**The Second Half of the Chessboard**

There
is a story loved by mathematicians, of an ancient emperor of

In fact the first half of the
chessboard would have to accommodate a heap of rice weighing 100,000 kilograms,
but it is on the second half of the board that the amount of rice reaches
astronomical proportions; literally, because if the rice grains on the whole
board were placed end-to-end they would reach to the nearest star (Alpha Centauri)
and back again. Piled up, the heap of rice would be larger than ^{63} =
9,223,372,036,854,775,808 grains of rice, more than two billion times as much
as on the whole of the first half of the board.

To
get back to aphids, parthenogenesis coupled with viviparity has given them the
ability to double their population size is less than 3 days, given favourable
conditions, so that starting from a single aphid, in about three months they would
be beginning to populate the second half of the chessboard. In a year they would be nearing the end of
their *second* chessboard, and we would
not just be “knee-deep in aphids”, the whole world would be buried under a
layer of aphids nearly 150 km thick** ^{
}**(Harrington 1994).

Let’s take a closer look at how aphids can reproduce so quickly, by using a diagram to illustrate the succession of generations in a parthenogenetic aphid, and compare it with that in an insect that reproduces sexually. The drastic effect of the telescoping of generations by the aphid can be clearly seen. We start with one egg of a parthenogenetically-reproducing aphid and one of the sexually-reproducing insect, suppose that both take exactly the same time to develop through embryonic and nymphal stages to adulthood, and also that both produce 50 progeny per female individual in each generation. In a short time the number of aphids is several orders of magnitude greater than the number of the sexually-reproducing insect. The difference between the two is further increased by the fact that, whereas all the aphids are females, we must assume that 50% of the individuals produced by the other insect are males, which make no contribution to its fecundity.

**Fig.
1 **Comparison of the powers of
multiplication of **A**, an aphid
reproducing parthenogenetically and **B**,
a sexually-reproducing insect, assuming that both produce 50 offspring per
female and that embryo, nymph and adult stages occupy the same period of time
in each case. The figures show the potential number of descendants in
successive generations starting from a single individual.

Of course, the comparison is not really
as simple as this because the aphid is producing its young continuously, so
that the generations rapidly start to overlap, whereas in most sexually
reproducing insects the generations remain discrete, at least in temperate
climates. It is this overlapping of generations that causes particular problems
for the ecologists, in their attempts to explain how aphid populations do not
rapidly escalate out of control, and bury the world in aphids.

**Reference:**

Harrington, R. (1994) Aphid layer. *Antenna ***18**: 50-51.