Aposematic signalling in prey-predator systems: determining evolutionary stability when prey populations consist of a single species

Aposematism is the signalling of a defence for the deterrence of predators. We presently focus on aposematic organisms that exhibit chemical defences, which are usually signalled by some type of brightly coloured skin pigmentation (as is the case with poison frog species of the Dendrobatidae family), although our treatment is likely transferable to other forms of secondary defence. This setup is not only a natural one to consider but also opens up the possibility for rich mathematical modelling: the strength of aposematic traits (signalling and defence) can be unambiguously realised using variables that are continuously quantifiable, independent from one another and which together define a two-dimensional strategy space wherein the aposematic behaviour of any one organism can be represented by a single point. We presently develop an extensive mathematical model in which we explore the joint co-evolution of aposematic traits within the context of evolutionary stability. Even though empirical and model-based studies are conflicting regarding how aposematic traits are related to one another in nature, the majority of works allude to a positive correlation. We presently suggest that both positively and negatively correlated combinations of traits can achieve evolutionarily stable outcomes and further, that for a given level of signal strength there can be more than one optimal level of defence. Our findings are novel and pertinent to a sizeable body of physical evidence, which we discuss.