Suppose that a population model is the logistic model with harvesting term h, a constant.

dN/dt=rN(1-(alpha)*(N))-h

r,h,alpha +ve constants

1.)assume r>4h(alpha).find and classify the steady states N_1 and N_2 where N_1<N_2.

I can find steady states of normal differential equations and sub in x=whatever when f(x)=0 but i dont have a clue what this is asking of me. any useful sites would be much appreciated also.

In general if dy/dx = f(y)

Then the steady state solutions are the zeroes of f(y)

Let a =alpha

For dN/dt = rN(1-aN) - h

Steady states are found when rN(1-aN) - h = 0

Expanding -arN^2 + rN - h = 0

Use the good ol' quadratic formula

N = 1/2a + sqrt(r^2 - 4arh)/2ar

this will clean up to 1/2a + sqrt(1-4ah/r)/2a

or N = [1 + sqrt(1-4ah/r)]/2a

since r > 4ah the solutions are real

By considering a graph of dN/dt vs N you should be able to determine N1 unstable and N2 stable

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