Fixation in large populations: a continuous view of a discrete problem

We study fixation in large, but finite, populations with two types, and dynamics governed by birth-death processes. By considering a restricted class of such processes, we derive a continuous approximation for the probability of fixation that is valid beyond the weak-selection (WS) limit. From the continuous approximations, we then obtain asymptotic approximations for evolutionary dynamics with at most one equilibrium, in the selection-driven regime, that does not preclude a weak-selection regime. As an application, we study the fixation pattern when the infinite population limit has an interior Evolutionary Stable Strategy (ESS): (i) we show that the fixation pattern for the Hawk and Dove game satisfies what we term the one-half law: if the Evolutionary Stable Strategy (ESS) is outside a small interval around \sfrac{1}{2}, the fixation is of dominance type; (ii) we also show that, outside of the weak-selection regime, the long-term dynamics of large populations can have very little resemblance to the infinite population case; in addition, we also present results for the case of two equilibria. Finally, we present continuous restatements valid for large populations of two classical concepts naturally defined in the discrete case: (i) the definition of an \textsc{ESS}_N strategy; (ii) the definition of a risk-dominant strategy. We then present two applications of these restatements: (i) we obtain an asymptotic definition valid in the quasi-neutral regime that recovers both the one-third law under linear fitness and the generalised one-third law for $d$-player games; (ii) we extend the ideas behind the (generalised) one-third law outside the quasi-neutral regime and, as a generalisation, we introduce the concept of critical-frequency; (iii) we recover the classification of risk-dominant strategies for $d$-player games

A non-standard evolution problem arising in population genetics

We study the evolution of the probability density of an asexual, one locus population under natural selection and random evolution. This evolution is governed by a Fokker-Planck equation with degenerate coefficients on the boundaries, supplemented by a pair of conservation laws. It is readily shown that no classical or standard weak solution definition yields solvability of the problem. We provide an appropriate definition of weak solution for the problem, for which we show existence and uniqueness. The solution displays a very distinctive structure and, for large time, we show convergence to a unique stationary solution that turns out to be a singular measure supported at the endpoints. An exponential rate of convergence to this steady state is also proved.Comment: 16 pages. Proposition 4 statement and proof corrected. Also a number of typos has been fixe

Discussion on the energy content of the galactic dark matter Bose-Einstein condensate halo in the Thomas-Fermi approximation

We show that the galactic dark matter halo, considered composed of an axionlike particles Bose-Einstein condensate \cite{pir12} trapped by a self-graviting potential \cite{boh07}, may be stable in the Thomas-Fermi approximation since appropriate choices for the dark matter particle mass and scattering length are made. The demonstration is performed by means of the calculation of the potential, kinetic and self-interaction energy terms of a galactic halo described by a Boehmer-Harko density profile. We discuss the validity of the Thomas-Fermi approximation for the halo system, and show that the kinetic energy contribution is indeed negligible.Comment: 10 pages, 2 figures; v.2: 11 pages, 3 figures, references added, matches published version in JCA

The continuous limit of the Moran process and the diffusion of mutant genes in infinite populations

We consider the so called Moran process with frequency dependent fitness given by a certain pay-off matrix. For finite populations, we show that the final state must be homogeneous, and show how to compute the fixation probabilities. Next, we consider the infinite population limit, and discuss the appropriate scalings for the drift-diffusion limit. In this case, a degenerated parabolic PDE is formally obtained that, in the special case of frequency independent fitness, recovers the celebrated Kimura equation in population genetics. We then show that the corresponding initial value problem is well posed and that the discrete model converges to the PDE model as the population size goes to infinity. We also study some game-theoretic aspects of the dynamics and characterize the best strategies, in an appropriate sense.Comment: 40 pages, 14 figures, submitte

