xia@cc.helsinki.fi (12/02/90)
Uncertainty of paternity can select against paternal care Trivers (1972) proposed that uncertainty of paternity would favour male desertion. This hypothesis was later criticized by Maynard-Smith (1978) on the groud that if a male is uncertain of his paternity in the current batch of young, he is equally uncertain of his paternity in the future young. Therefore uncertainty of paternity alone cannot select against paternal care. With the same, but perhaps independent, reasoning, Krebs and Davies (1987) reached the same conclusion. I will demonstrate in this paper that this reasoning, insightful as it is, is unfortunately fallacious. In order to expose the fallacy in Maynard-Smith (1978) and Krebs and Davies (1987), let me represent their reasoning in symbolic form. Suppose an avian species in which females always provide maternal care to their young of fixed clutch size of w (same as in Maynard-Smith 1977). Denote P2 and P1 as probability of young surviving to adulthood with, and without, paternal care, respectively (P2>P1). If a male deserts, then he can mate n extra times (n>0). Let Pc be certainty of paternity (0=<Pc=<1). Given these assumptions, a paternal male and a deserting male will have equal fitness if [1] 0.5*w*P2*Pc = 0.5*w*P1(1+n)*Pc, where the term on the left side of equation [1] is the fitness of the paternal gene and the term on the right is the fitness of the deserting gene. Cancelling out identical terms, we have [2] P2 = P1*(1+n), or P2 [2'] P1= -----. (1+n) Apparently, Pc, which is found on both sides of equation [1], cancells itself out, i.e., Pc does not play any role in determining whether a male should provide paternal care or not. This is the reasoning that leads Maynard-Smith (1978) and Krebs and Davies (1987) to conclude that uncertainty of paternity alone does not select against paternal care. The reasoning is correct as long as Pc is a constant, no matter what specific value Pc takes between 0 and 1. But Pc is not a constant by definition. The uncertainty of paternity implies not that Pc is some specific value less than 1, but that Pc is a random variable varying between 0 and 1, with its own mean and variance. This makes a difference because now a paternal male and a deserting male will have equal fitness only if [3] P2*Pc1 = P1*Pc1+P1*Pc2+...+P1*Pcn+1, or [4] P2*Pc1 = P1*(Pc1+Pc2+...+Pcn+1). If Pc1=Pc2=...=Pcn+1, then equation [4] is reduced to equation [2]. But now that Pc is a random variable, not a constant, equations [4] and [2] are no longer the same. The difference between equations [2] and [4] can be seen by substitute P1 with P2/(1+n) (see equation [2']) into equation [4]. Now we have P2 [5] P2*Pc1 = ----- * (Pc1+Pc2+...+Pcn+1), or (n+1) (Pc1+Pc2+...+Pcn+1) [6] P2*Pc1 = P2 * --------------------. (n+1) With equation [6], it becomes clear that the fitness variable (for the deserting gene) represented by the term on the right of the equation is not the same as the fitness variable (for the paternal gene) on the left. The two fitness variables have the same arithmatic mean, but the one on the right is (n+1) times less variable than that on the left. Because mean fitness of a gene over N generations is the geometric mean (Gillespie 1977 and literature cited therein), not arithmatic mean, over these N generations, the paternal gene loses in a finite population because of greater fluctuation of its fitness over generations. Readers not familiar with selection for reduced fluctuation of fitness over time should consult Seger and Brockmann (1987 and literature cited therein). In summary, uncertainty of paternity increases fitness variance of the paternal gene relative to the deserting gene and therefore can select against paternal care. So Trivers (1972) has been wrongly criticized by Maynard-Smith (1978) and Krebs and Davies (1987) for his assertion that uncertainty of paternity can select against paternal care. LITERATURE CITED Gillespie, J. H. 1977. Natural selection for variance in offspring numbers. Am. Nat. 111:1010-1014. Krebs, J. R. and N. B. Davies. 1987. An introduction to behavioural ecology. Blackwell Scientific, Oxford. Maynard-Smith, J. 1977. Parental investment: a prospective analysis. Anim. Behav. 25:1-9. Maynard-Smith, J. 1978. The evolution of sex. Cambridge University Press, Cambridge. Seger, J. and H. J. Brockmann. 1987. What is bet-hedging? Oxford Surv. Evol. Biol. 4:182-211. Trivers, R. L. 1972. Parental investment and sexual selection. In: Sexual selection and the descent of man (Ed. by B. Campbell). Heinemann, London. -------------- Xuhua Xia Department of Zoology University of Helsinki SF-00100 Helsinki Finland