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(Received 27 August 1985; revised 2 October 1985; MS. number: sc-273)

Convergent Models: Evidence of a Robust Theory of Infanticide

In a recent paper, Glass et al. (1985) used a game theoretical formulation to determine the frequency with which adult males would be expected to pursue either of two alternative reproductive stra- tegies, infanticidal or non-infanticidal, under several different conditions. Al though these auth- ors referred to similar work on this topic by myself and co-workers (Chapman & H a u s f a t e r 1979; Hausfater et al. 1982; Hausfater 1984), they implied that their own analysis was a more general formulation and thereby 'subsumed as special cases' all previous quantitative analyses of infanti- cide. However, I suggest that the model of Glass et al. (1985) does not constitute a more general analysis than earlier formulations, but rather repre- sents an alternative derivation using graphical techniques (and a different notational system) of certain previously published equilibrium equa- tions. The present note served to demonstrate the mathematical equivalence of these various equa- tions as well as to clarify several points concerning the assumptions and conclusions of earlier models.

In their analysis, Glass et al. (1985) determined

the equilibrium frequency of infanticide among adult males in each of four different evolutionary scenarios. In case 1, males pursuing an infanticidal reproductive strategy were assumed to have an absolute (i.e. not frequency-dependent) reproduc- tive advantage compared with males pursuing a non-infanticidal strategy. In case 2, non-infantici- dal males were assumed to have the absolute advantage. Thus in mathematical terms, these two cases are indistinguishable and it should be obvious that the equilibrium in either case is complete fixation of the more advantageous strategy (Haus- later et al. 1982; Glass et al. 1985).

However, f rom both a mathematical and a biological standpoint, two additional cases exam- ined by Glass et al. (1985) are much more interest- ing; these are cases which considered the possibility that a reproductive advantage might be obtained by males pursuing whichever strategy was momen- tarily more common (case 3), or momentari ly more rare (case 4), in their population. Thus in case 3, infanticidal males were assumed to obtain a repro- ductive advantage only when most of the males in the populat ion were infanticidal. However, if most males in the population were non-infanticidal, then infanticidal males were presumed to be at a disad- vantage. Conversely, infanticidal males in case 4 were assumed to obtain a reproductive advantage only when a majority of males in the population were non-infanticidal. Otherwise, infanticidal males were presumed to be at a reproductive disadvantage compared to their non-infanticidal counterparts.

In fact, either of these cases will result in an equilibrium between infanticidal and non-infanti- cidal males, although such an equilibrium is stable only in the latter case (Hausfater et al. 1982; Glass et al. 1985). In their graphic and game theoretic formulation, Glass et al. (1985) obtained the equilibrium proportion, P*, of non-infanticidal males for both cas e 3 and case 4 in terms of two quantities A1 and A2 such that

p , _ A1 --Al +A2

In turn, A1 and A2 were themselves simple linear functions of the fitness of infanticidal (I) and non- infanticidal (N) males. Specifically, A 2 equalled ( N - I ) and provided a measure of the relative advantage (if any) of being non-infanticidal when other males were non-infanticidal. Conversely, A1 was intended to measure the relative advantage (if any) of being infanticidal when most other males in a populat ion were infanticidal. In such infanticidal populations, both infanticidal and non-infanticidal males were considered likely to lose some propor- tion of their own offspring to the ravages of

618 Animal Behaviour, 34, 2

infanticidal successors. The proportional reduction in fitness experienced by infanticidal and non- infanticidal males under these conditions was denoted by Q' and Q, respectively and thus AI was calculated as (Q'I-QN).

The formulae for both AI and A2 c a n be directly rewritten in the notational system used by Haus- fater et al. (1982). Specifically, in that system, the variables Ru and RN~ denoted the expected repro- ductive success of infanticidal and non-infanticidal males, respectively, when either type of male was replaced by an infanticidal successor. Thus, (R~-/2N~) in the latter notation is mathematically equivalent to (Q'I-QN), and hence A~, in the notational system of Glass et al. (1985). Similarly, Hausfater et al. (1982) used the variables kiN and RNN to denote the expected reproductive success of infanticidal and non-infanticidal males when either type of male was replaced by a non-infanticidal successor. Thus, (RNN--RIN) is mathematically equivalent to (N-- I), and hence A2, in the model of Glass and co-workers.

