Mathematical Genetics: A Hybrid Seed for Educators to Sow

A principal area of applications of mathematical ideas is the study of population genetic systems. Such investigations have inspired the development of mathematical statistics, insights for certain classes of stochastic processes of diffusion type and intriguing algebraic structures. In this paper a series of simple mathematical models for evolutionary theory and inbreeding systems are set forth. Included are the Hardy‐Weinberg law and models of selection balance. We also treat several special inbreeding processes for selfing (self fertilization), sib mating, incompatibility systems and imprinting. Calculations of inbreeding coefficients using the notion of ‘identity by descent’ are made. Some classical simple stochastic models exposing the concept of genetic drift are highlighted.     

Polymorphic evolution sequence and evolutionary branching

We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence (TSS), is called polymorphic evolution sequence (PES). Then we introduce a scaling of the size of mutations and we study the PES in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. This phenomenon corresponds to the situation where the population, initially essentially single modal, is driven by the selective forces to divide into two separate subpopulations. To this end we finely analyze the asymptotic behavior of three-dimensional competitive Lotka?CVolterra systems.     

Analysis of the reproduction operator in an artificial bacterial foraging system

In his seminal paper published in 2002, Passino pointed out how individual and groups of bacteria forage for nutrients and how to model it as a distributed optimization process, which he called the Bacterial Foraging Optimization Algorithm (BFOA). One of the major driving forces of BFOA is the reproduction phenomenon of virtual bacteria each of which models a trial solution of the optimization problem. During reproduction, the least healthier bacteria (with a lower accumulated value of the objective function in one chemotactic lifetime) die and the other healthier bacteria each split into two, which then starts exploring the search place from the same location. This keeps the population size constant in BFOA. The phenomenon has a direct analogy with the selection mechanism of classical evolutionary algorithms. In this letter we provide a simple mathematical analysis of the effect of reproduction on bacterial dynamics. Our analysis reveals that the reproduction event contributes to the quick convergence of the bacterial population near optima.     

Mutation rates and equilibrium selection under stochastic evolutionary dynamics

We examine an evolutionary model in which the mutation rate varies with the strategy. Bergin and Lipman (Econometrica 64:943–956, 1996) show that equilibrium selection using stochastic evolutionary processes depends on the specification of mutation rates. We offer a characterization of how mutation rates determine the selection of Nash equilibria in 2 × 2 symmetric coordination games for single and double limits of the small mutation rate and the large population size. We prove that the restrictions on mutation rates which ensure that the risk-dominated equilibrium is selected are the same for both orders of limits.     

Multifactorial inheritance with selection

We present an alternative model for multifactorial inheritance. By changing the way the malformation (and selection) is determined from the genetic information, we arrive at a model that can be properly handled in the mathematical sense. This includes the proof of population convergence and computation of conditional malformation probabilities in a closed form. We also present a comparison to similar models and results of fitting our model to Hungarian data.     

Random collision model for random genetic drift and stochastic difference equation

Summary At first we introduce a simple stochastic difference equation, to simulate random sampling drift in population genetics, which is naturally obtained from a random collision model. Next, we introduce a random collision model to simulate overdominance model in population genetics. We assume in a time interval °t, a random collision of four particles, which represents overdominant selection, takes place at a certain probability, where a particle corresponds to a gene. We assume that mutation takes place by some rate and assume that every new mutation is different from extant alleles. We estimate mean heterozygosity by our simulation method and compare it with the result obtained by using a stochastic difference equation for overdominance model. The Institute of Statistical Mathematics     

广义抽象进化算法的性质分析

There has been a growing interest in mathematical models to character the evolutionary algorithms. The best-known one of such models is the axiomatic model called the abstract evolutionary algorithm (AEA), which unifies most of the currently known evolutionary algorithms and describes the evolution as an abstract stochastic process composed of two fundamental abstract operators: abstract selection and evolution operators. In this paper, we first introduce the definitions of the generalized abstract selection and evolution operators. Then we discuss the characterization of some parameters related to generalized abstract selection and evolution operators. Based on these operators, we finally give the strong convergence of the generalized abstract evolutionary algorithm. The present work provides a big step toward the establishment of a unified theory of evolutionary computation.     

