Steady states/systems of differential equations/Phase portraits

[sid9221]

http://dl.dropbox.com/u/33103477/2007%2010b.png

I'm can't get my head around this question, there doesn't seem to be enough information to compute a steady state ?

Any ideas on how to begin ??

 

[spamiam]

Hi sid9221,

First of all, can't you solve explicitly for p(t) and r(t)? The ODE for r(t) should look very familiar.

This problem seems a bit strange. Typically to find the steady states you set ##\frac{dp}{dt} = 0## and ##\frac{dr}{dt} = 0##. These two conditions give you steady states ##(p_*, r_*)##. What do you get in this case?

It seems very fishy to me that ##\frac{dp}{dt}## and ##\frac{dr}{dt}## don't seem to depend on p at all...

Could we see part a) of the question, too?

 

[sid9221]

Part a) is completely unrelated.

It's some nonsense about the population of fish...

 

[spamiam]

Part a) is completely unrelated.

It's some nonsense about the population of fish...

The situations may be different, but are the models similar?

Did you solve for p(t) and r(t) or find the condition given by setting the derivatives to zero?

 

[sid9221]

http://dl.dropbox.com/u/33103477/Untitled.png

 

[spamiam]

Thanks for posting the full problem.

Anyway, if you set ##\frac{dp}{dt}=0## and ##\frac{dr}{dt} = 0##, what must r equal?

Also, if ##r'(t) = - \beta \, r(t)##, what function must ##r(t)## be?