Path: utzoo!attcan!utgpu!watserv1!watmath!uunet!bionet!uwm.edu!wuarchive!zaphod.mps.ohio-state.edu!usc!snorkelwacker.mit.edu!bloom-beacon!eru!hagbard!sunic!news.funet.fi!hydra!hylka!xia
From: xia@cc.helsinki.fi
Newsgroups: bionet.population-bio
Subject: Uncertainty of paternity and paternal care
Message-ID: <4130.2759095b@cc.helsinki.fi>
Date: 2 Dec 90 14:02:03 GMT
Organization: University of Helsinki
Lines: 112


    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


Brought to you by Super Global Mega Corp .com