On the evolution of the timing of reproduction.

In this paper, we study the evolution of the per capita rate of reproduction as a function of time in the modelling framework introduced by Eskola and Geritz [Eskola, H.T.M., Geritz, S.A.H., 2007. On the mechanistic derivation of various discrete-time population models, Bull. Math. Biol. 69, 329-346]. We assume that the total number of juveniles one adult individual can produce is a finite constant, and we study how this number should be distributed during the season, when certain interaction and mortality processes are also included in the model. If aggressive interactions between the juveniles are not included in the model, evolution is simply optimizing, and the optimal reproductive strategy is always a single Dirac delta-peak within the season. If aggressive interactions between the juveniles are included, an evolutionarily stable strategy can consist of not only one or two delta-peaks, but also of continuous reproduction during the season. Using this approach, we have also derived conditions under which the classical population dynamical models of Beverton and Holt [Beverton, R.J.H., Holt, S.J., 1957. On the dynamics of exploited fish populations. Fisheries Investigations, Ser. 219], Hassell [Hassell, M.P., 1975. Density-dependence in single-species populations. J. Animal Ecology 44, 283-295] and Ricker [Ricker, W.E., 1954. Stock and recruitment. J. Fisheries Res. Board Can. 11, 559-623] are evolutionarily stable.