Nutrient equation derivation

An example of prediction equations for intake that may be used in conjunction with feeding standards is provided by the UK Agricultural Research Council s Technical Committee on Responses to Nutrients, which constructed a series of equations for predicting the intake of grass silage by cattle (both dairy and beef) fed on silage and concentrates. The following is an example of the equations derived for beef cattle ... 

Essentially all organic matter in the ocean is ultimately derived from inorganic starting materials (nutrients) converted by photosynthetic algae into biomass. A generalized model for the production of plankton biomass from nutrients in seawater was presented by Redfield, Ketchum and Richards (1963). The schematic "RKR" equation is given below ... 

We seek to write differential equations for this model, and begin by considering just one organism growing in the chemostat. (A more complete derivation can be found elsewhere see e.g. Herbert, Elsworth, and Telling [HET].) The rate of change of the nutrient can be expressed as... 

Consider two populations, with densities Xx andA 2, competing for a single nutrient of concentration S in the chemostat. Competition occurs in the sense that each population consumes nutrient and so makes it unavailable for the competitior. The average amount of stored nutrient per individual of population Xx is denoted by Qx, and for population X2 by 02-Following the derivation of Section 2, we have the following equations ... 

It follows that deep seawater contains nutrients from two sources. First, it may contain nutrients that were present with the water when it sank from the surface. These are called "preformed nutrients". Second, it may contain nutrients derived by the in situ remineralization of organic particles. These are called oxidative nutrients. The oxidative nutrients can be estimated from the RKR equation. From this model, we might expect the four dissolved chemical species (O2, CO2, NO3, PO4) to vary in seawater according to the proportions predicted. The key to understanding these remineralization reactions is the parameter Apparent Oxygen Utilization (AOU), defined as ... 

Steele (1956) (6) found that the steady-state assumption did not apply to the seasonal variation of the phytoplankton population. Instead, he used two volume segments to represent the upper and lower water levels and kept the time derivatives in the equations. Thus, both temporal and spatial variations were considered. In addition, the differential equations for phytoplankton and zooplankton concentration were coupled so that the interactions of the populations could be studied, as well as the nutrient-phytoplankton dependence. The coefficients of the equations were not functions of time, however, so that the effects of time-varying solar radiation intensity and temperature were not included. The equations were numerically integrated and the results compared with the observed distribution. Steele applied similar equations to the vertical distribution of chlorophyll in the Gulf of Mexico (7). 

The first term In equation (14), representing the gross rate of biomass production, Is Identical with the function Monod (25) originally adopted "to express conveniently the relation between exponential growth rate and concentration of an essential nutrient." Such a rectangular hyperbolic function has been derived many times from various reaction mechanisms (26-30). but none has addressed the present case of continuous culture systems where y j and K have been observed to vary with temperature and dilution rate. 

The metabolic pools defined in the denominator of the nutrient-response equation can be viewed as intermediate metabolites. Thus, the logarithmic derivative of each metabolic pool with respect to nutrient intake provides an estimate of the sensitivity of the pool to nutrient intake. Figure 7 shows... 

In the trade, potassium-containing compounds or salts that are applied primarily as a source of this nutrient or that find use in industry are referred to as potash. The name derives from an early production method in which potassium carbonate, leached from wood ashes, was crystallized by evaporating the leachate in large iron pots. The salt potassium chloride (muriate, muriate of potash, or KCl) is now the major source of the element (95%) other important salts are potassium sulfate (potassium sulphate, K2SO4, or sulphate of potash), potassium magnesium sulfates of varying K/Mg ratios, and potassium nitrate (KNO3). The fertilizer industry expresses the potassium content of fertilizer salts in terms of the potassium oxide (K2O) content, not as the K content. Muriate thus contains 60 percent + of K2O—which equates with 49.8 percent-h of potassium (K). 

---