Fisher Diffusion Research Articles

Duality Between the Two-Locus Wright–Fisher Diffusion Model and the Ancestral Process with Recombination

Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Duality Between the Two-Locus Wright–Fisher Diffusion Model and the Ancestral Process with Recombination

Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Wright–Fisher diffusion with negative mutation rates

We study a family of n-dimensional diffusions, taking values in the unit simplex of vectors with nonnegative coordinates that add up to one. These processes satisfy stochastic differential equations which are similar to the ones for the classical Wright-Fisher diffusions, except that the "mutation rates" are now nonpositive. This model, suggested by Aldous, appears in the study of a conjectured diffusion limit for a Markov chain on Cladograms. The striking feature of these models is that the boundary is not reflecting, and we kill the process once it hits the boundary. We derive the explicit exit distribution from the simplex and probabilistic bounds on the exit time. We also prove that these processes can be viewed as a "stochastic time-reversal" of a Wright-Fisher process of increasing dimensions and conditioned at a random time. A key idea in our proofs is a skew-product construction using certain one-dimensional diffusions called Bessel-square processes of negative dimensions, which have been recently introduced by Going-Jaeschke and Yor.

Change of measure in the lookdown particle system

We perform various changes of measure in the lookdown particle system of Donnelly and Kurtz. The first example is a product type h-transform related to conditioning a Generalized Fleming–Viot process without mutation on coexistence of some genetic types in remote time. We give a pathwise construction of this h-transform by just “forgetting” some reproduction events in the lookdown particle system. We also provide an intertwining relationship for the Wright–Fisher diffusion and explicit the associated pathwise decomposition. The second example, called the linear or additive h-transform, concerns a wider class of measure-valued processes with spatial motion. Applications include a simple description of the additiveh-transform of the Generalized Fleming–Viot process and an immortal particle representation for the additive h-transform of the Dawson–Watanabe process.

A mean-reverting SDE on correlation matrices

We introduce a mean-reverting SDE whose solution is naturally defined on the space of correlation matrices. This SDE can be seen as an extension of the well-known Wright–Fisher diffusion. We provide conditions that ensure weak and strong uniqueness of the SDE, and describe its ergodic limit. We also shed light on a useful connection with Wishart processes that makes understand how we get the full SDE. Then, we focus on the simulation of this diffusion and present discretization schemes that achieve a second-order weak convergence. Last, we give a possible application of these processes in finance and argue that they could easily replace and improve the standard assumption of a constant correlation.

An explicit transition density expansion for a multi-allelic Wright–Fisher diffusion with general diploid selection

Characterizing time-evolution of allele frequencies in a population is a fundamental problem in population genetics. In the Wright–Fisher diffusion, such dynamics is captured by the transition density function, which satisfies well-known partial differential equations. For a multi-allelic model with general diploid selection, various theoretical results exist on representations of the transition density, but finding an explicit formula has remained a difficult problem. In this paper, a technique recently developed for a diallelic model is extended to find an explicit transition density for an arbitrary number of alleles, under a general diploid selection model with recurrent parent-independent mutation. Specifically, the method finds the eigenvalues and eigenfunctions of the generator associated with the multi-allelic diffusion, thus yielding an accurate spectral representation of the transition density. Furthermore, this approach allows for efficient, accurate computation of various other quantities of interest, including the normalizing constant of the stationary distribution and the rate of convergence to this distribution.

Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory

The allele frequency spectrum has attracted considerable interest for the simultaneous inference of the demographic and adaptive history of populations. In a recent study, Evans et al. (2007) developed a forward diffusion equation describing the allele frequency spectrum, when the population is subject to size changes, selection and mutation. From the diffusion equation, the authors derived a system of ordinary differential equations (ODEs) for the moments in a Wright–Fisher diffusion with varying population size and constant selection. Here, we present an explicit solution for this system of ODEs with variable population size, but without selection, and apply this result to derive the expected spectrum of a sample for time-varying population size. We use this forward-in-time-solution of the allele frequency spectrum to obtain the backward-in-time-solution previously derived via coalescent theory by Griffiths and Tavaré (1998). Finally, we discuss the applicability of the theoretical results to the analysis of nucleotide polymorphism data.

