Evolutionarily stable investment in secondary defences

1. Previous workers have suggested that the evolutionarily stable strategy (ESS) for investment in antipredator defences, such as toxins, will critically depend on the nature of expression of the defence. Specifically, it has been suggested that if the different levels of a defence are best described as a continuous variable, then this will lead to pure ESSs with all individuals in a population adopting similar defence levels; whereas defences that can only take on discrete levels will lead to mixed ESSs (featuring variation in defence within the population). 2. Our principal aim is to determine the validity of these viewpoints, and examine how the pure and mixed strategies predicted by the two types of defences can be reconciled with practical and philosophical difficulties in defining any given defence unambiguously as continuous or discrete. 3. We present the first model of a continuously varying defence that is solved explicitly for evolutionarily stable strategies. 4. We are able to demonstrate analytically, that the model always has a unique ESS, which is always pure. This strategy may involve all members of the population adopting no defence, or all members of the population making the same non-zero investment in defence. 5. We then modify our model to restrict the defence to a number of discrete levels and demonstrate that the unique ESS in this case can be either pure or mixed. We further argue that the mixed ESS can be a combination of no more than two defence levels, and the two levels in a mixed ESS must be nearest neighbour levels in an ordered list of the levels that the defence can take. 6. This, in turn, means that the mixed ESS will be practically identical to a pure ESS if the discrete defence is fine-grained.