Friday, 29 July 2011

If a model ain't broken, why fix its assumptions?

One of the greatest success stories of behavioural ecology starts with a model for optimal mating behaviour in dungflies. Geoffrey A. Parker (2010, p. 268) recollects his fascination with animal behaviour as follows: 
"So when I began my PhD, staring hard at the intense melee of dungflies swarming on a cowpat, I was utterly fascinated, and never thought that what I was doing was odd." (Parker 2010, p. 268) 
Most students doing field work probably know the bemused looks of passers-by. Parker's lowly dungfly studies, however, became the starting point of an exceptionally successful research career including sperm competition, evolution of anisogamy (small sperm, large eggs rather than equally sized gametes), animal contest, sexual conflict etc. all contributing to the "behavioural-ecology explosion" (Parker 2010). 
   The paper proposing the seminal optimization model was Parker and Stuart (1976). In a more general context, optimization models are also called the marginal value theorem, because an individual in question should leave a resource patch, which could be a host or feeding site as well as a female, when the gain in fitness return sinks to a certain margin.
   Parker observed the behaviour of male dungflies at cowpats and distinguished three patterns: (1) searching for females, (2) copulating with females, and (3) guarding females after copulation, while the females lay their eggs. Feeding and other behavioural patterns take place elsewhere, not near cowpats. The post-copulatory guarding is necessary, because the last male to mate with a female fertilizes more that 80% of her eggs. So a male that does not guard at all will jeopardize its reproductive success. The average values of time spent in the respective behaviours were: searching = 140 min, copulating = 35.5 min, and guarding = 16.5 min. 
   When reading the paper as a PhD, I stumbled across and odd assumption that was just proclaimed without giving further reasons for it.
"The mean search cost S for a new female is approximately 140 min (Parker 1970b), to which must be added the 16.5 min guarding time during oviposition." (Parker and Stuart 1976, p. 1060)
Hence there are 156.5min searching + guarding against 35.5min copulating. 

   I found this strikingly counter-intuitive. From biological functions of the behavioural patterns, I would have argued that copulating is an investment of time and sperm into the eggs of the fertilized female, while the guarding that follows is an investment of time into the same eggs of the same female. Guarding protects the previous investment made during copulation from being ousted by a later male, because the rule is that the last male to mate fertilizes more than  80% of the eggs. Therefore, I would have intuitively added the guarding time to the copula duration and traded this off against the search time.
   The figure below illustrates the model:






 Legend: There are two curve fits. The solid curve gives the best exponential fit to gain in terms of the portion of eggs fertilized (gain data stem from Parker 1970):
                                                gain = 1 – e -C/16,                
where -C/16 is an exponent, C being the actual copula duration (35.5min). This curve fit predicts and optimal copula duration of Copt = 41.4min (see Parker and Stuart 1976). The dotted curve is an approximation that is parabolic first and constant later:  
                                             gain = (0.95C/45exp2)(90-C),   for C < 45min 
                                             gain = 0.95),                            for C > 45min. 
This curve fit predicts Copt = 39.9min (Parker and Stuart 1976).



There is a roughly five minutes discrepancy between the model predictions (41.4 and 39.9 minutes respectively) and the observed average (35.5 minutes).
   Two things are clear from the graph. Firstly, if the behavioural patterns were apportioned the other way round (copula duration + guarding time vs. searching time), there would be no acceptable fit between model and data at all. The tangent would hit the X-axis at -140 minutes and necessarily touch the curves below 41.4 or 39.9 minutes respectively. While that may seem closer to the target of 35.5 minutes at first sight, the target would move to 52 minutes by virtue of adding the guarding time to the copula duration. That is, adding the guarding to the copulation time would have destroyed everything. Secondly, if the guarding time was simply taken out of the trade-off, the model fit would become almost perfect. 
   My suggestion to was that there should be two trade-offs, one of searching time vs. copula duration and a second of guarding time vs. some measure of competition from other males. In personal communications Geoffrey Parker and Kate Lessells stated that the convention of adding guarding to searching is warranted by the model’s assumption that mate guarding can be treated as a constant. My hunch is that this has to do with the balance all modelers need to strike between the generality, realism, and precision of a model. My suggestion would have improved the precision of the model. By requiring a second trade-off for mate guarding, however, it is probably seen as less simple or general. 
   While I still cannot bring myself to accept that mathematical could run contrary to biological reasoning, the whole issue only shows that science advances by tinkering, working with whatever works and not halting to question assumptions as long as the applied research tools do their job. Consequently, nobody gave a damn when I called an assumption into question that was simply claimed in 1976, but never explained or justified since. The quote below, though taken from another context, seems entirely fitting.
"There were opportunities for explaining all sorts of things, and to question the basic premises would have been a waste of time." (Ghiselin 1988, p. 13)
P.S.: Unless, of course, someone comes up with empirical data showing that mate guarding is not constant but varies with competition.


References
  • Ghiselin MT (1988) The evolution of sex: a history of competing points of view. In: Michod RE, Levin BR (eds) The Evolution of Sex: an examination of current ideas. Sinauer Associates, Sunderland, pp. 7-23.
  • Parker G.A. 1970: Sperm competition and its evolutionary effect on copula duration in the fly Scatophaga stercoraria. J. Insect Physiol. 16: 1301-1328.
  • Parker GA, Stuart RA (1976) Animal behavior as a strategy optimizer: evolution of resource assessment strategies and optimal emigration thresholds. American Naturalist 110:1055-1076.
  • Parker GA (2010) In celebration of questions, past, present and future. In: Székely T, Moore AJ, Komdeur J (eds) Social Behaviour. Genes, Ecology and Evolution. Cambridge University Press, Cambridge, pp. 268-270




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