Substituting the 'translated' versions of A~ and A2 into the equilibrium equation of Glass et al. (1985), one obtains

p , = (/~II - - RNI)

(~n - -~N~) + (~N~ - ~IN) Recall that the model of Glass et al. (1985) yielded equilibria in terms of the proportion of non- ififanticidal males in a population. Obviously, in this case ( I - P * ) then gives the proportion of infanticidal males in a population and it was in this latter form that equilibria were presented in the model of Hausfater et al. (1982).

Subtracting the above equilibrium frequency from 1 (written as {RII-- RNI-~ RNN-- kiN} over itself) and changing the sign of both the numerator and denominator, one finds that the equilibrium proportion, P*', of infanticidal males in a popula- tion under cases 3 and 4 above is equal to

P*' = (1 - P*) = _ RNI --/~il + KIN -- -/~NN

which is precisely equation (11) as given by Haus- fater et al. (1982). Furthermore, in discussing this equation, these last authors called attention to the fact that the equivalent result could be obtained by graphical or game theoretical approaches, as has now been admirably demonstrated by Glass et al. (1985).

Several other points concerning previous quanti- tative models of infanticide also need to be clari- fied. First, Glass et al. (1982, page 388) attribute to me and my co-workers the conclusion that strictly non-infanticidal populatiohs 'can never exist'. In fact, we reached just the opposite conclusion and

clearly stated that strictly non-infanticidal popula- tions were at equilibrium according to our model, although we also showed that such populations were highly susceptible to invasion by infanticidal males (Hausfater et al. 1982). Nevertheless, our statement that such populations are susceptible to invasion by infanticidal males should not be taken to mean that all real-life primate populations, or those of other species, will actually have been so invaded.

Second, Glass et al. (1985) suggest that an erroneous assumption was made by Chapman & Hausfater (1979) in their model of langur monkey (Presbytis entellus) infanticide. Specifically, one of the basic assumptions of that model was that an infanticidal male, upon introduction into an other- wise strictly non-infanticidal population, would in all cases be at a reproductive advantage compared to his non-infanticidal counterparts. This assump- tion was in turn based on the idea that the death of an infant due to infanticide (or other causes) would almost always result in a reduced waiting time to the next conception for both its mother and the infanticidal male. In contrast, Glass et al. (1985) argue the null case; namely, that the death of an unweaned infant would be expected to have 'little if any impact on the speed at which a female returns to estrus ~ (page 388). Yet a variety of empirical studies have shown this latter assertion to be patently untrue, not only for langur monkeys (Vogel & Loch 1984), but for many other non- human primate species as well (Altmann et al. 1978; Hausfater & Hrdy 1984).

Finally, it should be noted that Levins (1966), among others, has argued that when two or more models of different structure and assumptions lead to the same conclusion, then that conclusion can be considered particularly 'robust'. Thus the present case of convergent results obtained from graphic, game theoretic and algebraic models should per- haps be viewed as a subtle, yet positive indication that behavioural ecology is now in possession of a robust theory of infanticide.

I thank F. Breden, C. Gerhardt, S. Kirmeyer and Bw. Aardwolf for their careful reading of an earlier version of this note and N.S.F. and N.I.H. for support of my research on infanticide.

GLENN HAUSFATER Division of Biological Sciences, University of Missouri, Columbia, MO 65211, U.S.A.

References Altmann, J., Altmann, S. & Hausfater, G. 1978. Primate

infant's effects on mother's future reproduction. Science, N.Y., 201, 1028-1030.

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Chapman, M. & Hausfater, G. 1979. The reproductive consequences of infanticide in langurs: a mathematical model. Behav. Ecol. Sociobiol., 5, 227~40.

Glass, G., Holt, R. & Slade, N. 1985. Infanticide as an evolutionarily stable strategy. Anim. Behav., 33, 384- 391.

Hausfater, G. 1984. Infanticide in langurs: strategies, counterstrategies and parameter values. In: Infanticide: Comparative and Evolutionary Perspectives (Ed. by G. Hausfater & S. Hrdy), pp. 257~82. New York: Aldine/ DeGruyter.

