.\nMaynard Smith, J. (1982) Evolution and the Theory of Games. \ Cambridge University Press, New York.\nPrice, G. R. (1995) The nature of \ selection. ", StyleBox["Journal of Theoretical Biology", FontSlant->"Italic"], " 175: 389-396[written circa 1971].\nMacevicz, S. & Oster, G. (1976) \ Modeling social insect populations II: optimal reproductive strategies in \ annual eusocial insect colonies. ", StyleBox["Behavioral Ecology and Sociobiology", FontSlant->"Italic"], " 1: 265-282.\nMcNamara, J. M., Houston, A. I. & Webb, J. N. (1994) Dynamic \ kin selection. ", StyleBox["Proceedings of the Royal Society of London Series B-Biological \ Sciences", FontSlant->"Italic"], " 258: 23-28.\nMotro, U. (1983) Optimal rates of dispersal. 3. \ Parent-offspring conflict. ", StyleBox["Theoretical Population Biology", FontSlant->"Italic"], " 23: 159-168.\nReeve, H.K. & Ratnieks, F.L.W. (1993) Queen-queen conflicts \ in polygynous societies: mutual tolerance and reproductive skew. ", StyleBox["Queen Number and Sociality in Insects ", FontSlant->"Italic"], "(L. Keller, ed.), pp. 45-85. Oxford University Press, Oxford.\nReuter, M. \ & Keller, L. (2001) Sex ratio conflict and worker production in eusocial \ Hymenoptera. ", StyleBox["American Naturalist", FontSlant->"Italic"], " 158: 166-177.\nTrivers, R. L. & Hare, H. (1976) Haplodiploidy and the \ evolution of the social insects. ", StyleBox["Science", FontSlant->"Italic"], " 191: 249-263.\nWenseleers, T. & Ratnieks, F.L.W. Towards a general theory \ of conflict: the sociobiology of mendelian segregation. Submitted. 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(*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)