Help! Finding fixed points in ODEs.

[Joanna1594]

How to find fixed points of a dynamical system that has those properties?
1) dy/dt = β(y/N) * x - γy
2) dx/dt = −β(y/N) *x + γy .

Gamma and beta are positive numbers. I am supposed to use the fact that x+y=N (both equations sum to 0). It is probably pretty simple, but I'm stuck. I know I just need to substitute dy/dt and dx/dt for 0, but I can't seem to get rid of N...

 

[HallsofIvy]

First. why do you need to "get rid of N"? Can't the fixed points depend on N? Second, how do you know that "x+ y= N"? Since "both equations sum to 0" x+ y must be a constant but how do you know that constant is N? If you do in fact know that you can "get rid of N" by replacing N with x+ y.

\(\displaystyle \beta xy/N- \gamma y= y(x/N- \gamma)= 0\) so either \(\displaystyle y= 0\) or \(\displaystyle x= \gamma N= \gamma(x+ y)\), and then \(\displaystyle y= \frac{\gamma- 1}{\gamma}x\).

 

[Joanna1594]

Thank you so much for your reply!! How do I know it - from the task (attached). I just replaced x and y for [A] and , because B kept me confusing it with beta... I got the same results as you (except that you missed beta at some point), but I'm not sure if it makes sense in light of what they write in the task. They say: "You might find it useful to write your results in terms of R0=beta/gamma, but I don't seem to get it anywhere (it is in the task number 2).

 

[Joanna1594]

Plus, unfortunately, getting this exact R0 form is required for the next step, which is the bifurcation plot: So as long as I am stuck with those fixed points I can't proceed any further...