On the botanic model of plant growth with intermediate vegetative–reproductive stage

The application of dynamic optimization to mathematical models of ontogenic biological growth has been the subject of much research [see e.g. Cohen, 1971. J. Theor. Biol. 33, 299–307]. Kozłowsky and Ziółko [1988. Thor. Popul. Biol. 34, 118–129] and Ziółko and Kozłowski [1995. IEEE Trans. Automat. Contr. 40(10), 1779–1783] presented a model with gradual transition from vegetative to reproductive growth. The central point of their model is a mixed state-control constraint on the rate of reproductive growth, which leads to a mixed vegetative–reproductive growth period. Their model is modified here in order to take into account the difference of photosynthesis use efficiency when energy is accumulated in the vegetative and in the reproductive organs of a plant, respectively. The simple assumption on correlation between photosynthesis and temperature permits us to modify the model in a form that is useful for changing climate. Unfortunately, the mathematical solution of the optimal control problem in Kozłowsky and Ziółko (1988) and Ziółko and Kozłowski (1995) is incorrect. The strict mathematical solution is presented here, the numerical example from Kozłowski and Ziółko (1988) is solved, and the results are compared. The influence of the length of the season and the relative photosynthesis use efficiency, as well as of the potential sink demand of the reproductive organs, on the location and duration of the mixed vegetative–reproduction period of growth is investigated numerically. The results show that the mixed growth period is increased and shifted toward the end of the season when the lengths of the season is increased. Additional details of the sensitivity analysis are also presented.