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Heredity 56 (1986) 119— 121 The Genetical Society of Great Britain Received 20 May 1985

An optimisation principle for sex allocation in a temporally varying environment*

Eric L. Charnov Departmentof , University of Utah, Salt Lake City, UT 84112, U.S.A.

This paper extends the theory of sex allocation to the case of temporal variation in male against options. In particular, it derives a product theorem for the phenotypic equilibrium allocation to versus in a cosexual .

Sex allocation refers to the of the frequency of AB at time T, r ( T + 1) the in dioecious , time (and order) of sex frequency one year (here a generation) later. Fol- change in sequential and the allo- lowing Bull (1981) and Charnov (1979a), the fol- cation of resources to male versus female function lowing linear equation approximates the gene in simultaneous hermaphrodites (Charnov, 1982). frequency dynamics. These problems share common features of Ii /m(P)f(P)\ frequency dependence of fitness (Fisher, 1930), (T+1)e(T) (1) and the phenotypic equilibrium (the ESS of May- nard Smith, 1982) may often be shown to satisfy a certain general optimality principle (Charnov, where m andf refer to the production of pollen 1979a). The problem of temporal variation in male (m)andseed (f) as functions of the resource input against female reproductive opportunities, and its (randF) for the AA common type and the mutant impact on the ESS sex allocation, has been little (see Charnov, 1979b, 1982; and Charnov and Bull, 1985 for further discussion of the details of produc- explored (Werren and Charnov, 1978; Seger, 1983; = Bull, 1981; Stubblefield, 1985). Here, I develop an tion). A simple example is mxrandf= ESS argument for a simultaneous (1— r).ywith n,x, andy>O (Charnov, 1979b). in a temporally varying environment; the ESS allo- If the mandf functions do not vary through cation will be shown to satisfy an optimality prin- time,theESS r (Charnov eta!., 1976)is that value ciple. which maximises: Consider the following infinite sized hermaph- H=log(m f). (2) roditic plant ; generations are discrete and is at random with each individual It is easy to show this using relation (1). Rewrit- having R resources to allocate to either pollen or ing (1) as (T+1)=A e(T), we wish to find r production. At a locus controlling sex alloca- such that A 1 for all i (i.e., the mutant is always tion, let the common homozygote (AA) allocate r selected against). For P near r this can be found proportionof R to pollen, 1 —r toseeds. We con- by the procedure of putting aA/aP =0,setting P =r sider the evolutionary dynamics of a rare dominant and solving for r. This first order condition finds mutant (B) whose bearers allocate P proportion the critical point; it is an ESS provided the tradeoff of R to pollen production. While the mutant is of m versus! is convex (Charnov, 1982; Charnov rare we need only consider the spread of the et aL,1976).If we apply this product relation, heterozygote (AB) to know if risstable to invasion equation 2,to our simple example, H = by some new sex allocation (the P). Let s( T) be log(r'-(1—r)yx)and aH/ar=0 when r= n/(1+n) (n must be<1 for this to be a stable *Thispaper is dedicated to Julia Laylander. point). 120 ERIC L. CHARNOV

