This model shows chaos, not randomness.

 Adjust the growth rate (r) to 3. Run the simulation three times in a row.
While the pattern is unpredictable (chaotic), every run the results are the same (deterministic, not random).
Adjusting the starting population even a little (by a hundredth), with the growth rate (r) at 3, will cause a drastic change in the final outcome. 

This is sensitive dependence on initial conditions, also known as the butterfly effect.