On the Karlin–Kimura approaches to the Wright–Fisher diffusion with fluctuatingselection

The goal of this work is a comparative study of two Wright–Fisher-like diffusionprocesses on the interval, one due to Karlin and the other one due to Kimura. Eachmodel accounts for the evolution of one two-locus colony undergoing randommating, under the additional action of selection in a random environment. In otherwords, we study the effect of disorder on the usual Wright–Fisher model withfixed (nonrandom) selection. There is a drastic qualitative difference betweenthe two models and between the random and nonrandom selection hypotheses.We first present a series of elementary stochastic models and tools that are needed toconduct this study in the context of diffusion process theory, including Kolmogorovbackward and forward equations, scale and speed functions, classification of boundaries,and Doob transformation of sample paths using additive functionals. In this spirit, webriefly revisit the neutral Wright–Fisher diffusion and the Wright–Fisher diffusion withnonrandom selection.With these tools at hand, we first deal with the Karlin approach to the Wright–Fisherdiffusion model with randomized selection differentials. The specificity of thismodel is that in the large population case, the boundaries of the state space arenatural and hence inaccessible, and so quasi-absorbing only. We supply somelimiting properties pertaining to times of hitting of points close to the boundaries.Next, we study the Kimura approach to the Wright–Fisher model with randomizedselection, which may be viewed as a modification of the Karlin model, using an appropriateDoob transform which we describe. This model also has natural boundaries, but they turnout to be much more attracting and sticky than in Karlin’s version. This leads to a fasterapproach to the quasi-absorbing states, to a larger time needed to move from the vicinity ofone boundary to the other and to a local critical behavior of the branching diffusionobtained after the relevant Doob transformation.

A coalescent dual process in a Moran model with genic selection

A coalescent dual process for a multi-type Moran model with genic selection is derived using a generator approach. This leads to an expansion of the transition functions in the Moran model and the Wright–Fisher diffusion process limit in terms of the transition functions for the coalescent dual. A graphical representation of the Moran model (in the spirit of Harris) identifies the dual as a strong dual process following typed lines backwards in time. An application is made to the harmonic measure problem of finding the joint probability distribution of the time to the first loss of an allele from the population and the distribution of the surviving alleles at the time of loss. Our dual process mirrors the Ancestral Selection Graph of [Krone, S. M., Neuhauser, C., 1997. Ancestral processes with selection. Theoret. Popul. Biol. 51, 210–237; Neuhauser, C., Krone, S. M., 1997. The genealogy of samples in models with selection. Genetics 145, 519–534], which allows one to reconstruct the genealogy of a random sample from a population subject to genic selection. In our setting, we follow [Stephens, M., Donnelly, P., 2002. Ancestral inference in population genetics models with selection. Aust. N. Z. J. Stat. 45, 395–430] in assuming that the types of individuals in the sample are known. There are also close links to [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38–54]. However, our methods and applications are quite different. This work can also be thought of as extending a dual process construction in a Wright–Fisher diffusion in [Barbour, A.D., Ethier, S.N., Griffiths, R.C., 2000. A transition function expansion for a diffusion model with selection. Ann. Appl. Probab. 10, 123–162]. The application to the harmonic measure problem extends a construction provided in the setting of a neutral diffusion process model in [Ethier, S.N., Griffiths, R.C., 1991. Harmonic measure for random genetic drift. In: Pinsky, M.A. (Ed.), Diffusion Processes and Related Problems in Analysis, vol. 1. In: Progress in Probability Series, vol. 22, Birkhäuser, Boston, pp. 73–81].

On Pólya–Friedman random walks

The Pólya process is an urn scheme arising in the context of contagion spreading. It exhibits unstable persistence effects. The Friedman urn process is dual to the Pólya one with antipersistent stabilizing effects. It appears in a safety campaign problem. A Pólya–Friedman urn process is investigated with a tuning persistence parameter extrapolating the latter two extreme processes. The study includes the diffusion approximations of both the Pólya–Friedman proportion process and the population gap random walk. The structure of the former is a generalized Wright–Fisher diffusion appearing in population genetics. The correlation structure of the latter presents an anomalous character at a critical value of the persistence parameter.

On Wright–Fisher diffusion and its relatives

We present a series of elementary stochastic models arising from population genetics.Various mathematical aspects are investigated at the intuitive level. Special emphasis is puton the diffusion method. The focus is on the Wright–Fisher diffusion model and itsvariations, describing the forward evolution of one colony undergoing random sampling,possibly under the additional forces arising from mutation and selection with or withoutdominance. Some aspects of their dual coalescents obtained while running the diffusionprocess backward in time are also investigated.