Note that the product relation summarises what In the product m f male and female are treated we find in looking at aA/a?=0 when P= r. Now, as equivalent means for reproduction (which they suppose that the m and f relations are functions are). In a temporally changing environment, we of r (and P) and variables which alter through simply retain this equivalence on average. time. Returning to our simple example, y, x, or n For illustration let us apply (6) to the special may change from year to year. Suppose that they case of are random variables whose vaues are set each m = year. Further suppose that the individual cannot x track the temporal variation: what now is the ESS f=y. (1—r) sex allocation? In an unchanging environment, the mutant sex allocation (i.e., AB using P) was selec- where x, y and n are now temporally varying ted against if A <1; the comparable condition for random variables. Now, letting a bar (—) denote our temporally varying environment is E log A <0 average: (Karlin and Liberman, 1974) where E refers to E log (m .f)=E[log (x. r" y (1— r))J expectation. I suggest that we simply apply this new stability criterion in place of the previous one; =log x+ il log r+log y+log (1— r). we can thus find the new ESS r by putting Thus: (9ElogA )/aP =0,setting P =rand solving for r. Now, E log A may be written as follows: 3E log(mf)0ni0 ar r 1—r ElogA=JJ...Jg(xi...xn) or n log ([-+]) dx1,...dx. (3) 1+n (7) where x, refers to a temporally varying random Stochastic variations in y and x have no impact variable which is a parameter in the m and f on the ESS r; indeed the ESS r evolves to simply functions and where g(x1 ... x)is the joint pdf reflect the average n, quite independent of the for the x71. To search for the ESS r we first set amount of temporal variability in n. 9E(logA)/aP=0, then set P=r. This gives: This optimisation principle (just as eq. 2) requires that the hermaphroditism itself be stable, 11 1 i1am/r 0=iI I g(x1...x)—i for example, against pure males or . The .3) J 2Lm deterministic theory (Charnov et a!., 1976; Char- nov, 1982) gives the appropriate stability condi- (4) tions. For our special case, this is n < 1 (diminish- +L]dX1...dX. ing returns on input to pollen: Charnov, 1979b). But, this is simply the condition*: Forthe special case extended to include the tem- poral variability, the comparable condition is ñ < =0. (5\ 1. Conditions for hermaphroditism with time vary- E (+L)mf ing parameters to be stable to invasion by pure sexes are a straightforward generalisation using Condition (5) is the first order condition for E log A <0 in place of A <1. the ESS r to maximise It is simple to extend these hermaphrodite E[log(m •ffl. (6) results to encompass factors affecting male versus female reproductive success in addition to produc- Note how this optimisation principle is simply tion of ; for example, time variation in a logical generalisation of equation 2, which pollinating agents which must carry pollen away applies to a temporally constant environment. To if it is to gain fertilisations (Charnov and Bull, put this result in words, consider first the unchang- 1985). This brief note develops the basic generali- ing environment. The ESS is to maximise sation [max: E log (m .f)]independent of these log (m f); this principle follows because every added complications. gets half of its nuclear genes from male As a final question, we may ask just what repro- function (m) and half from female function (f). ductive factors would be expected to show time * Equation(5)wouldalso follow from using EA< Ias the variation; recall that it is time variation in male stability criterion (and carrying out the same differentiation). compared to female. For instance, if total plant SEX ALLOCATION IN A TEMPORALLY VARYING ENVIRONMENT 121 reproductive resource (R) changes from year to REFERENCES year but this does not affect, in our simple example, the shape of the male gain relation (does BULL,j.j. 1981.Sex ratio when fitness varies. Heredity, 46,9-26. not change n), such variation does not affect sex CHARNOV, E. L. 1979a. The genetical evolution of patterns of sexuality: Darwinian fitness. Amer. Natur., 113, 465-480. allocation. At least three quite changeable things CHARNOv, E. L. 1979b. Simultaneous hermaphroditism and may affect sex allocation. First, total plant resource . Proc.Natn.Acad. Sci. USA, 76,2480-2484. (R) which, contrary to the above illustration, could CHARNOV, a L. 1982. The Theory of Sex Allocation. Princeton well be expected to alter the male gain relation Univ. Press, Princeton, N.J., USA. CHARNOV, E. L. ANDBULL,J. i. 1985. Sex allocation and (i.e., change n). Second, time variation in availabil- pollinator attraction in cosexual . Jour. Theor. Biol., ity of pollinating agents which ought also to affect (in press). n (also perhaps x and y; but note that this does CHARNOV, E. L., MAYNARD SMITH, .1. AND BULL, J. .i. 1976. not affect the ESS allocation). Thirdly, seed disper- Why be an hermaphrodite? Nature, 263, 125-126. sal agents may vary in effectiveness from year to FISHER, R. A. 1930. The Genetical Theory of Natural Selection (reprint 1958). Dover, N.Y. year and alter f (see Charnov and Bull, 1985 for KARLIN, S. AND LIEBERMAN, U. 1974. Random temporal vari- further development). ation in selection intensities: case of large population sizes. It is surprising that the ESS result in equation Theor. Pop. Biol., 6, 355-382. 7 contains only the time average of the variable n; MAYNARD SMITH, .t. 1982. Evolution and the Theory of Games. while the power model (i.e., m cx: r'1) is specific, the Cambridge Univ. Press, Cambridge, U.K. SEGER, .i. 1983. Partial bivoltinism may cause alternating sex- shape parameter n allows a great many shapes to ratio biases that favor eusociality. Nature, 301, 59—62. be discussed. Other models for m and! should be STUBBLEFIELD, .i. w. 1985. Sex allocation in bivoltine popula- studied but my guess is that in general temporal tions: Unbeatable investment ratios and implications for variance in shape will play a minor role and the the origin of eusociality. (Submitted to Jour. Theor. BioL) WERREN, J. H. AND CHARNOV, E. L. 1978. Facultative sex ratio average shape will dominate the ESS. and population dynamics. Nature, 272, 349-350.

Acknowledgments Conversations with J. J. Bull were of much help in thinking about these ideas. Robin Baker helped me to expand the biological discussion.