Notation Question (Differential Equations)

[dkotschessaa]

I want to show, in a talk, how to build the Predator-Prey (Lotka Volterra) model. It starts off with the "Population Law of Mass Action" i.e. "At time t the rate of change of one population due to interaction with another is proportional to the product of the two populations at that time t"


Is it an abuse of notation to write this (for populations x (predator) and y(the prey)) as:

$\frac{dx}{dt} \propto xy$
$\frac{dy}{dt} \propto xy$



I think it's ok, I've just never seen this written for a differential equation.

The next step would be to write:

$\frac{dx}{dt} = axy$
$\frac{dy}{dt} = bxy$

For some a and b, then start showing what happens when x starts eating y.

-Dave K

 

[pasmith]

I want to show, in a talk, how to build the Predator-Prey (Lotka Volterra) model. It starts off with the "Population Law of Mass Action" i.e. "At time t the rate of change of one population due to interaction with another is proportional to the product of the two populations at that time t"


Is it an abuse of notation to write this (for populations x (predator) and y(the prey)) as:

$\frac{dx}{dt} \propto xy$
$\frac{dy}{dt} \propto xy$



I think it's ok, I've just never seen this written for a differential equation.

That is potentially confusing; the full Lotka-Volterra equations include other terms, so it is not actually true that dx/dt \propto xy etc.

I would start with <br /> \frac{dx}{dt} = rx + p_{\mathrm{interact}} \\<br /> \frac{dy}{dt} = st + q_{\mathrm{interact}}<br /> and then proceed immediately to <br /> p_{\mathrm{interact}} = axy, \\<br /> q_{\mathrm{interact}} = bxy.<br /> I would also make all the coefficients positive and introduce minus signs where necessary.

The important points are that (1) the rate of change due to interaction is proportional to the product of the populations, and (2) the sign of the coefficient depends on who is eating who.

 

[dkotschessaa]

That is potentially confusing; the full Lotka-Volterra equations include other terms, so it is not actually true that dx/dt \propto xy etc.

I would start with <br /> \frac{dx}{dt} = rx + p_{\mathrm{interact}} \\<br /> \frac{dy}{dt} = st + q_{\mathrm{interact}}<br /> and then proceed immediately to <br /> p_{\mathrm{interact}} = axy, \\<br /> q_{\mathrm{interact}} = bxy.<br /> I would also make all the coefficients positive and introduce minus signs where necessary.

The important points are that (1) the rate of change due to interaction is proportional to the product of the populations, and (2) the sign of the coefficient depends on who is eating who.

did you mean $\frac{dy}{dt} = sy + q_{\mathrm{interact}}$ in your second example?

That's a very good way of explaining it though. Thank you.

-Dave K