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The equations are generally of the form, as quoted from "A Modern Introduction to Differential Equations 2nd edition by Henry Ricardo":

$$\frac{dx}{dt} = a_1x-a_2xy, \frac{dy}{dt}=-b_1y+b_2xy$$

**My issue:** The $xy$ terms represent the number of possible interactions between two species. However, they only represent the number of possible *one-on-one* interactions between the two species. In order to account for *all* the possible interactions, such as $(x-1)$ predators acting on $2$ preys, shouldn't we arrive at $$\sum_{k=1}^x\sum_{j=1}^y {x \choose k}{y \choose j} = (2^x-1)(2^y-1)$$ and thus $$\frac{dx}{dt} = a_1x-a_2(2^x-1)(2^y-1), \frac{dy}{dt}=-b_1y+b_2(2^x-1)(2^y-1)?$$

Doesn't this make the number of interactions proportional not to the product of the number of predators and prey, but to their exponentiation?