. 2019 Jul 16;9:10264. doi: 10.1038/s41598-019-46706-7

Table 4.

Equations’ parameters, specifications, assumptions, and data sources used for nutrient balance calculations.

Eq.	Param.	Definition/Variables represent (Specifications, assumptions and data sources)
1	$Q_{m}^{n}$	Total quantity of nutrient n in excreta in municipality m, where n represents nutrient (N, P or K), and $m \in M$ represents municipality. Here $m$ is the set of all municipalities.
	$E_{l m}$	Total number of individuals, where $l \in L$ represents source (livestock type or human), and m represents municipality. Here L is the set of sources (i.e., livestock and human). The municipal total human population was obtained for the year 2007 for 290 municipalities from⁵⁴ Municipal total livestock population for 2007 was obtained from⁵⁵ for each of the following animal types: dairy cows, cows for calf production, heifers, bulls and steers, calves under 1 year, rams and ewes, lambs, breeding boars, breeding sows, fattening pigs 20 kg and over, piglets under 20 kg, poultry, laying chickens, broilers, turkeys and horses. Data for horses were not available for 2007, so we used these 2003 data⁵⁵. Poultry and pigs achieve their final weight and are slaughtered in less than a year. Therefore, annual records collected annually do not reflect the entire year of production. However, once slaughtered these animals were replaced by new animals. This assumption allowed us to use annual excretion rates⁵⁶.
	$e_{l}^{n}$	Excretion per individual, where n represents nutrient (N, P or K), and l represents source (livestock type or human) We obtained the Swedish specific coefficients of N, P and K excretion for most livestock from⁵⁶. The coefficients were given per animal per year. We obtained the nutrient excretion rates of fowls and breeding boars from⁵⁸ We obtained the Swedish specific coefficients of N, P and K for human excreta from⁵⁷. The coefficients were given per human per year.
	$v_{l}^{n}$	Constant for a gaseous loss, where n represents nutrient (N, P or K), and l represents source (livestock type or human) We used the Swedish specific storage loss of N of manure from⁵⁹. We assumed that manure from sheep, horses, broilers, and turkeys was stored as semisolid ( $v$ = 0.2), while other kinds of manure were stored as a slurry ( $v$ = 0.1). These storage losses are given for a covered storage with an opening for the pump. We assume the same storage loss for human excreta as we used for excreta stored as slurry.
2	$C_{m}^{n}$	Total crop fertilizer needs in a municipality, where $n$ represents nutrient (N, P, and K), and m represents municipality
	$A_{t m}$	Cropped area (hectares), where $t \in Γ$ represents crop/crop group, and $m$ represents municipality. Here $Γ$ is the set of crop groups. Municipal total arable land use for 2007 was obtained from⁶⁰ for each of the following crops/crop groups: winter wheat, spring wheat, rye, winter barley, spring barley, oats, triticale, mixed grain, field peas for cooking, fodder peas, vetches and field beans, green peas, white beans, green fodder, utilized ley for hay and pasture, ley for seeds, table potatoes, potatoes for processing of starch, sugar beets, winter rape, spring rape, winter turnip rape, spring turnip rape, oil flax, horticulture plants, other crops, energy forest, fallow land, other untilled arable land, and unspecified arable land. Other crops include white mustard and oilseed crops⁶⁰. We disaggregated hay and grazed pasture land uses on arable land by assuming the fraction of cut vs. grazed pasture at the county was representative of the municipalities it encompassed⁶⁵. For Blekinge county, we used Skåne statistics. We assumed unspecified arable land was used to grow vegetables. Note that although we do include grazing land that is on arable land, we do not include grazing areas that are part of what is known as agricultural land but not arable land. This is because there are no good statistics on how much of this extensive, and usually low production land, is actively used and how many, and how intensively animals use it (or what type of animals).
	$R_{t m}^{n}$	Recommended fertilizer rate of a nutrient per hectare, where $n$ represents nutrient (N, P, and K), and t represents crop/crop group, and m represents municipality We selected application rates that matched 2007 Swedish crop yields (Statistics Sweden, 2017b) for the following crops: Winter wheat, Spring wheat, Rye, Winter barley, Spring barley, Oats, Mixed grain, Peas, Field beans, Corn, Table potatoes, Potatoes for processing of starch, Sugar beets, Winter rape, Spring rape, Winter turnip rape, Spring turnip rape, Oil flax, Arable grassland We obtained agro-climate specific fertilizer recommendations of N, and soil class specific fertilizer recommendations of P_, and K from⁵⁶. For the following crops, we had to use some other sources or a different strategy to select a fertilizer recommendation: Mixed grain = average of all grain crops, Green fodder = average of maize and ley hay, Oil flax = average of oilseed crops, Other crops = average of oilseed crops, Ley hey = We assumed a mixture of grasses and legumes (20% white clover) and 3 expected cuts, Ley seed⁶¹ where we use the average of all recommendations to different ley seeds, Horticulture plants⁶³, Energy forests⁶², Vegetables⁶³, where we use the average of all recommendations for different vegetables. In⁵⁶ for Norrland N recommendation to winter wheat, rye oats and triticale were missing. For these crops, we used the same recommendations as they were given for Norra Götaland & Svealand. In No data soil classes of P-AL and K-AL, we used average fertilizer recommendations of a crop for all soil classes combined. Note that because we use fertilizer recommendations and not crop demand, we are implicitly accounting for N fixation by legumes. Because we are only looking at one year, we are not accounting for multi-crop or multi-year rotations (e.g., green fertilizers).
3	$B_{m}^{n}$	Balance of nutrient n in municipality m, where n represent nutrient N, P, or K
4	SWEⁿ	National balance of nutrient n for Sweden.
4	SFⁿ	Nutrient n sold in synthetic fertilizers at the national scale. We obtained synthetic fertilizer data at the national scale from⁷³ which was given as total consumption of N, P and K in the country.