Economic ideas and equilibrium analysis have many fascinating applications in biology. Popular discussions of natural selection and biological fitness often take it for granted that animal traits are selected for the benefit of the species. Modern thinking in biology emphasizes that individuals (or strictly speaking, genes) are the unit of selection. A mutant gene that induces an animal to behave in such a way as to help the species at the expense of the individuals that carry that gene will soon be eliminated, no matter how beneficial that behavior is to the species.
A good illustration is a paper in the Journal of Theoretical Biology, 1979, by H. J. Brockmann, A. Grafen, and R. Dawkins, called “Evolutionarily Stable Nesting Strategy in a Digger Wasp.” They maintain that natural selection results in behavioral strategies that maximize an individual animal’s expected rate of reproduction over the course of its lifetime. According to the authors, “Time is the currency which an animal spends.”
Females of the digger wasp Sphex ichneumoneus nest in underground burrows. Some of these wasps dig their own burrows. After she has dug her burrow, a wasp goes out to the fields and hunts katydids. These she stores in her burrow to be used as food for her offspring when they hatch. When she has accumulated several katydids, she lays a single egg in the burrow, closes off the food chamber, and starts the process over again. But digging burrows and catching katydids is time-consuming. An alternative strategy for a female wasp is to sneak into somebody else’s burrow while she is out hunting katydids. This happens frequently in digger wasp colonies. A wasp will enter a burrow that has been dug by another wasp and partially stocked with katydids. The invader will start catching katydids, herself, to add to the stock. When the founder and the invader finally meet, they fight. The loser of the fight goes away and never comes back. The winner gets to lay her egg in the nest.
Since some wasps dig their own burrows and some invade burrows begun by others, it is likely that we are observing a biological equilibrium in which each strategy is as effective a way for a wasp to use its time for producing offspring as the other. If one strategy were more effective than the other, then we would expect that a gene that led wasps to behave in the more effective way would prosper at the expense of genes that led them to behave in a less effective way.
Suppose the average nesting episode takes 5 days for a wasp that digs its own burrow and tries to stock it with katydids. Suppose that the average nesting episode takes only 4 days for invaders. Suppose that when they meet, half the time the founder of the nest wins the fight and half the time the invader wins. Let D be the number of wasps that dig their own burrows and let I be the number of wasps that invade the burrows of others. The fraction of the digging wasps that are invaded will be about 5/4 I/D. (Assume for the time being that 5/4 I/D < 1.) Half of the diggers who are invaded will win their fight and get to keep their burrows. The fraction of digging wasps who lose their burrows to other wasps is then 1/2 5/4 I/D = 5/8 I/D. Assume also that all the wasps who are not invaded by other wasps will successfully stock their burrows and lay their eggs.
(a) Then the fraction of the digging wasps who do not lose their burrows is just ____________
Therefore over a period of 40 days, a wasp who dug her own burrow every time would have 8 nesting episodes. Her expected number of successes would be ____________
(b) In 40 days, a wasp who chose to invade every time she had a chance would have time for 10 invasions. Assuming that she is successful half the time on average, her expected number of successes would be ___________ Write an equation that expresses the condition that wasps who always dig their own burrows do exactly as well as wasps who always invade burrows dug by others. ________________
(c) The equation you have just written should contain the expression I/D. Solve for the numerical value of I/D that just equates the expected number of successes for diggers and invaders. The answer is ____________
(d) But there is a problem here: the equilibrium we found doesn’t appear to be stable. On the axes below, use blue ink to graph the expected number of successes in a 40-day period for wasps that dig their own burrows every time where the number of successes is a function of I/D. Use black ink to graph the expected number of successes in a 40-day period for invaders. Notice that this number is the same for all values of I/D. Label the point where these two lines cross and notice that this is equilibrium. Just to the right of the crossing, where I/D is just a little bit bigger than the equilibrium value, which line is higher, the blue or the black? _____________ At this level of I/D, which is the more effective strategy for any individual wasp? _____________ Suppose that if one strategy is more effective than the other, the proportion of wasps adopting the more effective one increases. If, after being in equilibrium, the population got joggled just a little to the right of equilibrium, would the proportions of diggers and invaders return toward equilibrium or move further away? _________________
(e) The authors noticed this likely instability and cast around for possible changes in the model that would lead to stability. They observed that an invading wasp does help to stock the burrow with katydids. This may save the founder some time. If founders win their battles often enough and get enough help with katydids from invaders, it might be that the expected number of eggs that a founder gets to lay is an increasing rather than a decreasing function of the number of invaders. On the axes below, show an equilibrium in which digging one’s own burrow is an increasingly effective strategy as I/D increases and in which the payoff to invading is constant over all ratios of I/D. Is this equilibrium stable? __________