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## #1 2010-10-02 09:28:01

ToastCrunch
Member
Registered: 2010-10-02
Posts: 1

### Equilibrium Population Stability

How do I find if equilibrium populations are stable?

For example, dN/dt = aN(f-BN)  / (1+agN)

That's Smith's model of population growth.

I know that the problem boils down to dN/dt and that I want to see whether when dN/dt=0 whether population tends to approach N near that time (stable) or not.

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## #2 2010-10-02 10:56:40

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Equilibrium Population Stability

A steady state is stable if d²N/dt² is negative for that value of N.

You can see why this is, because it means that for a value of N just less than a steady state, dN/dt would be positive (and so N would be increasing towards the steady state), and for an N just more than a steady state, dN/dt would be negative and so N would be decreasing towards the steady state.

Similarly, a steady state is definitely unstable if d²N/dt² is positive for that N.

If the second derivative is 0, then that test doesn't conclude anything.

Why did the vector cross the road?
It wanted to be normal.

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