I have a problem with the Lotka-Volterra equations themselves. I believe that they might be wrong. Here is my reasoning - I would appreciate it if someone could find a flaw in it!

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

**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

and thus

?

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