Black and Yellow

Source: https://www.pexels.com/photo/black-and-yellow-snake-65296/

Why does black and yellow signal danger?  Is it just an arbitrary convention or is there some explanation for this particular choice of colours?

The answer may lie in Game Theory.  We can think of evolution as an iterative game played by the following 3 players:
  1. The genome of poisonous prey-like animals
  2. The genome of non-poisonous prey
  3. The genome of predators
Most non-poisonous prey camouflage themselves to some degree or another, for example by adopting the colour of chlorophyll or the colour of mud.  Poisonous animals often do the exact opposite and choose colours that stand out, like yellow and black.

Why shouldn't poisonous animals use camouflage colours too?  It is often claimed that this is done to help the predators for spot them, but that doesn't make much sense.  If the snake shown in the picture were green and brown it would still be possible for predators to learn not to eat it because of its distinctive markings, but it would also be spotted less often.

A more plausible reason for the colour scheme is that any other would fail to be an Evolutionary Stable Strategy.  Let's take an example. Suppose a poisonous insect was coloured with green and brown markings. Since this is good camouflage there would be zero downside to a non-poisonous species to mimic it. Eventually individuals from non-poisonous species with the markings would massively outnumber individuals from the poisonous species, and brave predators would start eating insects with those markings. If, on the other hand, the poisonous species chose a colour scheme that made it stick out, there would be a cost associated with mimicry, and in the Nash equilibrium the proportion of species that mimic the markings would be smaller.

The diagram below assumes that the poisonous species is coloured black and yellow and describes qualitatively what would happen over time to i) the proportion of non-poisonous species that "fake" toxicity by mimicking the markings, and ii) the proportion of predators that are prepared to risk eating insects that look similar.

The horizontal line running through the middle marks the threshold at which there are so many predators eating black and yellow insects that the advantage in mimicry no longer exceeds the disadvantage of being more easily spotted.  The vertical line marks the threshold at which there are so many mimics that the risk to a predator is sufficiently low to make it worthwhile taking a bite.

If the poisonous insect chose a colour that did not make it stand out then the horizontal line would be at 100%.  The result would be that the Nash equilibrium would move to the top right corner (100%, 100%).  At that point there would be zero advantage to the poisonous species in having its markings and, eventually, natural selection would try out an alternative colour scheme.

In game theory any signalling without an associated cost is known as "cheap talk". An example is where you go to an auction and before the bidding starts you go around telling everyone "I have a lot of money and I really want this item". No one will believe you because, well, you would have said that wouldn't you! However, if before the bidding starts you pull out a Havana cigar and light it with a $100 note, that's not cheap talk, and will probably cause a number of your competitors to quietly back off.

POSTSCRIPT

The diagram above suggests that the basin of attraction for the Nash equilibrium is the entire space. Although this may be the case it's not necessary that the trajectories spiral inwards.  Closed loop solutions are also possible meaning that the genomes may change cyclically instead of converging.

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