## Analysis of the Simple Genetic Algorithm on the
Single-peak and Double-peak Landscapes

Alden H. Wright
Computer Science Dept.
Univ. of Montana
Missoula, MT 59812
wright@cs.umt.edu
http://www.cs.umt.edu/u/wright/wright.htm
Jonathan E. Rowe
School of Computer Science
The University of Birmingham
UK
j.e.rowe@cs.bham.ac.uk
James R. Neil
School of Computer Science
The University of Birmingham
UK
j.r.neil@cs.bham.ac.uk
Abstract
We compare the behavior of a GA with and without crossover.
A simple GA with crossover can have two stable fixed points
(bistability) on the single-peak landscapes for string lengths at
least 8. There are catastrophic transitions from one to two and back
to one stable fixed point as the mutation rate increases. Crossover
also causes the fixed-point population distribution to change.
For a fixed point near the peak, there are fewer copies of the
optimum string, more copies of near-optimum strings, and less
copies of far-from-optimum strings. An example is given where
average population fitness decreases to the minimum possible
population fitness for the with-crossover GA, while the average
population fitness increases for the without-crossover GA. The
primary tool in obtaining these results is the Vose dynamical system
model.