Birth–death fixation probabilities for structured populations

Abstract

This paper presents an adaptation of the Moran birthdeath model of evolutionary processes on graphs. The present model makes use of the full population state space consisting of 2 N binary-valued vectors, and a Markov process on this space with a transition matrix defined by the edge weight matrix for any given graph. While the general case involves solution of 2 N -2 linear equations, symmetry considerations substantially reduce this for graphs with large automorphism groups, and a number of simple examples are considered. A parameter called graph determinacy is introduced, measuring the extent to which the fate of any randomly chosen population state is determined. Some simple graphs that suppress or enhance selection are analysed, and comparison of several examples to the Moran process on a complete graph indicates that in some cases a graph may enhance selection relative to a complete graph for only limited values of the fitness parameter.

Fixation probabilities for simple digraphs

“…Much of this effort has been directed to the study of fixation probabilities for single mutants randomly introduced into an otherwise genetically homogeneous population. This work shows that population structure can enhance or suppress selection relative to drift [7][8][9][10] and that whether selective effects are enhanced or suppressed may depend on both fitness and initial placement [11,12].…”

“…Other than equation (1.1), few analytic results are known: Broom & Rychtář [19] obtained partial results for path graphs and found a closed-form solution for the single-vertex fixation probability of star graphs; Zhang et al [25] computed k-vertex fixation probabilities for star graphs; and Voorhees [26] gave a solution for the single-vertex fixation probability of the complete bipartite graph K s,n . In addition to computation of fixation probabilities, various other questions have been seriously investigated.…”

“…Broom & Rychtář [19] showed that the solution of this system of equations is unique. In [11], a different but equivalent approach is taken. Two vectors a(v) and b(v) are defined by a(v) = v · W and b(v) = v · W, where v i = 1 − v i .…”

“…In general, however, calculation of the birth-death fixation probability for a graph with N vertices involves solution of 2 N − 2 linear equations in an equivalent number of unknowns [11,19]; hence, most studies seek asymptotic approximations [20][21][22], employ methods of 'lumping' vertices into equivalence classes [23] or rely on numerical computations [12,20,24]. Other than equation (1.1), few analytic results are known: Broom & Rychtář [19] obtained partial results for path graphs and found a closed-form solution for the single-vertex fixation probability of star graphs; Zhang et al [25] computed k-vertex fixation probabilities for star graphs; and Voorhees [26] gave a solution for the single-vertex fixation probability of the complete bipartite graph K s,n .…”

Markov Chain Aggregation for Agent-Based Models

“…See Claussen and Traulsen (2005); Traulsen et al (2005);Nowak (2006) for treatments of the associated Moran process. While early studies Maruyama (1974); Slatkin (1981) had indicated that population structure has no or only little effect on the model behavior, it has recently been shown that population structure can have a significant influence (Liberman et al, 2005;Nowak, 2006;Shakarian et al, 2012;Voorhees and Murray, 2013;Voorhees, 2013, among many others). The setting -sometimes referred to as evolutionary graph theory (Liberman et al, 2005) -is usually as follows: suppose the is a population of N individuals with fitness 1; suppose that a mutant with fitness r is introduced in one of the individuals; what is the probability that the mutant invades the entire population?…”