/mhj 12 Finite Population Models of Dynamic Environments in Population Genetics Anthony M.L. Liekens Biomedical Imaging and Informatics Introduction Population genetics studies the dynamics of frequencies of genes in a population, as the population evolves according to the neo-darwinian process of evolution. Commonly, these models are based on static selection pressure environments. Enhancements of finite population models are constructed to study the influence of selection and reproduction parameters in these dynamic environments, such as environments with alternating fitness functions or antagonistic co-evolution. Models and Methods Static Models  Markov model of finite population models can be constructed by enumerating all possible populations and computing the generational transition probabilities among these states  The Markov model can be written as a probability transition matrix T. If x denotes the current distribution, then Tx denotes the distribution at the next generation  Fixed point distribution over this set of possible populations gives the expected behavior of the population. The fixed point or limit behavior can be found be computing the eigenvector with corresponding eigenvalue 1 of T Dynamic Models  Several matrices T 1 , T 2 , ..., T n represent generational transition matrix of a population, according to n different fitness environments, respectively  Matrix T n tn …T 2 t2 T 1 t1 represents the probability transition matrix for t1+t2+...+tn generations. The limit behavior of this system gives us a basis to study the influence of selection and reproduction parameters in a deterministically dynamic environment  Similarly, we can build stochastically dynamic environments and study the influence of parameters on this model Co-Evolutionary Models  We assume 2 or more interacting populations, with their states spaces and transition probabilities  The Cartesian product of these Markov models gives us a Markov model of coupled populations, in which co-evolution (anatagonistic or cooperative) can be studied  We are specifically interested in the differences between haploidy and diploidy in dynamic environments. In the co-evolutionary model, we let both systems evolve against each other to study their relative performance Computational Challenges Huge matrices  Transition matrices of one population tend to be huge  Combinations of these matrices (as required to build dynamic and co-evolutionary models) are even larger  Computationally difficult to put these matrices in memory, or time expensive to use compression Results Dynamic Environments  Diploid populations tend to gather more fitness, as compared to haploid populations, under similar reproductive conditions (e.g. Figure), as environments change quickly Co-Evolutionary Environments  In haploid versus diploid games, diploid populations tend to gather more fitness as a strictly positive dominance coefficient is assumed  Limit behavior predicts large standard deviations. Hence, it is difficult to find similar results with empirical studies (e.g. by doing simulations) Conclusions Markov models of evolution under static selection pressure can be coupled to form models of evolution in dynamic or co-evolutionary environments More work is required to study the importance of ploidy in dynamic environments. We specifically want to study the set of parameters for which haploidy or diploidy performs best References: [1] A.M.L. Liekens, H.M.M. ten Eikelder, P.A.J. Hilbers, Modeling and Simulating Diploid Simple Genetic Algorithms, in Foundations of Genetic Algorithms VII, Malaga, Spain, 2003 [2] A.M.L. Liekens, H.M.M. ten Eikelder, P.A.J. Hilbers, Finite Population Models of Dynamic Optimization with Alternating Fitness Functions, in EvoDOP workshop, GECCO 2003, Chicago, United States, 2003 [3] A.M.L. Liekens, H.M.M. ten Eikelder, P.A.J. Hilbers, Finite Population Models of Co-Evolution and their Application to Haploidy versus Diploidy, in GECCO 2003, Chicago, United States, 2003