J.M.C. Hutchinson, J.M. McNamara & I.C. Cuthill, 1993. Song, sexual selection, starvation and strategic handicaps. Animal Behaviour 45:1153–1177.

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Update (no longer maintained, I'm afraid)

This paper has been quite widely cited, so I cannot mention all the relevant papers. First, some papers taking the theoretical work further:
Hutchinson, J.M.C. & McNamara, J.M. 2000. Ways to test stochastic dynamic programming models empirically. Animal Behaviour 59: 665–676.
Hutchinson, J.M.C. 2002. Two explanations of the dawn chorus compared: how monotonically changing light levels favour a short break from singing. Animal Behaviour 64: 527–539.
Kondoh, M. & Ide, J.Y. 2003. Evolution of periodicity in insect mate-seeking bheaviour: a male-female coevolutionary game model. Animal Behaviour 65: 1013–1020.
Rob Thomas has conducted several empirical tests of the models:
Thomas, R.J. 1999. Two tests of a stochastic dynamic programming model of daily singing routines in birds. Animal Behaviour 57: 277–284.
Thomas, R.J. 1999. The effect of variability in the food supply on the daily singing routines of European Robins: a test of a stochastic dynamic programming model. Animal Behaviour 57: 365–369.
Thomas, R. J. & Cuthill, I.C. 2002. Body mass regulation and the daily singing routines of European Robins. Animal Behaviour 63: 285–292.
Thomas, R.J. 2002. The costs of singing in Nightingales. Animal Behaviour 63: 959–966.
Thomas, R.J. 2002. Seasonal changes in the nocturnal singing routines of Common Nightingales Luscinia megarhynchos. Ibis 144 E105–E112.
Thomas, R.J., Cuthill, I.C., Goldsmith, A.R., Cosgrove, D.F., Lidgate, H.C. & Burdett Proctor, S.L. 2003. The trade-off between singing and mass gain in a daytime-singing bird, the European Robin. Behaviour 140: 387–404.
Other empirical work relating song output to energetic cost and mass changes includes the following. Naturally I find the Hasselquist & Bensch (2008) study in the wild more reassuring than earlier lab-based studies of energetic costs of bird song.
Godfrey, J.D. & Bryant, D.M. 2000. State-dependent behaviour and energy expenditure: an experimental study of European Robins on winter territories. Journal of Animal Ecology 69: 301–313.
Lucas, J.R., Schraeder, A. & Jackson, C. 1999. Carolina Chickadee (Aves, Paridae, Poecile carolinensis) vocalization rates: effects of body mass and food availability under aviary conditions. Ethology 105: 503–520.
Oberweger, K. & Goller, F. 2001. The metabolic cost of birdsong production. J. Experimental Biology 204: 3379–3388.
Franz, M. & Goller, F. 2003. Respiratory patterns and oxygen consumption in singing zebra finches. J. Experimental Biology 206: 967–978.
Ward, S., Speakman, J.R. & Slater, P.J.B. 2003. The energy cost of song in the Canary, Serinus canaria. Animal Behaviour 66: 893–902.
Ward, S., Lampe, H.M. & Slater, P.J.B. 2004. Singing is not energetically demanding for Pied Flycatchers, Ficedula hypoleuca. Behavioral Ecology 15: 477–484.
Ward, S. & Slater, P.J.B. 2005. Raised thermoregulatory costs at exposed song posts increase the energetic cost of singing for Willow Warblers Phylloscopus trochilus. Journal of Avian Biology 36: 280–286.
Barnett, A.C. & Briskie, J.V. 2007. Energetic state and the performance of dawn chorus in Silvereyes (Zosterops lateralis). Behavioral Ecology and Sociobiology 61: 579–587.
Hasselquist, D. & Bensch, S. 2008. Daily energy expenditure of singing Great Reed Warblers Acrocephalus arundinaceus. J. Avian. Biol. 39:384–388.
There are also a number of recent papers addressing whether song output is an honest signal of quality and correlates with reproductive success. These include:
Hoi-Leitner, M., Nechtelberger, H. & Hoi, H. 1995. Song rate as a signal for nest site quality in Blackcaps (Sylvia atricapilla). Behavioral Ecology and Sociobiology 37: 399–405
Otter, K., Chruszcz, B., Ratcliffe, L. 1997. Honest advertisement and song output during the dawn chorus of Black-capped Chickadees. Behavioral Ecology 8: 167–173.
Poesel, A., Foerster, K. & Kempenaers, B. 2001. The dawn song of the Blue Tit Parus caeruleus and its role in sexual selection. Ethology 107: 521–531.
