HOEMWORK COSC 120  WINTER 2017
Distributed: 2/2/2017
Due: CHANGE DATE TO 2/14/2017

Write three Python scripts to show the convergence of the following models.

(1) Babylonian.  Compute sqrt(2000) in 20 steps (aka ticks).
x(t+1) = 0.5 * (x(t) + (a / x(t) )
x0 = a  (or x0 = (a + a/2) / 2

(2) Compute pi using the following expansion for 20 ticks.
See https://en.wikipedia.org/wiki/Pi

(use floating point arithmetic)

pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 +  . . .

(3) This models predator (y) and prey (x) population.
r is growth rate of prey  (0 <= r <= 1.0)
d is death rate of predator ( 0 <= d <= 1.0)
b is a tuning parameter to determine how quickly d (death rate of predator) changes as 
	y (prey population) increases.
c is a tuning parameter to determine how quickly r (growth rate of prey) changes as 
	x (predator population increases.
K is carrying capacity of the environment.

x(t+1) = x(t) + r * x(t)*( 1 - x(t)/K) - (1 - 1/ (b * y(t) + 1) ) * x(t)
y(t+1) = y(t) - d * y(t) + c * x(t) y(t)

Run the model for 300 ticks with:
r = b = 0.5
d = c = 1
K = 20
x0 = y0 = 1