On the stochastic evolution of finite populations

This work is a systematic study of discrete Markov chains that are used to describe the evolution of a two-types population. Motivated by results valid for the well-known Moran (M) and Wright-Fisher (WF) processes, we define a general class of Markov chains models which we term the Kimura class. It comprises the majority of the models used in population genetics, and we show that many well-known results valid for M and WF processes are still valid in this class. In all Kimura processes, a mutant gene will either fixate or become extinct, and we present a necessary and sufficient condition for such processes to have the probability of fixation strictly increasing in the initial frequency of mutants. This condition implies that there are WF processes with decreasing fixation probability --- in contradistinction to M processes which always have strictly increasing fixation probability. As a by-product, we show that an increasing fixation probability defines uniquely an M or WF process which realises it, and that any fixation probability with no state having trivial fixation can be realised by at least some WF process. These results are extended to a subclass of processes that are suitable for describing time-inhomogeneous dynamics. We also discuss the traditional identification of frequency dependent fitnesses and pay-offs, extensively used in evolutionary game theory, the role of weak selection when the population is finite, and the relations between jumps in evolutionary processes and frequency dependent fitnesses

On the Classification of Finite Semigroups and RA-loop with the Hyperbolic Property

We classify the finite semigroups S, for which all the Z-orders O of the rational Q-algebra QS, is such that the unit group U(O) is hyperbolic. We also classify the RA-loops L, for which the unit loop U(ZL) does not contain any free abelian subgroup of rank two.Comment: 5 pages; this is part of the second and third chapters of the second author's PhD. Thesis. The last theorem was revised, as well as added some definitions and reference

Free Groups in Quaternion Algebras

In \cite{jpsf} we constructed pairs of units $u,v$ in $\Z$-orders of a quaternion algebra over \Q (\sqrt{-d}), $d \equiv 7 \pmod 8$ positive and square free, such that $$ is free for some $n\in \mathbb{N}$. Here we extend this result to any imaginary quadratic extension of $\ \mathbb{Q}$, thus including matrix algebras. More precisely, we show that $$ is a free group for all $n\geq 1$ and $d>2$ and for $d=2$ and all $n\geq 2$. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also established and used to prove that two certain units generate a free semigroup but that, in this case, the Ping-Pong Lemma can not be applied to show that the group they generate is free.Comment: 10 pages, article presented in conferences: Algebra School, Brasilia-Brazil, Brasilia National University (july-2010); Summer 2009 Meeting of CMS in Groups and Hopf Algebras section, St. Jonh's-Canada, Memorial University of Newfoundland (June-2009); Groups, Rings and Group Rings, Ubatuba-Brazil (july-2008

Hyperbolicity of Orders of Quaternions Algebras and Group Rings

For a given divison algebra of the quaternions we construct two types of units: Pell units and Gauss units. If K is a rational quadratic extension and G is a finite group, we classify R and G, s.t., the unit group U(RG) of augmentation one is hyperbolic. In particular we give necessary and sufficient conditions when G is the quaternion group of order 8. In this case, the hyperbolic boundary is isomorphic to the 2-dimensional sphere. As a result we reach a class of groups of one end, that are not virtually free.Comment: This report is part of the first chapter of the second author's PhD. Thesis. It was included a result in main theorem and a sketch of its proof before it, as well as two new reference

Alternative algebras with the hyperbolic property

We investigate the structure of an alternative finite dimensional \Q-algebra $\mathfrak{A}$ subject to the condition that for a $\Z$-order $\Gamma \subset \mathfrak{A}$, and thus for every $\Z$-order of $\mathfrak{A}$, the loop of units of \U (\Gamma) does not contain a free abelian subgroup of rank two. In particular, we prove that the radical of such an algebra associates with the whole algebra. We also classify $RA$-loops $L$ for which $\mathbb{Z}L$ has this property. The classification for group rings is still an open problem.Comment: Third author's Ph.D. (parcial); generalization of those results for the non-associative case, 9 pps. Conf.: Geometry, Topology, Algebra and Number Theory, Steklov Math. Inst., Moscow-Russia (Aug, 2010); XVIII Latin American Colloquium of Algebra, S\~ao Paulo-Brazil (July, 2009); Algebras, Representation and Applications, Ubatuba-Brazil (aug., 2007); Group and Group Rings, Bedlewo-Poland (2005

Hyperbolicity of Semigroup Algebras II

In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra %over a field of characteristic zero, contains a ${\mathbb{Z}}$-order with hyperbolic unit group. In this paper we complete this classification by handling the case in which the semigroup is semi-simple.Comment: In doi:10.1016/j.jalgebra.2008.03.015 it was, partially, classified the finite semigroups S which the semigroup algebra QS has the hyperbolic property. In this article we complete this classificatio