The effect of adaptive behaviour on the stability of population dynamics

I consider how adaptive changes in behaviour with population size affect th e stability of the population dynamics. In any given year the behavioural r ule of a member of a single-species population is determined by the value o f a certain trait. I allow for the possibility that this trait value can ch ange from year to year. The number of descendants left in one year's time b y an individual depends on its trait value, the values of other population members and the population size. The population dynamics is modelled as the change in population size from one year to the next. I focus on a populati on that is at a fixed point of the dynamics and in which members adopt the evolutionarily stable trait value for that equilibrium size. I compare the stability of the population dynamics under the following two assumptions ab out the dependence of trait values on population size: (i) trait values do not change from that at the equilibrium size, and (ii) trait values change so as to be evolutionarily stable for the current size. In a range of examp les, I show that adaptive behaviour tends to destabilise population dynamic s in the sense that stability under assumption (ii) implies stability under assumption (i). In other words, the region of parameter space for which th ere is instability under an adaptive response contains the region of instab ility under no response. Various equivalent general criteria for this to ho ld are given.