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.