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== Jacob-Monod Model ==

Another approach to modeling the interaction between prey and
predators was developed to account as well for organisms (such as
bacteria) taking up nutrients. There is a limited uptake rate that
such organisms are capable of, and the next model accounts for
limited uptake rates. Suppose now that <math>x</math> denotes the size of a
population of feeders and that they are feeding on a chemical
species of concentration <math>y\ .</math> These are usually taken to represent
concentrations of feeders and nutrients in solution rather than
"head counts". The '''Jacob-Monod model''' is
:<math>\dot x = \frac{V y}{K+y}x</math>
:<math>\dot y = -\frac 1Y \frac{V y}{K+y}x</math>
where 
* <math>V</math> is the uptake velocity, 
* <math>K</math> is the saturation constant, and
* <math>Y</math> is the yield of <math>x</math> per unit <math>y</math> taken up. 
Note that when <math>y=K\ ,</math> the uptake velocity is <math>V/2\ ,</math> half the maximum; in practice, <math>y=K</math> is taken as a tipping point: 
If <math>y<K\ ,</math> then uptake is ignored. This is the canonical model of nutrient uptake (nutrient <math>y</math> is taken up by species <math>x</math>), and it underlies many calculations in biology,
microbiology, and food engineering (digestion, beer, etc.). This
model was discovered independently in several diverse applications.
It is akin to the [[Haldane-Briggs model]] and [[Michaelis-Menten model]] in
[[biochemistry]], the Jacob-Monod model in microbial ecology, and the
[[Beverton-Holt model]] in fisheries. It serves as one of the important
building blocks in studies of complex [[biochemical reaction]]s and in
[[ecology]] (Smith and Waltman 1997).

The quantity <math>C=x +Y y</math> is conserved (since the derivative with
respect to time of the right hand side is zero) where
<math>C=x(0)+Yy(0)\ .</math> Substituting this into the first equation gives
:<math>\dot x =\frac{V (C-x)}{Y K+(C-x)}x\ .</math>
The solutions can be found by quadratures, and these show that if
<math>x(0)>0\ ,</math> then <math>x(t)\to C</math> as <math>t\to\infty\ ,</math> at which point the nutrient has been depleted.

A typical use of this model is to describe a continuous-flow growth
device, such as a [[chemostat]], where there is continuous removal of nutrient and feeders
and a continuous supply of fresh nutrient. The Jacob-Monod model is used to describe such a bacterial growth device, for example to determine conditions for a sustained dynamic equilibrium to exist by balancing growth due to uptake of nutrient with wash out of feeders.

Some predator-prey models use terms similar to those appearing in the Jacob-Monod model to describe the rate at which predators consume prey.  More generally, any of the data in the Lotka-Volterra model can be taken to depend on prey density as appropriate for the system being studied. This is referred to as a functional response, an idea that is introduced and discussed by C. S. Holling (1959). Several different forms of functional response have been used in population models, but the Jacob-Monod form, also called the Holling type 2 form by ecologists, is one of the more common ones. Many other investigations of predator-prey models have involved functional responses. For example, <math>r</math> is replaced by <math>r \ln(K/x)</math> (Gompertz, 1825), <math>r(K-x)/(K+\epsilon x),</math> (Smith, 1963), and <math>r\left((K/x)^g-1\right)</math> with <math>0< g \le 1\ ,</math> (Rosenzweig, 1971). These and other functional responses are also discussed in May (1974). Such mechanisms in the Lotka-Volterra model can stabilize or destabilize the system, for example resulting in predator extinction or in co-existence of prey and predators. This is in contrast to the plurality of cycles predicted by the original Lotka-Volterra model.