Gil, D. & Gahr, M. 2002. The honesty of bird song: multiple constraints for multiple traits. Trends in Ecology and Evolution 17: 133–141.
Verhulst, S. 2003. Song as a signal to negotiate a sexual conflict? Animal Biology 53: 159–171.
Gorissen, L., Snoeijs, T., Duyse, E.V. & Eens, M. 2005. Heavy metal polution affects dawn singing bheaviour in a small passerine bird. Oecologia 145: 504–509.
Berg, M.L., Beintema, N.H., Welbergen, J.A. & Komdeur, J. 2005. Singing as a handicap: the effects of food availability and weather on song output in the Australian Reed Warbler Acrocephalus australis. Journal of Avian Biology 36: 102–109.
Parker, T.H., Barr, I.R. & Griffith, S.C. 2006. The Blue Tit’s song is an inconsistent signal of male condition. Behavioral Ecology 17: 1029–1040.
Poesel, A., Kunc, H.P., Foerster, K., Johnsen, A. & Kempenaers, B. 2006. Early birds are sexy: male age, dawn song and extrapair paternity in Blue Tits, Cyanistes (formerly Parus) caeruleus. Animal Behaviour 72: 531–538.
van Oort, H., Otter, K.A., Fort, K.T. & Holschuh, C.I. 2006. Habitat quality, social dominance and dawn chorus song output in Black-capped Chickadees. Ethology 112: 772–778.
Dolan, A.C., Murphy, M.T., Redmond, L.J., Sexton, K. & Duffield, D. 2007. Extrapair paternity and the opportunity for sexual selection in a socially monogamous passerine. Behavioral Ecology 18: 985–993.
Dalziell, A.H. & Cockburn, A. 2008. Dawn song in Superb Fairy-wrens: a bird that seeks extrapair copulations during the dawn chorus. Animal Behaviour 75: 489–500.
Murphy, M.T., Sexton, K., Dolan, A.C. & Redmond, L.J. 2008. Dawn song of the Eastern Kingbird: an honest signal of male quality? Animal Behaviour 75: 1075–1084.
Compatible with our finding that when males sing should be driven by when females listen is that females were more responive to playback at times within an Amazonian dawn chorus when males of that species sung:
Luther, D.A. 2008. Signaller: receiver coordination and the timing of communication in Amazonian birds. Biol. Lett. 4:651–654.
Similarly a paper on crickets supports our modelling by showing that male singing routines broadly echo when females are most active. There is also a suggestion that males of all qualities "pull out all the stops" to sing when females are most active (15.00-20.00), so that the difference in output between males of different qualities is LESS then than earlier.
Jacot, A., Scheuber, H., Holzer, B., Otti, O. & Wrinkhof M.W.G. 2008. Diel variation in a dynamic sexual display and its association with female mate-searching behaviour. Proc. Roy. Soc. B 275: 579–585.
Some other papers partly relevant from crickets and frogs (further papers use SDP to model attendance of a frog chorus through the season):
Judge, K.A. Ting, J.J. & Gwynne, D.T. 2008. Condition dependence of male life span and calling effort in a field cricket. Evolution 62: 868–878.
Judge, K.A. & Brooks, R.J. 2001. Chorus participation by male Bullfrogs, Rana catesbeiana: a test of the energetic constraint hypothesis. Animal Behaviour 62: 849–861.
A paper on the dusk chorus in a nocturnally feeding owl found less male response to playback at this time than others. This tends to conflict with our model, which implied usually a larger surplus of food relative to requirements following the enforced period of starvation:
Hardouin, L.A., Robert, D. & Bretagnolle, V. 2008. A dusk chorus in a nocturnal bird: support for mate and rival assessment functions. Behav. Ecol. Sociobiol. 62:1909–1918.
A couple of papers supporting some of the other explanations for the dawn chorus:
Brown, T.J. & Handford, P. 2003. Why birds sing at dawn: the role of consistent song transmission. Ibis 145: 120–129.
Amrhein, V., Kunc, H.P. & Naguib, M. 2004. Non-territorial Nightingales prospect territories during the dawn chorus. Proc. Roy. Soc. B (Suppl.) 271: S167–S169.
Lastly, for a somewhat unsympathetic review see:
Staicer, C. A., Spector, D. A. & Horn, A. I. 1996. The dawn chorus and other diel patterns in acoustic signaling. In: Ecology and Evolution of Acoustic Communication in Birds (Ed. by D. E. Kroodsma & E. H. Miller), pp. 426–453. Ithaca, New York: Cornell University Press.

See also the update to my later paper on this topic.

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