Open Problems in the Spectral Analysis of Evolutionary Dynamics

Spectral analysis for a class of integral-difference operators: known facts, new results, and open problems

Discrete Dynamics in Nature and Society ◽

10.1155/s1026022604401022 ◽

2004 ◽

Vol 2004 (1) ◽

pp. 221-249 ◽

Cited By ~ 3

Author(s):

Yuri B. Melnikov

Keyword(s):

Spectral Analysis ◽

Present State ◽

Statistical Physics ◽

State Of The Art ◽

Difference Operators ◽

Open Problems ◽

Nonequilibrium Statistical Physics

We present state of the art, the new results, and discuss open problems in the field of spectral analysis for a class of integral-difference operators appearing in some nonequilibrium statistical physics models as collision operators. The author dedicates this work to the memory of Professor Ilya Prigogine, who initiated this activity in 1997 and whose interesting and most enlightening advices had gudided the author during all these years.

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Evolutionary dynamics of cancer: From epigenetic regulation to cell population dynamics—mathematical model framework, applications, and open problems

Science China Mathematics ◽

10.1007/s11425-019-1629-7 ◽

2020 ◽

Vol 63 (3) ◽

pp. 411-424

Author(s):

Jinzhi Lei

Keyword(s):

Mathematical Model ◽

Population Dynamics ◽

Cell Population ◽

Epigenetic Regulation ◽

Evolutionary Dynamics ◽

Model Framework ◽

Open Problems ◽

Cell Population Dynamics

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Spectral analysis of transient amplifiers for death–birth updating constructed from regular graphs

Journal of Mathematical Biology ◽

10.1007/s00285-021-01609-y ◽

2021 ◽

Vol 82 (7) ◽

Author(s):

Hendrik Richter

Keyword(s):

Spectral Analysis ◽

Structural Properties ◽

Evolutionary Dynamics ◽

Mixed Population ◽

Fixation Probability ◽

Regular Graphs ◽

Central Question ◽

Random Walks On Graphs ◽

Determining Factor ◽

Coalescence Times

AbstractA central question of evolutionary dynamics on graphs is whether or not a mutation introduced in a population of residents survives and eventually even spreads to the whole population, or becomes extinct. The outcome naturally depends on the fitness of the mutant and the rules by which mutants and residents may propagate on the network, but arguably the most determining factor is the network structure. Some structured networks are transient amplifiers. They increase for a certain fitness range the fixation probability of beneficial mutations as compared to a well-mixed population. We study a perturbation method for identifying transient amplifiers for death–birth updating. The method involves calculating the coalescence times of random walks on graphs and finding the vertex with the largest remeeting time. If the graph is perturbed by removing an edge from this vertex, there is a certain likelihood that the resulting perturbed graph is a transient amplifier. We test all pairwise nonisomorphic regular graphs up to a certain order and thus cover the whole structural range expressible by these graphs. For cubic and quartic regular graphs we find a sufficiently large number of transient amplifiers. For these networks we carry out a spectral analysis and show that the graphs from which transient amplifiers can be constructed share certain structural properties. Identifying spectral and structural properties may promote finding and designing such networks.

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