scholarly journals A MARKOV PROCESS OF GENE FREQUENCY CHANGE IN A GEOGRAPHICALLY STRUCTURED POPULATION

Genetics ◽  
1974 ◽  
Vol 76 (2) ◽  
pp. 367-377
Author(s):  
Takeo Maruyama
Keyword(s):  
Genetic Variation ◽  
Markov Process ◽  
Diffusion Process ◽  
Gene Frequency ◽  
Transition Probabilities ◽  
Large Population ◽  
Sample Path ◽  
Time Parameter ◽  
Frequency Change ◽  
Structured Population

ABSTRACT A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value.

scholarly journals CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

2008 ◽  
Vol 51 (2) ◽  
pp. 337-362 ◽  
Author(s):  
Torben Fattler ◽  
Martin Grothaus
Keyword(s):  
Boundary Condition ◽  
Markov Process ◽  
Diffusion Process ◽  
Lebesgue Measure ◽  
Dirichlet Form ◽  
Measure Zero ◽  
Particle Systems ◽  
Lebesgue Measure Zero ◽  
Compact Set ◽  
Singular Interactions

AbstractWe give a Dirichlet form approach for the construction and analysis of elliptic diffusions in $\bar{\varOmega}\subset\mathbb{R}^n$ with reflecting boundary condition. The problem is formulated in an $L^2$-setting with respect to a reference measure $\mu$ on $\bar{\varOmega}$ having an integrable, $\mathrm{d} x$-almost everywhere (a.e.) positive density $\varrho$ with respect to the Lebesgue measure. The symmetric Dirichlet forms $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ we consider are the closure of the symmetric bilinear forms\begin{gather*} \mathcal{E}^{\varrho,a}(f,g)=\sum_{i,j=1}^n\int_{\varOmega}\partial_ifa_{ij} \partial_jg\,\mathrm{d}\mu,\quad f,g\in\mathcal{D}, \\ \mathcal{D}=\{f\in C(\bar{\varOmega})\mid f\in W^{1,1}_{\mathrm{loc}}(\varOmega),\ \mathcal{E}^{\varrho,a}(f,f)\lt\infty\}, \end{gather*}in $L^2(\bar{\varOmega},\mu)$, where $a$ is a symmetric, elliptic, $n\times n$-matrix-valued measurable function on $\bar{\varOmega}$. Assuming that $\varOmega$ is an open, relatively compact set with boundary $\partial\varOmega$ of Lebesgue measure zero and that $\varrho$ satisfies the Hamza condition, we can show that $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ is a local, quasi-regular Dirichlet form. Hence, it has an associated self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and diffusion process $\bm{M}^{\varrho,a}$ (i.e. an associated strong Markov process with continuous sample paths). Furthermore, since $1\in D(\mathcal{E}^{\varrho,a})$ (due to the Neumann boundary condition) and $\mathcal{E}^{\varrho,a}(1,1)=0$, we obtain a conservative process $\bm{M}^{\varrho,a}$ (i.e. $\bm{M}^{\varrho,a}$ has infinite lifetime). Additionally, assuming that $\sqrt{\varrho}\in W^{1,2}(\varOmega)\cap C(\bar{\varOmega})$ or that $\varrho$ is bounded, $\varOmega$ is convex and $\{\varrho=0\}$ has codimension at least 2, we can show that the set $\{\varrho=0\}$ has $\mathcal{E}^{\varrho,a}$-capacity zero. Therefore, in this case we can even construct an associated conservative diffusion process in $\{\varrho>0\}$. This is essential for our application to continuous $N$-particle systems with singular interactions. Note that for the construction of the self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and the Markov process $\bm{M}^{\varrho,a}$ we do not need to assume any differentiability condition on $\varrho$ and $a$. We obtain the following explicit representation of the generator for $\sqrt{\varrho}\in W^{1,2}(\varOmega)$ and $a\in W^{1,\infty}(\varOmega)$:$$ L^{\varrho,a}=\sum_{i,j=1}^n\partial_i(a_{ij}\partial_j)+\partial_i(\log\varrho)a_{ij}\partial_j. $$Note that the drift term can be singular, because we allow $\varrho$ to be zero on a set of Lebesgue measure zero. Our assumptions in this paper even allow a drift that is not integrable with respect to the Lebesgue measure.

