scholarly journals Fixation Probabilities of Evolutionary Graphs Based on the Positions of New Appearing Mutants

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Pei-ai Zhang
Keyword(s):  
Markov Chain ◽  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Chain Process ◽  
Fixation Probability ◽  
Spatial Structures ◽  
Single Level ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory

Evolutionary graph theory is a nice measure to implement evolutionary dynamics on spatial structures of populations. To calculate the fixation probability is usually regarded as a Markov chain process, which is affected by the number of the individuals, the fitness of the mutant, the game strategy, and the structure of the population. However the position of the new mutant is important to its fixation probability. Here the position of the new mutant is laid emphasis on. The method is put forward to calculate the fixation probability of an evolutionary graph (EG) of single level. Then for a class of bilevel EGs, their fixation probabilities are calculated and some propositions are discussed. The conclusion is obtained showing that the bilevel EG is more stable than the corresponding one-rooted EG.

scholarly journals Fixation probabilities in network structured meta-populations

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Sedigheh Yagoobi ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Evolutionary Ecology ◽  
Research Topic ◽  
Fixation Probability ◽  
Popular Approach ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory ◽  
Selection For ◽  
Carry Over ◽  
Regular Networks

AbstractThe effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two patch structure with unequal population size suppresses selection for low migration probabilities.

scholarly journals Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150334 ◽  
Author(s):  
Karan Pattni ◽  
Mark Broom ◽  
Jan Rychtář ◽  
Lara J. Silvers
Keyword(s):  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Process ◽  
Structured Populations ◽  
Fixation Probability ◽  
Finite Populations ◽  
Moran Model ◽  
Moran Process ◽  
Evolutionary Graph Theory ◽  
Associated Matrix

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.

scholarly journals Fixation probabilities in network structured meta-populations

2021 ◽  
Author(s):  
Sedigheh Yagoobi ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Evolutionary Ecology ◽  
Research Topic ◽  
Fixation Probability ◽  
Popular Approach ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory ◽  
Selection For ◽  
Carry Over ◽  
Regular Networks

The effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two-patch structure with unequal population size suppresses selection for low migration probabilities.

scholarly journals Martingales and fixation probabilities of evolutionary graphs

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130730 ◽  
Author(s):  
T. Monk ◽  
P. Green ◽  
M. Paulin
Keyword(s):  
Graph Theory ◽  
Random Walk ◽  
Large Population ◽  
Initial Population ◽  
Weak Selection ◽  
Simplifying Assumptions ◽  
Population Sizes ◽  
Fixation Probabilities ◽  
Evolutionary Graph Theory ◽  
Complete Bipartite Graphs

Evolutionary graph theory is the study of birth–death processes that are constrained by population structure. A principal problem in evolutionary graph theory is to obtain the probability that some initial population of mutants will fixate on a graph, and to determine how that fixation probability depends on the structure of that graph. A fluctuating mutant population on a graph can be considered as a random walk. Martingales exploit symmetry in the steps of a random walk to yield exact analytical expressions for fixation probabilities. They do not require simplifying assumptions such as large population sizes or weak selection. In this paper, we show how martingales can be used to obtain fixation probabilities for symmetric evolutionary graphs. We obtain simpler expressions for the fixation probabilities of star graphs and complete bipartite graphs than have been previously reported and show that these graphs do not amplify selection for advantageous mutations under all conditions.

scholarly journals Counterintuitive properties of the fixation time in network-structured populations

2014 ◽  
Vol 11 (99) ◽  
pp. 20140606 ◽  
Author(s):  
Laura Hindersin ◽  
Arne Traulsen
Keyword(s):  
Evolutionary Dynamics ◽  
Transition Probabilities ◽  
Structured Populations ◽  
Death Process ◽  
Fixation Time ◽  
Fixation Probability ◽  
Mutant Type ◽  
Starting Conditions ◽  
Fixation Probabilities ◽  
Regular Networks

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth–death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations.

EVOLUTIONARY GRAPHS WITH FREQUENCY DEPENDENT FITNESS

2009 ◽  
Vol 23 (04) ◽  
pp. 537-543 ◽  
Author(s):  
PU-YAN NIE ◽  
PEI-AI ZHANG
Keyword(s):  
Graph Theory ◽  
Large Population ◽  
Fixation Probability ◽  
Frequency Dependent ◽  
Evolutionary Graph Theory

Evolutionary graph theory was recently proposed by Lieberman et al. in 2005. In the previous papers about evolutionary graphs (EGs), the fitness of the residents in the EGs is in general assumed to be unity, and the fitness of a mutant is assumed to be a constant r. We aim to extend EG to general cases in this paper, namely, the fitness of a mutant is heavily dependent upon frequency. The corresponding properties for these new EGs are analyzed, and the fixation probability is obtained for large population.

scholarly journals Evolutionary graph theory derived from eco-evolutionary dynamics

2021 ◽  
pp. 110648
Author(s):  
Karan Pattni ◽  
Christopher E. Overton ◽  
Kieran J. Sharkey
Keyword(s):  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Graph Theory

scholarly journals An analysis of the fixation probability of a mutant on special classes of non-directed graphs

2008 ◽  
Vol 464 (2098) ◽  
pp. 2609-2627 ◽  
Author(s):  
M Broom ◽  
J Rychtář
Keyword(s):  
Evolutionary Dynamics ◽  
Directed Graphs ◽  
Homogeneous Structure ◽  
Fixation Probability ◽  
Weighted Graphs ◽  
Simple Graphs ◽  
Single Mutant ◽  
Fixation Probabilities ◽  
Similar Solutions ◽  
Initial Starting

There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al . (Lieberman et al . 2005 Nature 433 , 312–316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations.

scholarly journals Calibration of Transition Intensities for a Multistate Model: Application to Long-Term Care

Risks ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 37
Author(s):  
Manuel L. Esquível ◽  
Gracinda R. Guerreiro ◽  
Matilde C. Oliveira ◽  
Pedro Corte Real
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Long Term Care ◽  
Transition Probabilities ◽  
Markov Chain Model ◽  
Chain Model ◽  
Chain Process ◽  
Term Care ◽  
National Network

We consider a non-homogeneous continuous time Markov chain model for Long-Term Care with five states: the autonomous state, three dependent states of light, moderate and severe dependence levels and the death state. For a general approach, we allow for non null intensities for all the returns from higher dependence levels to all lesser dependencies in the multi-state model. Using data from the 2015 Portuguese National Network of Continuous Care database, as the main research contribution of this paper, we propose a method to calibrate transition intensities with the one step transition probabilities estimated from data. This allows us to use non-homogeneous continuous time Markov chains for modeling Long-Term Care. We solve numerically the Kolmogorov forward differential equations in order to obtain continuous time transition probabilities. We assess the quality of the calibration using the Portuguese life expectancies. Based on reasonable monthly costs for each dependence state we compute, by Monte Carlo simulation, trajectories of the Markov chain process and derive relevant information for model validation and premium calculation.

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