New Phytologist Supporting Information: Constrained growth flips the direction of optimal phenological responses among annual plants

We first show the influence of mortality on our model predictions as a robustness check, and second we show that the optimal control is of bang-bang type, with one unique switch, for annuals with constrained growth, fixed season length, and constant mortality. As a robustness check we extend the model with constant background mortality rate m. We assume that the vegetative part grows according to dV dt = u(t)F (V) with V (0) = V 0 , (S1.1) and that the reproductive part grows according to dR dt = (1 − u(t)F (V))s(t), (S1.2) where s(t) = e −mt (S1.3) is the survival function and m is the mortality rate. The reproductive mass depends on the survival, but the vegetative mass does not since we would otherwise account for the survival twice. According to Cohen (1976) the optimal control is a bang-bang control of reproduction. Below we provide an alternative proof of this fact, based on optimal control theory, showing also that there 1