scholarly journals GENETIC VARIATION IN A HETEROGENEOUS ENVIRONMENT. I. TEMPORAL HETEROGENEITY AND THE ABSOLUTE DOMINANCE MODEL

Genetics ◽  
1974 ◽  
Vol 78 (2) ◽  
pp. 757-770
Author(s):  
Philip W Hedrick
Keyword(s):  
Genetic Variation ◽  
Temporal Variation ◽  
Gene Frequency ◽  
Environmental Heterogeneity ◽  
Finite Population ◽  
Environmental Variation ◽  
Dominance Model ◽  
Infinite Population ◽  
The Absolute ◽  
Absolute Dominance

ABSTRACT The conditions for a stable polymorphism and the equilibrium gene frequency in an infinite population are compared when there is spatial or temporal environmental heterogeneity for the absolute dominance model. For temporal variation the conditions for stability are more restrictive and the equilibrium gene frequency is often at a low gene frequency. In a finite population, temporal environmental heterogeneity for the absolute dominance model was found to be quite ineffective in maintaining genetic variation and is often less effective than no selection at all. For comparison, the maximum maintenance for temporal variation is related to the overdominant model. In general, cyclic environmental variation was found to be more effective at maintaining genetic variation than where the environment varies stochastically. The importance of temporal environmental variation and the maintenance of genetic variation is discussed.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (03) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn (w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn , which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN –1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ 2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

Advertising Frequency Decisions in a Discrete Markov Process under a Budget Constraint

1998 ◽  
Vol 35 (3) ◽  
pp. 399-406 ◽  
Author(s):  
Bart J. Bronnenberg
Keyword(s):  
Markov Process ◽  
Transition Probabilities ◽  
Optimal Level ◽  
Long Term Effects ◽  
Carryover Effects ◽  
Advertising Budget ◽  
Managerial Implications ◽  
Functional Forms ◽  
Short And Long Term

The author studies the optimality of advertising pulsing under the assumption that demand follows a discrete and interpretable Markov process and that the advertising budget is constrained. The author develops two main results. First, when pulsing is optimal, the prevalence of advertising effects on switching or repurchasing affects the length of the pulse (shorter versus longer, respectively), as well as the optimal level of advertising. Second, the author identifies the functional forms of the short- and long-term effects of advertising in the discrete Markov process and shows that pulsing can be optimal if the transition probabilities are concave in advertising. As an alternative to a pure Markov carryover (if any), the author considers that carryover effects of advertising also might be caused by accumulation of memory for the advertisement. Although general results are difficult to obtain, the author analyzes one case of the compound dynamics of the Markov process and memory effects for advertising with results similar to the pure Markov process. Similarities and differences with continuous-time models, as well as managerial implications, are discussed.

On Sieved Orthogonal Polynomials II: Random Walk Polynomials

1986 ◽  
Vol 38 (2) ◽  
pp. 397-415 ◽  
Author(s):  
Jairo Charris ◽  
Mourad E. H. Ismail
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Orthogonal Polynomials ◽  
Transition Probabilities ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Markov Process ◽  
Image Position

A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities(1.1)satisfy(1.2)as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by

Limit theorems for processes such as the Markov branching process

1975 ◽  
Vol 12 (02) ◽  
pp. 289-297
Author(s):  
Andrew D. Barbour
Keyword(s):  
Markov Process ◽  
Residence Time ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Continuous Mapping ◽  
Random Variables ◽  
Time Parameter ◽  
Image Position ◽  
Independent Identically Distributed

LetX(t) be a continuous time Markov process on the integers such that, ifσis a time at whichXmakes a jump,X(σ)– X(σ–) is distributed independently ofX(σ–), and has finite meanμand variance. Letq(j) denote the residence time parameter for the statej.Iftndenotes the time of thenth jump andXn≡X(tb), it is easy to deduce limit theorems forfrom those for sums of independent identically distributed random variables. In this paper, it is shown how, forμ> 0 and for suitableq(·), these theorems can be translated into limit theorems forX(t), by using the continuous mapping theorem.

