MHB Lotka-Volterra equations mistake

  • Thread starter Thread starter kalish1
  • Start date Start date
  • Tags Tags
    Mistake
kalish1
Messages
79
Reaction score
0
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?
 
Physics news on Phys.org
kalish said:
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.

.
 
Thread 'Direction Fields and Isoclines'
I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
Back
Top