## 3 Materials and Methods
### The Modified Miami Model
** Including critical minimum temperature  
extend exponent by a term with MATnat  
**Step 3. ** Including evapotranspiration  
**Step 4. ** Validation  
determination of the constants for BMmax and coefficients of determination  
**Step 5. Combining both functions by the Minimum formulation**  
MAT is the minimum: MAT explains ABM  
MAP is the minimum: MAP explains ABM  
**Step 6. ** The maximum value of the  
Figure 2: ABM modeled as a function of MAT using the MAT function and showing the influence of MATnat: **(a)** ABM of Norway spruce and _Miscanthus_ without the influence of MATnat; **(b)** Norway spruce with a MATnat of 1 oC and _Miscanthus_ with a MATnat of 18 oC; ABM, modeled above-ground biomass production of a species in dry matter (Mg ha-1a-1); MAT mean annual temperature (oC); MATnat, mean annual temperature of the native distribution area of a species (oC). In addition, since the MAP equation is an exponential function, the mean annual evapotranspiration (E) has to be included in the exponent to achieve a change of gradient in the MAP graph (Equation 5). The linear function of E, as E represents the above-ground water loss compared to MAP as water input, suggests that the best formulation is to subtract E from MAP. A restriction in this formulation is that MAP minus E cannot be negative, as there is no negative biomass production. This restriction is solved by the logic formulation \"if the term is negative, set it to zero\". The final formulation of the modified Miami model is: the relationship between the modeled ABM and the MAT, including the MATnat for a single species on a specific site is given by:  
\[\text{ABM}_{\text{MAT}}=1.5\times\text{BM}_{\text{max}}/(1+\text{e}^{((\text{ MAT}+\text{MATnat})(\text{MAT}+1))\times(1.315-0.119\text{MAT})})\text{ (Mg ha}^{-1}\text{a}^{-1}) \tag{4}\]  
Figure 3: ABM modeled as a function of MAT using the MAP function with MAP = 900 mm and showing the influence of E: (**a**) ABM of Norway spruce and _Miscanthus_ without E; and (**b**) after including E in the MAP function. ABM, modeled above-ground biomass production of a species in dry matter (Mg ha-1a-1); MAT, mean annual temperature (c); MAP, mean annual precipitation (mm); E, mean annual evapotranspiration (mm). In addition, the relationship between ABM and the MAP including the restriction that E \(\leq\) MAP for a species on a specific site is given by:  
\[\mathrm{ABM}_{\mathrm{MAP}}=2\times\mathrm{BM}_{\mathrm{max}}\times(1-\mathrm{e} ^{-0.00066(\mathrm{MAP}-\mathrm{E})})\;(\mathrm{Mg\;ha^{-1}a^{-1}}) \tag{5}\]  
The relationship between E and the MAT is given by:  
\[\mathrm{E}=\mathrm{MAT}\times 31.4\;(\mathrm{mm}) \tag{6}\]  
The relationship between ABM, MAP and MAT is given by:  
\[\mathrm{ABM}=\mathrm{Min}\;(\mathrm{ABM}_{\mathrm{MAT}};\mathrm{ABM}_{ \mathrm{MAP}})\;(\mathrm{Mg\;ha^{-1}a^{-1}}) \tag{7}\]  
Where \(\mathrm{BM}_{\mathrm{max}}\) is the maximum above-ground biomass production in the dry matter of a species (Mg ha\({}^{-1}\)a\({}^{-1}\)); MAT\({}_{\mathrm{nat}}\) is the mean annual temperature of the native distribution area of a species (\({}^{\circ}\)C); E is the mean annual evapotranspiration (mm); MAT is the mean annual temperature (\({}^{\circ}\)C); MAP is the mean annual precipitation (mm). Furthermore, the model has two restrictions regarding MAT that have to be considered:  
1. MAT \(\geq\) 0 \({}^{\circ}\)C: this model gives an estimate of ABM, starting at a MAT of 0 \({}^{\circ}\)C. It is not valid for values below 0 \({}^{\circ}\)C.
2. MAT\({}_{\mathrm{nat}}\) \(>\) 0 \({}^{\circ}\)C: the MAT\({}_{\mathrm{nat}}\) must be positive because of the mathematical formulation.