Classification of steady states

[sid9221]

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

So am I right in saying the 2 steady states are:

$$N_1=\frac{h}{r}, N_2=\frac{1-h}{\alpha}$$

Now plugging in N_1 into the equation I get:

$$\frac{-h^2 \alpha}{r} < 0$$

So N_1 is stable.

But I can't quite figure out how to classify N_2

Any advice ?

 

[vela]

No, that's not correct. In steady state, the population isn't changing; that is, it's when dN/dt=0.

 

[sid9221]

Exactly what I was thinking, I was shying away from doing that as they did not look "pretty" and I do not know what a "harvesting term" is.

So the steady states should be:

$$N_1 = \frac{r-\sqrt{r(r-4\alpha h)}}{2ar}, N_2 = \frac{r+\sqrt{r(r-4\alpha h)}}{2ar}$$

N_1<N_2

?

 

[vela]

Yes, those are right.

The harvesting term is h. It could represent, say, the number of deer killed by hunting every year. The population is decreased by that amount every year.

 

[sid9221]

How do I classify these as if I sub them back into the differential equation I get 0 ??

 

[vela]

You need to find the correct criterion. You're trying to figure out what happens if the system is disturbed slightly from an equilibrium point. Will it move back toward that point or will it move away? It might help to plot the function dN/dt as a function of N.

 

[spamiam]

Hi sid9221,

Out of curiosity, did this question come from a textbook? James Murray has a good book on mathematical biology and parts are available on Google books. He discusses of linear stability here.

vela is exactly right, you need to consider the effect of a small perturbation at the steady state. In the end, all you really need to do is set $f(N) = \frac{d N}{dt}$ and consider $f'(N)$ evaluated at the steady states.