Convergence of analytic gradient-type systems with periodicity and its applications in Kuramoto models

We consider the convergence of gradient-type systems with periodic and analytic potentials. The main tool is the celebrated Łojasiewicz inequality which is valid for any analytic function. Our results show that the convergence of such systems with periodic and analytic potentials is unconditional to the initial data; in other words, any trajectory converges to some equilibrium. As direct applications, we can show that any trajectory converges to phase-locked state for the first- and second-order Kuramoto models on a symmetric network with attractive–repulsive forces and identical natural frequencies. In particular, the inertial Kuramoto model with identical oscillators converges to phase-locked state for any initial configuration.     

Evolutionary games and matching rules

This study considers evolutionary games with non-uniformly random matching when interaction occurs in groups of \(n\ge 2\) individuals using pure strategies from a finite strategy set. In such models, groups with different compositions of individuals generally co-exist and the reproductive success (fitness) of a specific strategy varies with the frequencies of different group types. These frequencies crucially depend on the matching process. For arbitrary matching processes (called matching rules), we study Nash equilibrium and ESS in the associated population game and show that several results that are known to hold for population games under uniform random matching carry through to our setting. In our most novel contribution, we derive results on the efficiency of the Nash equilibria of population games and show that for any (fixed) payoff structure, there always exists some matching rule leading to average fitness maximization. Finally, we provide a series of applications to commonly studied normal-form games.     

A best-response approach for equilibrium selection in two-player generalized Nash equilibrium problems

ABSTRACT

In this paper, we propose a best-response approach to select an equilibrium in a two-player generalized Nash equilibrium problem. In our model we solve, at each of a finite number of time steps, two independent optimization problems. We prove that convergence of our Jacobi-type method, for the number of time steps going to infinity, implies the selection of the same equilibrium as in a recently introduced continuous equilibrium selection theory. Thus the presented approach is a different motivation for the existing equilibrium selection theory, and it can also be seen as a numerical method. We show convergence of our numerical scheme for some special cases of generalized Nash equilibrium problems with linear constraints and linear or quadratic cost functions.     

SASEGASA: A New Generic Parallel Evolutionary Algorithm for Achieving Highest Quality Results

This paper presents a new generic Evolutionary Algorithm (EA) for retarding the unwanted effects of premature convergence. This is accomplished by a combination of interacting generic methods. These generalizations of a Genetic Algorithm (GA) are inspired by population genetics and take advantage of the interactions between genetic drift and migration. In this regard a new selection scheme is introduced, which is designed to directedly control genetic drift within the population by advantageous self-adaptive selection pressure steering. Additionally this new selection model enables a quite intuitive heuristics to detect premature convergence. Based upon this newly postulated basic principle the new selection mechanism is combined with the already proposed Segregative Genetic Algorithm (SEGA), an advanced Genetic Algorithm (GA) that introduces parallelism mainly to improve global solution quality. As a whole, a new generic evolutionary algorithm (SASEGASA) is introduced. The performance of the algorithm is evaluated on a set of characteristic benchmark problems. Computational results show that the new method is capable of producing highest quality solutions without any problem-specific additions.     

A mutation operator based on a Pareto ranking for multi-objective evolutionary algorithms

Evolutionary Algorithms, EA’s, try to imitate, in some way, the principles of natural evolution and genetics. They evolve a population of potential solutions to the problem using operators such as mutation, crossover and selection. In general, the mutation operator is responsible for the diversity of the population and helps to avoid the problem of premature convergence to local optima (a premature stagnation of the search caused by the lack of population diversity).     

Genetic learning in strategic form games

We analyze the learning behavior of a Simple Genetic Algorithm in symmetric 3 × 3 Strategic-Form-Games. In cases of contests within one population and also between two populations the behavior of the SGA is compared with the behavior of the replicator dynamics and is analyzed with respect to equilibrium concepts in evolutionary game theory. Furthermore conservative non-adaptive strings are added to the population which lead to convergence to an equilibrium even in “GA-deceptive” games where the equilibrium can not be reached by GAs using only selection and crossover.     

Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences

Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logical testing by proof-of-concept mathematical models. Here, we present a stochastic agent-based model and a related deterministic integrodifferential equation model for the evolution of a phenotype-structured population composed of asexually-reproducing and competing organisms which are exposed to novel environmental conditions. This setting has relevance to the study of biological systems where colonising asexual populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene expression can change the reproductive capacity of individuals within the population in the new environment. Simulations and analyses of our models clarify the conditions under which certain evolutionary paths are possible and illustrate that while epigenetic mechanisms may facilitate adaptation in asexual species faced with environmental change, they can also lead to a type of “epigenetic load” and contribute to extinction. Moreover, our results offer a formal basis for the claim that constant environments favour individuals with low rates of stochastic phenotypic variation. Finally, our model provides a “proof of concept” of the verbal hypothesis that phenotypic stability is a key driver in rescuing the adaptive potential of an asexual lineage and supports the notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic instability and biased epimutations, and steer an asexual population back from the brink of an evolutionary dead end.     

