scholarly journals Expansion of error thresholds for the Moran model

2021 ◽  
Author(s):  
Maxime Berger
Keyword(s):  
Moran Model

A genealogical approach to variable-population-size models in population genetics

1986 ◽  
Vol 23 (02) ◽  
pp. 283-296 ◽  
Author(s):  
Peter Donnelly
Keyword(s):  
Population Genetics ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Process Models ◽  
Moran Model ◽  
Fisher Model ◽  
Variable Population ◽  
Exchangeable Model ◽  
Necessary And Sufficient ◽  
Variable Population Size

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.

scholarly journals Duality and Asymptotics for a Class of Nonneutral Discrete Moran Models

2009 ◽  
Vol 46 (3) ◽  
pp. 866-893 ◽  
Author(s):  
Thierry Huillet ◽  
Martin Möhle
Keyword(s):  
Transition Matrix ◽  
Random Number ◽  
Dual Process ◽  
Probabilistic Interpretation ◽  
Uhlenbeck Process ◽  
Moran Model ◽  
Completely Monotone ◽  
Finite State ◽  
Selection Mechanisms ◽  
Finite State Space

A Markov chain X with finite state space {0,…,N} and tridiagonal transition matrix is considered, where transitions from i to i-1 occur with probability (i/N)(1-p(i/N)) and transitions from i to i+1 occur with probability (1-i/N)p(i/N). Here p:[0,1]→[0,1] is a given function. It is shown that if p is continuous with p(x)≤p(1) for all x∈[0,1] then, for each N, a dual process Y to X (with respect to a specific duality function) exists if and only if 1-p is completely monotone with p(0)=0. A probabilistic interpretation of Y in terms of an ancestral process of a mixed multitype Moran model with a random number of types is presented. It is shown that, under weak conditions on p, the process Y, properly time and space scaled, converges to an Ornstein–Uhlenbeck process as N tends to ∞. The asymptotics of the stationary distribution of Y is studied as N tends to ∞. Examples are presented involving selection mechanisms. results.

Looking forwards and backwards in a bisexual moran model

1989 ◽  
Vol 26 (04) ◽  
pp. 880-885 ◽  
Author(s):  
K. Kämmerle
Keyword(s):  
Diffusion Approximation ◽  
Large Population ◽  
Extinction Probability ◽  
Single Pair ◽  
Approximation Methods ◽  
Moran Model ◽  
Population Sizes ◽  
Limit Result

In this paper a bisexual Moran model is introduced. The population consists of N pairs of individuals. At times t = 1, 2, ·· ·two individuals are born, who ‘choose their parents randomly' and independently of each other. Then one of the pairs is removed and replaced by the two individuals born at that instant. The extinction probability of the descendants of a single pair and the number of ancestors of a whole generation are studied. A limit result for large population sizes has been derived by diffusion approximation methods.

Boundary-Driven Emergent Spatiotemporal Order in Growing Microbial Colonies

2018 ◽  
Author(s):  
Bhargav R. Karamched ◽  
William Ott ◽  
Ilya Timofeyev ◽  
Razan N. Alnahhas ◽  
Matthew R. Bennett ◽  
...  
Keyword(s):  
Collective Behavior ◽  
Mean Field ◽  
Population Level ◽  
Field Approximation ◽  
Cell Interactions ◽  
Boundary Effects ◽  
Mean Field Approximation ◽  
Bacterial Populations ◽  
Moran Model ◽  
Cell Cell

We introduce a tractable stochastic spatial Moran model to explain experimentally-observed patterns of rod-shaped bacteria growing in rectangular microfluidic traps. Our model shows that spatial patterns can arise as a result of a tug-of-war between boundary effects and modulations of growth rate due to cell-cell interactions. Cells align parallel to the long side of the trap when boundary effects dominate. However, when the magnitude of cell-cell interactions exceeds a critical value, cells align orthogonally to the trap’s long side. Our model is analytically tractable, and completely solvable under a mean-field approximation. This allows us to elucidate the mechanisms that govern the formation of population-level patterns. The model can be easily extended to examine various types of interactions that can shape the collective behavior in bacterial populations.

scholarly journals Recurrence Equations for the Probability Distribution of Sample Configurations in Exact Population Genetics Models

2010 ◽  
Vol 47 (03) ◽  
pp. 732-751 ◽  
Author(s):  
Sabin Lessard
Keyword(s):  
Mutation Rate ◽  
Overlapping Generations ◽  
Finite Population ◽  
The Other ◽  
Moran Model ◽  
Recurrence Equations ◽  
Sampling Formula ◽  
Fisher Model ◽  
Variation Distance ◽  
Exchangeable Model

Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.

scholarly journals Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population Model

2010 ◽  
Vol 47 (03) ◽  
pp. 713-731 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Markov Chain ◽  
Population Size ◽  
Side Effect ◽  
Population Model ◽  
Inversion Formula ◽  
Transition Matrix ◽  
Duality Relation ◽  
Moran Model ◽  
Fisher Model ◽  
Fixed Population

We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chain X is studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix of X and an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.

Fixation in bisexual models with variable population sizes

1997 ◽  
Vol 34 (02) ◽  
pp. 436-448 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Main Part ◽  
Sufficient Conditions ◽  
Moran Model ◽  
Fisher Model ◽  
Population Sizes ◽  
Variable Population ◽  
Allelic Type ◽  
Ultimate Homozygosity ◽  
Necessary And Sufficient ◽  
Forward Process

A general exchangeable bisexual model with variable population sizes is introduced. First the forward process, i.e. the number of certain descending pairs, is studied. For the bisexual Wright-Fisher model fixation of the descendants occurs, i.e. their proportion tends to 0 or 1 almost surely. The main part of this article deals with necessary and sufficient conditions for ultimate homozygosity, i.e. the proportion of an arbitrarily chosen allelic type tends to 0 or 1 almost surely. The results are applied to a bisexual Wright-Fisher model and to a bisexual Moran model.

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