## 3 Materials and Methods
### The Modified Miami Model  
In the Miami model, the global net primary production (NPP) is estimated using the limiting productivity level defined either by the mean annual temperature in degrees centigrade (MAT) or the mean annual precipitation in millimeters (MAP). This means that NPP is expressed as:  
\[\mathrm{NPP}_{\mathrm{MAT}}=30/(1+\mathrm{e}^{1.315-0.119\mathrm{MAT}})\,( \mathrm{Mg\leavevmode\
obreak\ ha^{-1}a^{-1}}) \tag{1}\]  
where MAT is the mean annual temperature (\({}^{\circ}\)C), or as:  
\[\mathrm{NPP}_{\mathrm{MAP}}=30\times(1-\mathrm{e}^{-0.00066\mathrm{MAP}})\,( \mathrm{Mg\leavevmode\
obreak\ ha^{-1}a^{-1}}) \tag{2}\]  
where MAP is the mean annual precipitation (mm). Each lower value of NPP is estimated from the two functions:  
\[\mathrm{NPP}=\mathrm{Min}(\mathrm{NPP}_{\mathrm{MAT}};\,\mathrm{NPP}_{ \mathrm{MAP}})\,(\mathrm{Mg\leavevmode\
obreak\ ha^{-1}a^{-1}}) \tag{3}\]  
Equations 1 and 2 are non-linear functions with a maximum productivity of 30 tons per hectare per year [14]. It is suggested that the NPP model can be transformed into a species-specific model by including two species-specific variables, the \"maximum above-ground biomass production in the dry matter of a species\" (BMmax) and the \"MAT of the native distribution area of a species\" (MATnat). Figure 1 gives an overview of the transformation of the Miami model. Moreover, it is supposed that evapotranspiration is linearly related to MAT [25]. The formulations of the two functions of the Miami model (Equations 1 and 2) support this type of modification. The exposed maxima of the two functions can be replaced by the BMmax found in the literature. As the Miami model was not designed for a single species, but for the collectivity of species on a single site, the MBmax has to be multiplied by a constant to avoid systematic underestimation of the biomass production of a single species. This constant is found by approximation of the modeled data to the literature data. Approximation during the validation process showed that forthe above-ground biomass production of a species in the dry matter (ABM) modeled by the MAT, multiplication of the BMmax by 1.5 (Equation 4) and for the ABM modeled by the MAP a multiplication of the MBmax by 2 (Equation 5) offer the best estimations by the model. The different BMmax potentials of single species leads to a fan of graphs differing only by a constant expressed by the differences in BMmax. This formulation is problematic for species with a high BMmax and adapted to warmer climates, such as _Paulownia_ spp. [26, 27]. Because of their weak frost resistance, such species can hardly survive under cooler conditions, and this leads to very low or no ABM at low MATs. This requirement can be fulfilled without changing the maximal values by extending the exponential MAT function (Equation 1) by MATnat within the exponent (Equation 4). The influence of MATnat expresses itself in an increasing gradient with an increase in MATnat, resulting in a shift of the zero-point in the direction of the inflexion point of the graph (Figure 2). This means that a higher MATnat leads to the start of biomass production at a higher temperature. The exact formulation of the term for MATnat was found by approximation during the validation process. **Step 1. ** Illustration of the modification of the Miami model in order to transform the net primary production (NPP) model into a species-specific ABM model. Modification of the MAT function is shown on the left and of the MAP function on the right. MAT, mean annual temperature (\({}^{\circ}\)C); MAP, mean annual precipitation (mm); NPP, global net primary production (Mg ha\({}^{-1}\)a\({}^{-1}\)); MATnat, mean annual temperature of the native distribution area of a species (\({}^{\circ}\)C); ABM, modeled above-ground biomass production of a species in dry matter (Mg ha\({}^{-1}\)a\({}^{-1}\)); BMmax, maximal above-ground biomass production in dry matter of a species found in the literature (Mg ha\({}^{-1}\)a\({}^{-1}\)); E, mean annual evapotranspiration (mm). **Step 2.