Non-equilibrium theory of the allele frequency spectrum

A forward diffusion equation describing the evolution of the allele frequency spectrum is presented. The influx of mutations is accounted for by imposing a suitable boundary condition. For a Wright–Fisher diffusion with or without selection and varying population size, the boundary condition is lim x ↓ 0 xf ( x , t ) = θ ρ ( t ) , where f ( · , t ) is the frequency spectrum of derived alleles at independent loci at time t and ρ ( t ) is the relative population size at time t. When population size and selection intensity are independent of time, the forward equation is equivalent to the backwards diffusion usually used to derive the frequency spectrum, but this approach allows computation of the time dependence of the spectrum both before an equilibrium is attained and when population size and selection intensity vary with time. From the diffusion equation, a set of ordinary differential equations for the moments of f ( · , t ) is derived and the expected spectrum of a finite sample is expressed in terms of those moments. The use of the forward equation is illustrated by considering neutral and selected alleles in a highly simplified model of human history. For example, it is shown that approximately 30% of the expected total heterozygosity of neutral loci is attributable to mutations that arose since the onset of population growth in roughly the last 150 , 000 years.

Extinction versus exponential growth in a supercritical super-Wright–Fisher diffusion

We study mild solutions u to the semilinear Cauchy problem ∂ ∂t u t(x)= 1 2 x(1−x) ∂ 2 ∂x 2 u t(x)+γu t(x)(1−u t(x)) (t⩾0),u 0(x)=f(x) with x∈[0,1], f a nonnegative measurable function and γ a positive constant. Solutions to this equation are given by u t= U tf , where ( U t) t⩾0 is the log-Laplace semigroup of a supercritical superprocess taking values in the finite measures on [0,1], whose underlying motion is the Wright–Fisher diffusion. We establish a dichotomy in the long-time behavior of this superprocess. For γ⩽1, the mass in the interior (0,1) dies out after a finite random time, while for γ>1, the mass in (0,1) grows exponentially as time tends to infinity with positive probability. In the case of exponential growth, the mass in (0,1) grows exponentially with rate γ−1 and is approximately uniformly distributed over (0,1). We apply these results to show that ( U t) t⩾0 has precisely four fixed points when γ⩽1 and five fixed points when γ>1, and determine their domains of attraction.

Finite difference solutions of reaction diffusion equations with continuous time delays

This paper is an extension of the monotone iterative methods for finite difference equations with discrete time delays to a class of nonlinear finite difference system with continuous time delays. The system under consideration is a finite difference approximation of a class of reaction diffusion equations with continuous time delays in the nonlinear reaction under either Dirichlet or Neumann-Robin boundary conditions. Various monotone iterative schemes, which depend on the property of the nonlinear reaction mechanism, are developed for the computation of numerical solutions. It is shown by the method of upper and lower solutions that the two sequences obtained from each iterative scheme converge monotonically from above and below, respectively, to a unique solution of the finite difference system. Applications are given to two model problems known as the diffusive logistic equation and the Fisher's diffusion equation in population genetics.

Forward and backward diffusion approximations for haploid exchangeable population models

The class of haploid population models with non-overlapping generations and fixed population size N is considered such that the family sizes ν 1,…, ν N within a generation are exchangeable random variables. A criterion for weak convergence in the Skorohod sense is established for a properly time- and space-scaled process counting the number of descendants forward in time. The generator A of the limit process X is constructed using the joint moments of the offspring variables ν 1,…, ν N . In particular, the Wright–Fisher diffusion with generator Af(x)= 1 2 x(1−x)f″(x) appears in the limit as the population size N tends to infinity if and only if the condition lim N→∞ E((ν 1−1) 3)/(N Var(ν 1))=0 is satisfied. Using the concept of duality, these convergence results are compared with the limit theorems known for the coalescent processes with simultaneous and multiple collisions arising when the models are considered backward in time. In particular the Wright–Fisher diffusion appears forward in time if and only if the Kingman coalescent appears backward in time as N tends to infinity. A commutative diagram leads to a full understanding of the model considered forward and backward in time for finite population size and in the limit as N tends to infinity.