Sensitivity Indices: Compute Normalized Forward SI for r,a

[shft600]

Homework Statement

Consider the population model with dx/dt=rx(1-x)-ax/(1+x). Compute the normalized forward sensitivity indices of the (positive) internal equilibrium with respect to r and a. The default parameter values are r=4, a=3.

Homework Equations

SI(x,t)=(∂x/∂t)*(t/u)

The Attempt at a Solution

I'm not even sure where to begin on this. I first followed the SI formula above, taking the partial's from the given problem with respect to r, then again with a. After that I input the r and a values, getting
SI(r)=4(1-x)
and
SI(a)=-3/(1+x)
The problem is, I literally have no idea if I'm even approaching the problem right. Should I be solving for some equilibrium first? I tried that and ended up with x*=1-sqrt(a/r), but that gets so completely messy when it comes time to take the partial derivatives that I can't believe a second year math course would ask me to do this for homework.

Anybody have any insight at all? Honestly, anything. The examples given in class always had more information, and were brutally simple.

 

[mighty2000]

First, let's find the internal equilibrium by setting dx/dt=0 and solving for x:

0 = rx(1-x)-ax/(1+x)
0 = x(r(1-x)-a/(1+x))
0 = x(r-rx-a)
x = r/(r+a)

Now, let's find the partial derivatives of x with respect to r and a:

∂x/∂r = a/(r+a)^2
∂x/∂a = -rx/(r+a)^2

Next, we can plug in the default parameter values (r=4, a=3) and the internal equilibrium (x=4/7) into the sensitivity index formula:

SI(r) = (∂x/∂r)*(t/u) = (3/49)*(4/7) = 12/343
SI(a) = (∂x/∂a)*(t/u) = (-4/7)*(4/7) = -16/49

Therefore, the normalized forward sensitivity indices for the internal equilibrium with respect to r and a are 12/343 and -16/49, respectively.