Discrete Logistic Growth

The discreet logistic growth is described by a simple update rule:

x_n+1 = x_n + r(1-x_n)x_n

Depending on r, this iteration may show very strange, and even chaotic behaviour.

This python script

# -*- coding: utf-8 -*-

"""

Created on Mon Nov  2 22:08:46 2015

@author: oliver

"""

import numpy as np

import matplotlib.pyplot as plt

def logGrowthIter(r,x):

    """

    returns next iteration. Takes r and x_n as arguments, returns x_{n+1}

    xn -> xn+1

    """

    return x + r*(1-x)*x

def logGrowthDiscreet(r,N,x0):

    """

    returns the first N iterations, starting with x0. Arguments: r,N,x0

    """

    X = np.zeros(N)

    X[0] = x0

    for i in range(1,N):

        X[i] = logGrowthIter(r,X[i-1])

    return X

if __name__ == '__main__':

    plt.figure()

    Npre = 1000

    Nplot = 50

    for r in np.linspace(1,3,1000):

        Xpre = logGrowthDiscreet(r,Npre,0.1)

        Xplot = logGrowthDiscreet(r,Nplot,Xpre[-1])

        #plt.plot([r]*Nplot,Xplot,'k.')

        plt.plot([r]*Nplot,Xplot,color='black',linestyle='none',markersize=1,marker='.')

    plt.axis([1,3,0,1.5])

    plt.draw()

    plt.show()

produces