""" An animation of the logistic function ============================================= """ #################################### # The logistic map and... rabbits? # ---------------------------------- # A commun example to understand the logistic map is to think of a pattern to simulate a population of rabbits # with the most simple conditions in a multiplicativ model. # Let's consider a group made of :math:`x_n` rabbits, where :math:`x_n` is a percentage of the maximum number of rabbits we can have. # Let's observe them every six months (time they need to be able to produce offsprings). # Then at the next observation, if we note :math:`r` the growth rate of the population, there will be :math:`rx_n` rabbits. # However, this modelisation is wrong because it assumes that none of the rabbits died or escaped its cage to go back to the wilderness. # So, a way to control this pattern is to multiply :math:`rx_n` by :math:`(1-x_n)`, a factor that, when :math:`x_n` approaches it's maximum, tends to :math:`0`. # # # Thus the equation of the logistic map is : # # .. math:: # # x_{n+1}=rx_n(1-x_n),\ \forall\, (x_0,r)\in [0,1]\times [0,4]. # # Let's see how the population evolves over time depending on the growth ratio. Let's begin with a growth ratio of :math:`2.5`. import chaoseverywhere as chaos import matplotlib.pyplot as plt plt.style.use('ggplot') chaos.logistic_draw(0.01, 2.5, 100, 100) hline = plt.axhline(y=1.5/2.5, color='black', ls=':', label=r'$y=\dfrac{2.5-1}{2.5}=0.6$') plt.legend(handles=[hline]) plt.show() ################################ # Evolution in term of r values # -------------------------------- # * For :math:`r\leq 1`, our bunnies die. The sequence tends to :math:`0`. # * For :math:`r\in [1,3]`, the population oscillates and then stabilizes to the value :math:`\frac{r-1}{r}`. # * For :math:`r\in [3,3.57]` the population oscillates between several values, there is no longer one attractor. # * For :math:`r\geq 3.57` there is almost surely a chaotic design. # # # .. raw:: html # # #