Question

If a population with harvesting rate h is modeled by

dx/do=9-x^2-h

find the bifurcation point for the equation.

Answer 1

** Definition:-A point (in a bifurcation diagram) where stability changes from stable to unstable is called a bifurcation point.

** To check for stability, if and are continuous, an equilibrium point is stable if and unstable if , provided .

Q. Solution:-

Given that

, where h is harvesting rate.

First, we need to find equilibrium point.

So, let .

To find equilibria,

To have a real roots ,

Now, to analyze the stability of the equilibria, we have

(stable equilibrium)

and (unstable equilibrium),​​​​​

provided .

But at , and it is the common point of the curves in the , where the stability changes from stable to unstable.

Hence, the bifurcation point for the equation is h=9,(i.e. x=0) in xh-plane.