The Discrete Logistic Equation

The discrete logistic equation is a crude model of discrete population growth. For example, it could model a species that has a single mating season so that population numbers are updated annually. There is also an annihilation parameter; if the population exceeds this value, then there will be total extinction the following year. Scaling so that the annihilation parameter is 1 we obtain the difference equation:

P_n+1=kP_n(1-P_n), where k is a positive constant.

Try the following constants and initial conditions to see different types of orbits:
k=0.5, P₀=0.5
k=1.25, P₀=0.5
k=2, P₀=0.5
k=3.2, P₀=0.8
k=3.5, P₀=0.88
k=3.98, P₀=0.95

Created by Dave Richeson using GeoGebra.