Lotka-Volterra equations mistake

[kalish1]

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":

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

**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?

 

[zzephod]

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":

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

**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?

No, the model assumes that the number of births of predators is proportional to the prey density, which is proportional to the number of prey in the system, and the number of predators present. Similarly the loss of prey due to predation is also proportional to the product.

If you double the number of predators you double the death rate of prey due to predation and you double the number of births of predators ...

You may not like the model, but that does not make it wrong, it is just a model.

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