Simple birth, death and saturation

When dealing with a single population, the standard modelling framework is 

Here r, the difference between birth and death rates is the maximum growth rate, since the effective population-dependent growth rate g(P) = r( 1 — P/K) decreases with P. K is the carrying capacity, identified with the maximum population that can be sustained by the available resources, that is the population value at which growth stops. 

The growth rate of P in (3.66) is maximum when P — 0 and very small populations always grow. Of course, if the population is 

Volterra (1926), in order to explain observations of changes in fish catches in the Adriatic Sea during the First World War, proposed a model in which the growth rate of a prey P in (3.65) is decreased proportionally to the presence of a predator Z. On the other hand the growth rate of a predator Z, negative when alone, is increased 

It turns out that this model is equivalent to the one derived by Lotka (1920) as arising from application of the law of Mass Action to the hypothetical chemical scheme  

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