Hausfater, G., Aref, S. & Cairns, S. 1982. Infanticide as an alternative male reproductive strategy in langurs: a mathematical model. J. theor. Biol., 94, 391412.

Hausfater, G. & Hrdy, S. (Eds) 1984. Infanticide: Com- parative and Evolutionary Perspectives. New York: Aldine/DeGruyter.

Levins, R. 1966. The strategy of model-building in population biology. Am. Scient., $4, 421431.

Vogel, C. &Loch, H. 1984. Reproductive parameters, adult-male replacements, and infanticide among free- ranging langurs (Presbytis entellus) at Jodhpur (Rajas- than), India. In: Infanticide: Comparative and Evolu- tionary Perspectives (Ed. by G. Hausfater & S. Hrdy), pp. 23%256. New York: Aldine/DeGruyter.

(Received 1 June 1985; revised 21 August 1985; MS. number." AS-345)

Models of Infanticide: A Reply to Hausfater

In the above note Hausfater (1986) discusses several issues raised by our paper (Glass et al. 1985 ) . However, his two major points appear to involve the equivalence of our equilibrial solutions and the causes of non-infanticidal populations. The following is intended to comment further on the similarities and differences in our models. Our model was designed as a non-specific, strategic analysis that would allow us to consider a variety of explanations that have been offered for infanticide. It was not intended to analyse a particular infanti- cidal system in detail. This contrasts with analyses by Hausfater and his co-workers (Chapman & Hausfater 1979; Hausfater etal . 1982; Hausfater 1984) which are tactical models, more closely tailored to examining the details of infanticidal behaviour in a particular species. Obviously, a strategic model cannot subsume the detailed character of a tactical model. Our claim that Hausfater's work could be viewed as a special case of an Evolutionarily Stable Strategy (ESS) arose because, if we limited the range of our parameters, we obtained his equilibrial results but, if we altered the range of our parameters, we obtained other workers' results.

When generalizing the predictions of tactical models, it is important to distinguish results that

are generally applicable from those that are idio- syncratic features of the organism for which the model was developed. We believe the expression for the equilibrial frequency of infanticidal beha- viour derived by Hausfater (1986) and ourselves is an example of the former, while most of the differences, including the persistence of non-infan- ticidal populations, are because of the latter.

In fact, the equilibrium depends on just two general assumptions shared by the models: (1) individual fitnesses are determined by pairwise interactions; and (2) these pairs of individuals are formed randomly from a large population (R. D. Holt, personal communication).

Let Wv(P) denote the fitness of an individual with phenotype P when it interacts with an indi- vidual using a different phenotype, P', and Wp (P) its fitness when interacting with an individual of its own type. The corresponding fitnesses of the alternative phenotypes are given by Wp(P') and Wv(P'). I fp is the fraction of the population with phenotype Pand dyads interact randomly, then the average fitness of individuals with pheontype P is

W(P) =p Wv(P) + (1 --p) Wv,(P) (1)

Similarly, the average fitness of individuals with the alternative phenotype p' is

W(P') =p We(P') + (1 --p) We,(P') (2)

Phenotypic fitnesses are frequency-dependent with a linear dependence on p (Glass et al. 1985), regardless of the pairwise fitness values. If the population is to be polymorphic at equilibrium, then all heritable phenotypes must have equal fitnesses (Slatkin 1978). Setting W(P)= W(P')gives an equilibrial frequency for P of

P* = (Wv,(P')- Wp,(P))/ (We(P) - Wv(e') + We,(P') - Wp,(P)) (3)

This is the expression as derived by Hausfater (1986). It should be stressed that the logic leading to this expression is general. It is not restricted to infanticide or any other behaviour. For example, equations (1) and (2) are the usual expression for genic fitnesses in randomly mating diploid popula- tions (Crow & Kimura 1970), where the pair of 'individuals' are the alleles at a single locus, and equation (3) describes equilibrial gene frequencies (Crow & Kimura 1970). Thus, the eqnilibrial frequencies predicted by our models are robust if the assumptions are met.

The primary difference in our models, which Hausfater discusses, is our explanations for non- infanticidal populations. We both recognize such populations exist, but in Hausfater's models these 'equilibrial' populations are highly susceptible to invasion and are never stable (Hausfater et al.