"""
Example from lecture on Sept 23, logistic growth
formulated and solved as a difference equation. 
"""


import numpy as np
import matplotlib.pyplot as plt
x0 = 100                      # initial population
rho = 0.05                    # growth rate 
R = 500                       # max population (carrying capacity)
N = 200                       # number of years

time = range(N + 1)
x = np.zeros(N + 1)

x[0] = x0
for n in time[1:]:
    x[n] = x[n - 1] + rho  * x[n - 1] * (1 - x[n - 1] / R)

plt.plot(time, x)
plt.xlabel('Time')
plt.ylabel('Population')
plt.show()