Lotka-Volterra equations mistake

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":

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

**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 $\displaystyle \sum_{k=1}^x\sum_{j=1}^y {x \choose k}{y \choose j} = (2^x-1)(2^y-1)$ and thus $\displaystyle \frac{dx}{dt} = a_1x-a_2(2^x-1)(2^y-1), \frac{dy}{dt}=-b_1y+b_2(2^x-1)(2^y-1)$?

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

Re: Lotka-Volterra equations mistake

Quote:

Originally Posted by

**abscissa** 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":

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

**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 $\displaystyle \sum_{k=1}^x\sum_{j=1}^y {x \choose k}{y \choose j} = (2^x-1)(2^y-1)$ and thus $\displaystyle \frac{dx}{dt} = a_1x-a_2(2^x-1)(2^y-1), \frac{dy}{dt}=-b_1y+b_2(2^x-1)(2^y-1)$?

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

The LV system is a model, nothing more. In this case a model that is amenable to solution. The didn't have computers to simulate differential equations back in the 1920s so closed form solutions were important.

Your system is also a model. Maybe it's more accurate maybe not. But it certainly seems to be much more difficult to solve if not impossible to find a closed form solution for.

I suspect LV were after the simplest model they could use that captured the essence of the predator prey problem.