In this paper we study the evolutionary dynamics of delayed maturation in semelparous individuals. We model this in a two-stage clonally reproducing population subject to density-dependent fertility. The population dynamical model allows multiple, cyclic and/or chaotic attractors, thus allowing it to illustrate how (i) evolutionary stability is primary a property of a population dynamical system as a whole, and (ii) that the evolutionary stability of a demographic strategy by necessity derives from the evolutionary stability of the stationary population dynamical systems it can engender, i.e., its associated population dynamical attractors. 

Our approach is based on numerically estimating invasion exponents, defined as the theoretical long-term average relative growth rates of different mutant population densities in the stationary environment defined by a resident population system. For some combinations of resident and mutant trait values we have to consider multivalued invasion exponents, which makes the evolutionary argument more complicated (and more interesting) than is usually envisaged. Multivaluedness occurs (i)  when more than one attractor is associated with the values of the residents' demographic parameters, or (ii) when the setting of the mutant parameters makes the descendants of a single mutant reproduce exclusively either in even or odd years, so that a mutant population is affected by either subsequence of the fluctuating resident densities only. 

Non-equilibrium population dynamics or random environmental noise select for strategists with a non-zero probability to delay maturation. When there is an evolutionary attracting pair of such a strategy and a population dynamical attractor engendered by it, this delaying probability is a Continuously Stable Strategy, this is an Evolutionarily Unbeatable Strategy which also is Stable in a long term evolutionary sense. Population dynamical, i.e., temporary, coexistence of delaying and non-delaying strategists is possible with non-equilibrium dynamics, but adding random environmental noise to the model destroys this coexistence. Adding random noise also shifts the CSS towards a higher probability of delaying maturation.