Logistic Model

**demode** · May 10th 2012, 06:21 AM

I need some help with the following problem:

My attempt

(a) I think to reach equilibrium it must satisfy

$\frac{dR}{dt} = 4000 \left( 1- \frac{4000}{130000} \right) - \alpha (4000)(160) = 0$

$\therefore \alpha = 6.05769 \times 10^{-3}$

Is this correct?

(b) When C=160 and R=4000, $\alpha RC = (6.05769 \times 10^{-3}) (4000)(160) = 3876.923$

So the contribution is that the population of the rabbits reduces by 3876.923 as the result of the interaction?

Could anyone show me how we can calculate how many rabbits did each cat eliminate per year? Any help would be greatly appreciated.

**mastermind2007** · May 11th 2012, 05:21 AM

Yes,
$ \alpha = \frac{63}{10400} and -\alpha RC = ... $

**HallsofIvy** · May 11th 2012, 08:56 AM

What definition of "equilibrium" are you using? My understanding of an "equilibrium solution" to a pair of equations is one in which both variables are constant. With C= 160, dC/dt is NOT 0 so I would not call this an equilibrium solution for any value of $\alpha$ .

**mastermind2007** · May 11th 2012, 10:35 AM

@ HallsofIvy:

Yes, it IS zero.
$1 - \frac{160}{160} = 0$

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