danryan.us

Start by identifying rates and amounts:

Rates: 0.2 new members per member per month; 40% attrition per month
Amounts: 10 new members each month

The general form of the difference equation here is

$members_{n+1} = (1 + 0.2 - 0.4) \times members_{n} + 10 = 0.8 \times members_{n} + 10$

We can plot this on a P_n+1 vs. P_n chart:

We calculate the equilibrium value in the normal way:
$P_{E} = \frac {b} {1-a} = \frac {10}{1 - 0.8} = 50$
We could also read this off the chart. And we can tell it is a stable equilibrium from the P_n+1 vs. P_n diagram since the difference equation line is shallower than the 45 degree line.

Finally, we should plot membership vs time for various starting values to see how this system behaves — we see that for initial values above 50 there is a gradual decline to 50 and for numbers below there is a gradual growth that tapers off at 50.

Excel File