scholarly journals Weak selection and the separation of eco-evo time scales using perturbation analysis

2021 ◽  
Author(s):  
Philip Gerlee
Keyword(s):  
Perturbation Theory ◽  
Evolutionary Dynamics ◽  
Replicator Equation ◽  
Full Model ◽  
Zeroth Order ◽  
Dynamical Models ◽  
Weak Selection ◽  
Frequency Dependent Selection ◽  
Outer Solution ◽  
Genotype Frequencies

AbstractWe show that under the assumption of weak frequency-dependent selection a wide class of population dynamical models can be analysed using perturbation theory. The inner solution corresponds to the ecological dynamics, where to zeroth order, the genotype frequencies remain constant. The outer solution provides the evolutionary dynamics and corresponds, to zeroth order, to a generalisation of the replicator equation. We apply this method to a model of public goods dynamics and show that the error between the composite solution, which describes the dynamics for all times, and the solution to the full model scales linearly with the strength of selection.

scholarly journals Negative frequency-dependent selection maintains coexisting genotypes during fluctuating selection

2019 ◽  
Author(s):  
Caroline B. Turner ◽  
Sean W. Buskirk ◽  
Katrina B. Harris ◽  
Vaughn S. Cooper
Keyword(s):  
Evolutionary Dynamics ◽  
Burkholderia Cenocepacia ◽  
Natural Environments ◽  
Fluctuating Selection ◽  
Frequency Dependent ◽  
Frequency Dependent Selection ◽  
Fluctuating Environment ◽  
Fluctuating Environments ◽  
Negative Frequency Dependent Selection ◽  
Selection For

AbstractNatural environments are rarely static; rather selection can fluctuate on time scales ranging from hours to centuries. However, it is unclear how adaptation to fluctuating environments differs from adaptation to constant environments at the genetic level. For bacteria, one key axis of environmental variation is selection for planktonic or biofilm modes of growth. We conducted an evolution experiment with Burkholderia cenocepacia, comparing the evolutionary dynamics of populations evolving under constant selection for either biofilm formation or planktonic growth with populations in which selection fluctuated between the two environments on a weekly basis. Populations evolved in the fluctuating environment shared many of the same genetic targets of selection as those evolved in constant biofilm selection, but were genetically distinct from the constant planktonic populations. In the fluctuating environment, mutations in the biofilm-regulating genes wspA and rpfR rose to high frequency in all replicate populations. A mutation in wspA first rose rapidly and nearly fixed during the initial biofilm phase but was subsequently displaced by a collection of rpfR mutants upon the shift to the planktonic phase. The wspA and rpfR genotypes coexisted via negative frequency-dependent selection around an equilibrium frequency that shifted between the environments. The maintenance of coexisting genotypes in the fluctuating environment was unexpected. Under temporally fluctuating environments coexistence of two genotypes is only predicted under a narrow range of conditions, but the frequency-dependent interactions we observed provide a mechanism that can increase the likelihood of coexistence in fluctuating environments.

scholarly journals Anisotropic Distribution Functions for the Elliptical Galaxy NGC 1600

1996 ◽  
Vol 171 ◽  
pp. 413-413
Author(s):  
Michael Matthias ◽  
Ortwin Gerhard
Keyword(s):  
Perturbation Theory ◽  
Elliptical Galaxy ◽  
Major Axis ◽  
Distribution Functions ◽  
Minor Axis ◽  
Dynamical Models ◽  
Ccd Photometry ◽  
Radial Anisotropy ◽  
Best Fitting ◽  
Integral Distribution

Three-integral (3I) dynamical models for NGC 1600 were constructed as follows: (i) Lucy-inversion of CCD photometry and gravitational potential as in Binney, Davies, Illingworth (ApJ 361, 78, 1990), assuming axisymmetry. (ii) Third integral by perturbation theory as in Gerhard & Saha (MN 261, 311, 1991). (iii) Two- and three-integral distribution functions as in Dehnen & Gerhard (MN 261, 311, 1993), assuming various anisotropy patterns. The kinematic results from these models are presented in Fig. 1. The best-fitting 3I model (solid line, right panels) has outward-increasing radial anisotropy on the major axis and is nearly isotropic on the minor axis. The M/L of the various 3I-models varies only slightly around M/L=6.2.

scholarly journals Parametric Excitation and Evolutionary Dynamics

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
Rocio E. Ruelas ◽  
David G. Rand ◽  
Richard H. Rand
Keyword(s):  
Differential Equation ◽  
Parametric Excitation ◽  
Variable Coefficient ◽  
Evolutionary Dynamics ◽  
Replicator Equation ◽  
Periodic Coefficients ◽  
Forcing Function ◽  
Governing Differential Equation ◽  
Social Applications ◽  
Dynamics Problems

Parametric excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. Evolutionary dynamics refers to a mathematical model of natural selection (the “replicator” equation) which involves a combination of game theory and differential equations. In this paper we apply perturbation theory to investigate parametric resonance in a replicator equation having periodic coefficients. In particular, we study evolution in the Rock-Paper-Scissors game, which has biological and social applications. Here periodic coefficients could represent seasonal variation. We show that 2:1 subharmonic resonance can destabilize the usual “Rock-Paper-Scissors” equilibrium for parameters located in a resonant tongue in parameter space. However, we also show that the tongue may be absent or very small if the forcing parameters are chosen appropriately.

scholarly journals Evolutionary dynamics in discrete time for the perturbed positive definite replicator equation

2021 ◽  
Vol 62 ◽  
pp. 148-184
Author(s):  
Amie Albrecht ◽  
Konstantin Avrachenkov ◽  
Phil Howlett ◽  
Geetika Verma
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Evolutionary Dynamics ◽  
Positive Definite ◽  
Genotype Distribution ◽  
Replicator Equation ◽  
System Matrix ◽  
Discrete Time System ◽  
Time System ◽  
Time Dynamics

The population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation. doi: 10.1017/S1446181120000140

scholarly journals Extinction rates in tumor public goods games

2017 ◽  
Author(s):  
Philip Gerlee ◽  
Philipp M. Altrock
Keyword(s):  
Population Growth ◽  
Evolutionary Dynamics ◽  
Replicator Dynamics ◽  
Evolutionary Game ◽  
Cell Types ◽  
Relative Fitness ◽  
Replicator Equation ◽  
Cellular Interactions ◽  
Cancer Evolution ◽  
Public Goods Games

AbstractCancer evolution and progression are shaped by cellular interactions and Darwinian selection. Evolutionary game theory incorporates both of these principles, and has been proposed as a framework to understand tumor cell population dynamics. A cornerstone of evolutionary dynamics is the replicator equation, which describes changes in the relative abundance of different cell types, and is able to predict evolutionary equilibria. Typically, the replicator equation focuses on differences in relative fitness. We here show that this framework might not be sufficient under all circumstances, as it neglects important aspects of population growth. Standard replicator dynamics might miss critical differences in the time it takes to reach an equilibrium, as this time also depends on cellular turnover in growing but bounded populations. As the system reaches a stable manifold, the time to reach equilibrium depends on cellular death and birth rates. These rates shape the timescales, in particular in co-evolutionary dynamics of growth factor producers and free-riders. Replicator dynamics might be an appropriate framework only when birth and death rates are of similar magnitude. Otherwise, population growth effects cannot be neglected when predicting the time to reach an equilibrium, and cell type specific rates have to be accounted for explicitly.

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