基于特殊选择的抽象进化算法的收敛性

There has been a growing interest in mathematical models to character the evolutionary algorithms. The best-known one of such models is the axiomatic model colled the abstract evolutionary algorithm. In this paper, we first introduce the definitions of the abhstract selection and evolution operators, and that of the abstract evolutionary algorithm, which describes the evolution as an abstract stochastic process composed of these two fundamental abstract operators. In particular, a kind of abstract evolutionary algorithms based on a special selection mechansim is discussed. According to the sorting for the state space, the properties of the single step transition matrix for the algorithm are anaylzed. In the end, we prove that the limit probability distribution of the Markov chains exists. The present work provides a big step toward the establishment of a unified theory of evolutionary computation.     

Adaptation in a stochastic multi-resources chemostat model

We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. Adaptive dynamics so far has been put on a rigorous footing only for direct competition models (Lotka–Volterra models) involving a competition kernel which describes the competition pressure from one individual to another one. We extend this to a multi-resources chemostat model, where the competition between individuals results from the sharing of several resources which have their own dynamics. Starting from a stochastic birth and death process model, we prove that, when advantageous mutations are rare, the population behaves on the mutational time scale as a jump process moving between equilibrium states (the polymorphic evolution sequence of the adaptive dynamics literature). An essential technical ingredient is the study of the long time behavior of a chemostat multi-resources dynamical system. In the small mutational steps limit this process in turn gives rise to a differential equation in phenotype space called canonical equation of adaptive dynamics. From this canonical equation and still assuming small mutation steps, we prove a rigorous characterization of the evolutionary branching points.     

A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations

In this note, we characterize the solution to a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection, and migration. Generalizing an approach based on the Hamilton–Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero). This method was initially used, for different problems arisen from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. A key point is a uniqueness property related to the weak KAM theory. This method allows us to go further than the Gaussian approximation commonly used by biologists, and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics.     

Asymptotic Dynamics in Populations Structured by Sensitivity to Global Warming and Habitat Shrinking

How to recast effects of habitat shrinking and global warming on evolutionary dynamics into continuous mutation/selection models? Bearing this question in mind, we consider differential equations for structured populations, which include mutations, proliferation and competition for resources. Since mutations are assumed to be small, a parameter ε is introduced to model the average size of phenotypic changes. A well-posedness result is proposed and the asymptotic behavior of the density of individuals is studied in the limit ε→0. In particular, we prove the weak convergence of the density to a sum of Dirac masses and characterize the related concentration points. Moreover, we provide numerical simulations illustrating the theorems and showing an interesting sample of solutions depending on parameters and initial data.     

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both mathematical finance and bio-mathematics. More precisely, we show that under a dissipative condition on the drift coefficient and super-linear growth condition on the diffusion coefficient the BEM scheme converges with strong order of a half. This type of convergence gives theoretical foundations for efficient variance reduction techniques for Monte Carlo simulations. We support our theoretical results with relevant examples, such as stochastic population models and stochastic volatility models.     

On the effect of selection in genetic algorithms

To study the effect of selection with respect to mutation and mating in genetic algorithms, we consider two simplified examples in the infinite population limit. Both algorithms are modeled as measure valued dynamical systems and are designed to maximize a linear fitness on the half line. Thus, they both trivially converge to infinity. We compute the rate of their growth and we show that, in both cases, selection is able to overcome a tendency to converge to zero. The first model is a mutation‐selection algorithm on the integer half line, which generates mutations along a simple random walk. We prove that the system goes to infinity at a positive speed, even in cases where the random walk itself is ergodic. This holds in several strong senses, since we show a.s. convergence, L^p convergence, convergence in distribution, and a large deviations principle for the sequence of measures. For the second model, we introduce a new class of matings, based upon Mandelbrot martingales. The mean fitness of the associated mating‐selection algorithms on the real half line grows exponentially fast, even in cases where the Mandelbrot martingale itself converges to zero. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 185–200, 2001