GENETIC LOADS IN HETEROGENEOUS ENVIRONMENTS

Genetics ◽  
1975 ◽  
Vol 80 (3) ◽  
pp. 621-635
Author(s):  
Charles E Taylor
Keyword(s):  
Population Structure ◽  
Genetic Variation ◽  
Gene Frequency ◽  
Phase Plane ◽  
Genetic Load ◽  
Sufficient Conditions ◽  
Heterogeneous Environments ◽  
Niche Space ◽  
Necessary And Sufficient ◽  
Existence Of Equilibria

ABSTRACT A model of population structure in heterogeneous environments is described with attention focused on genetic variation at a single locus. The existence of equilibria at which there is no genetic load is examined.—The absolute fitness of any genotype is regarded as a function of location in the niche space and the population density at that location. It is assumed that each organism chooses to live in that habitat in which it is most fit ("optimal habitat selection").—Equilibria at which there is no segregational load ("loadless equilibria") may exist. Necessary and sufficient conditions for the existence of such equilibria are very weak. If there is a sufficient amount of dominance or area in which the alleles are selectively neutral, then there exist equilibria without segregational loads. In the N, p phase plane defined by population size, N, and gene frequency, p, these equilibria generally consist of a line segment which is parallel to the p axis. These equilibria are frequently stable.

General solidarity theorems for semi-Markov processes

1972 ◽  
Vol 9 (04) ◽  
pp. 789-802
Author(s):  
Choong K. Cheong ◽  
Jozef L. Teugels
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Transition Probabilities ◽  
Large Set ◽  
Semi Markov Process ◽  
Regularly Varying ◽  
Specific Pair ◽  
Semi Markov Processes

Let {Zt, t ≧ 0} be an irreducible regular semi-Markov process with transition probabilities Pij (t). Let f(t) be non-negative and non-decreasing to infinity, and let λ ≧ 0. This paper identifies a large set of functions f(t) with the solidarity property that convergence of the integral ≧ eλtf(t)Pij (t) dt for a specific pair of states i and j implies convergence of the integral for all pairs of states. Similar results are derived for the Markov renewal functions Mij (t). Among others it is shown that f(t) can be taken regularly varying.

Genetic variation in fall cold hardiness in coastal Douglas-fir in western Oregon and Washington

2006 ◽  
Vol 84 (7) ◽  
pp. 1110-1121 ◽  
Author(s):  
J. Bradley St. Clair
Keyword(s):  
Genetic Variation ◽  
Cold Hardiness ◽  
Environmental Gradients ◽  
Large Population ◽  
Common Garden ◽  
Cold Season ◽  
Douglas Fir ◽  
Bud Burst ◽  
Population Differences ◽  
Bud Set

Genetic variation in fall cold damage in coastal Douglas-fir ( Pseudotsuga menziesii (Mirb.) Franco var. menziesii ) was measured by exposing excised branches of seedlings from 666 source locations grown in a common garden to freezing temperatures in a programmable freezer. Considerable variation was found among populations in fall cold hardiness of stems, needles, and buds compared with bud burst, bud set, and biomass growth after 2 years. Variation in fall cold hardiness was strongly correlated (r = 0.67) with cold-season temperatures of the source environment. Large population differences corresponding with environmental gradients are evidence that natural selection has been important in determining genetic variation in fall cold hardiness, much more so than in traits of bud burst (a surrogate for spring cold hardiness), bud set, and growth. Seed movement guidelines and breeding zones may be more restrictive when considering genetic variation in fall cold hardiness compared with growth, phenology, or spring cold hardiness. A regional stratification system based on ecoregions with latitudinal and elevational divisions, and roughly corresponding with breeding zones used in Oregon and Washington, appeared to be adequate for minimizing population differences within regions for growth and phenology, but perhaps not fall cold hardiness. Although cold hardiness varied among populations, within-population and within-region variation is sufficiently large that responses to natural or artificial selection may be readily achieved.

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