Time to the convergence of evolution in the space of population states
Phenotypic evolution of two-element populations with proportional selection and normally distributed mutation is considered. Trajectories of the expected location of the population in the space of population states are investigated. The expected location of the population generates a discrete dynamical system. The study of its fixed points, their stability and time to convergence is presented. Fixed points are located in the vicinity of optima and saddles. For large values of the standard deviation of mutation, fixed points become unstable and periodical orbits arise. In this case, fixed points are also moved away from optima. The time to convergence to fixed points depends not only on the mutation rate, but also on the distance of the points from unstability. Results show that a population spends most time wandering slowly towards the optimum with mutation as the main evolution factor.
- Chorążyczewski A., Galar R. and Karcz-Dulęba I. (2000): Considering phenotypic evolution in the space of population states. - Proc. 5th Conf. Neural Networks and Soft Computing, Zakopane, Poland, pp. 615-620.
- Galar R. (1985): Handicapped individua in evolutionary processes. -Biol. Cybern., Vol. 51, No. 1, pp. 1-9.
- Galar R. and Karcz-Dulęba I. (1994): The evolution of two: An example of space of states approach. - Proc. 3rd Ann. Conf. Evolutionary Programming, San Diego, CA, pp. 261-268.
- Karcz-Dulęba I. (2000): Dynamics of evolution of population of two in the space of population states. The case of symmetrical fitness functions. -Proc. 4th Nat. Conf. Evol. Algorithms and Global Optimization, Lądek Zdrój, Poland, pp. 115-122 (in Polish).
- Karcz-Dulęba I. (2002): Evolution of two-element population in the space of population states: Equilibrium states for asymmetrical fitness functions, In: Evolutionary Algorithms and Global Optimization (J. Arabas, Ed.). -Warsaw: Warsaw University of Technology Press, pp. 35-46.
- Karcz-Dulęba I. (2004): Asymptotic behavior of discrete dynamical system generated by simple evolutionary process. - Int. J. Appl. Math. Comp.Sci., Vol. 14, No. 1, PP. 79-90.
- Prugel-Bennett A. (1997): Modeling evolving populations. - J. Theor. Biol., Vol. 185, No. 1, pp. 81-95.
- Vose M.D. and Wright A. (1994): Simple Genetic Algorithms with Linear Fitness. - Evolut. Comput., Vol. 2, No. 4, pp. 347-368.
- Vose M.D. (1999): The Simple Genetic Algorithm. Foundations and Theory. - Cambridge, MA: MIT Press.