## Growth and the Limiting Factors

The relationship between growth v = dw/dt (w is weight) and the concentration of the limiting factor is described by the Michaelis-Menten equation, which is also used to give the relation between the concentration of a substrate and the rate of a biochemical reaction:

where v is the rate (e.g., growth rate), k a rate constant, s the concentration of a substrate, and ks the half-saturation constant. The dimensions of ks and s are the same: kgm~ or kgm~ . Notice that eqn [1] corresponds to a zero-order reaction when s>> ks and a first-order reaction when ks >> s. Growth follows a zero-order reaction when the resources are abundant, while eqn [1] describes the influence of limiting resources on the growth rate (s << ks).

The situation in nature is often not so simple, as two or more resources may be limiting simultaneously. This can be described by v = Kr (Ni/(ksi + Ni))(N2/(ks2 + N2)) [2]

where N1 and N2 are the nutrient concentrations, K r is a rate constant, and ks1 and ks2 are the half-saturation constants related to N1 and N2.

This equation will often limit the description of growth too much and is in disagreement with many observations. Equation [3] seems to overcome these difficulties:

This expression is in accordance with Liebig's minimum law. Another possibility would be to apply the average of two or more limiting factors, for instance, for two limiting factors (nutrients):

v = Kr [(Ni/(ksi + Ni)) + (N2/(ks2 + N2))]/2 [4] Finally, it is also possible to use the following expression:

If one element (nutrient) is limiting, the growth equation [i] can be used. It is often the case for lakes, where phosphorus is the limiting nutrient. In coastal areas both phosphorus and nitrogen may be limiting at different parts of the year and it is necessary to apply eqn [2], [3],

[4], or [5]. Experience has shown that eqns [3] and [4] have given the most promising results in the sense that it has been possible to obtain better calibration and validation for models that are based on these two equations.

The Michaelis-Menten equation is extensively applied in ecological models for nutrient-limiting plants' growth, for grazing, and for predation. The equation takes into consideration the needs for growth in form of nutrients or food and the availability of the resources. It is obvious that interactions between growth of organisms and the resources determining the growth are frequently describing the actual situation in an ecosystem, which explains the frequent application of the MichaelisMenten equations. The same equation is applied frequently in biochemistry to describe the kinetics of biochemical enzymatic processes. The enzymes correspond to the organisms and the substrates correspond to the nutrients or food.

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