## 3 Results  
Models including end-of-the-season values of ungulate density performed systematically better than those including previous-season values (Table 2). Hence, we only present here the results from the first type of model (although we also provide the results and diagnostic plots of the second type of models in Table 12 and Figure 13, Appendix A). All models well fitted the data (R 2 > 0.90, deviance explained > 92%; Table 2) and fulfilled (or showed only slight departures from) the assumptions of residuals' normality and homoscedasticity (Figure 12, Appendix A). No model showed relevant signs of temporal autocorrelation (Figure 12, Appendix A). Variation in the length of the phenological cycle (days until NDVI peak) and the rainfall accumulated by then was significantly associated with the production of all four vegetation types. These effects were only additive (main factors significant, interaction not significant) for one vegetation type (shrubland) and synergistic (significant factors and interactions) for the other three (Table 3). The two domestic ungulates differed strongly in their effects on different marsh vegetation types, which were always negative: saltmarsh production decreased significantly with horse density, bulrush marsh production decreased significantly with cattle density, and grassland production decreased significantly with both. Wild ungulate species had limited effects on vegetation production: red deer density was positively associated with bulrush marsh production, and fallow deer density showed a non-linear association with shrubland production. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Time of ungulate census**} & \multicolumn{2}{c}{**Saltmarsh**} & \multicolumn{2}{c}{**Bulrush marsh**} & \multicolumn{2}{c}{**Shrubland**} & \multicolumn{2}{c}{**Grassland**} \\ \cline{2-7}  & **AIC** & **R2** & **AIC** & **R2** & **AIC** & **R2** & **AIC** & **R2** \\ \hline Start of growth season & \(-195.5\) & \(0.91\) & \(-231.1\) & \(0.90\) & \(-325.1\) & \(0.94\) & \(-228.5\) & \(0.90\) \\ End of growth season & \(-193.6\) & \(0.90\) & \(-256.7\) & \(0.93\) & \(-331.7\) & \(0.94\) & \(-227.1\) & \(0.89\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary table for the comparison between models based on ungulate censuses before or after the plant’s growth season. Bold type indicates significantly better fits (\(\Delta\)AIC > 2). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{Rainfall} & \multicolumn{2}{c}{**Saltmarsh**} & \multicolumn{2}{c}{**Bulrush Marsh**} & \multicolumn{2}{c}{**Shrubland**} & \multicolumn{2}{c}{**Grassland**} \\ \cline{2-7}  & **F(2.7/.4) = 6.32** & **F(3.0/.4) = 19.9** & **F(1.9/.4) = 5.91** & **F(2.27/.4) = 6.03** \\  & \(p<1.6\times 10^{-6}\) & \(p<2\times 10^{-16}\) & \(p<4.2\times 10^{-5}\) & \(p<3.26\times 10^{-6}\) \\ \hline \hline \multirow{2}{*}{Phenology} & \multicolumn{2}{c}{**F(1.5/.4) = 2.29**} & \multicolumn{2}{c}{**F(2.16/.4) = 9.59**} & \multicolumn{2}{c}{**F(1.02/.0) = 53.89**} & \multicolumn{2}{c}{**F(1.45/.4) = 6.69**} \\  & \(p<0.0042\) & \(p<8.3\times 10^{-7}\) & \(p<1.7\times 10^{-6}\) & \(p<0.0074\) \\ \hline \hline \multirow{2}{*}{Rainfall“Phenology} & \multicolumn{2}{c}{**F(5.29,16) = 1.33**} & \multicolumn{2}{c}{**F(1.26,16) = 0.37**} & \multicolumn{2}{c}{F(0.66,16) = 0.09} & \multicolumn{2}{c}{F(7.28,16) = 2.54} \\  & \(p<0.0021\) & \(p<0.019\) & \(p>0.12\) & \(p<8.51\times 10^{-5}\) \\ \hline \hline \multirow{2}{*}{Horse} & \multicolumn{2}{c}{**F(1.37/.4) = 18.2**} & \multicolumn{2}{c}{F(1.45,4) = 10.6} & \multicolumn{2}{c}{F(0.40,4) = 3.17} & \multicolumn{2}{c}{F(0.72/.4) = 5.94} \\  & \(p<0.0006\) & \(p>0.0835\) & \(p>0.16\) & \(p<0.043\) \\ \hline \hline \multirow{2}{*}{Cattle} & \multicolumn{2}{c}{F(0.43,4) = 0.328} & \multicolumn{2}{c}{**F(0.97/.4) = 21.4**} & \multicolumn{2}{c}{F(0.46,4) = 0.60} & \multicolumn{2}{c}{F(0.68,4) = 1.89} \\  & \(p>0.14\) & \(p<2\times 10^{-16}\) & \(p>0.10\) & \(p<0.027\) \\ \hline \hline \end{tabular}
\end{table}