• Corpus ID: 238857291

Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics

@inproceedings{Alfaro2021ConfiningIE,
  title={Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics},
  author={Matthieu Alfaro and Pierre Gabriel and Otared Kavian},
  year={2021}
}
We consider nonlinear mutation selection models, known as replicator-mutator equations in evolutionary biology. They involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the principal eigenelement of the underlying linear operator. The novelties compared to the literature on these models are about the case of symmetric mutations: Université de Rouen Normandie, CNRS, Laboratoire de Mathématiques Raphaël… 

References

SHOWING 1-10 OF 35 REFERENCES
Mathematical Properties of a Class of Integro-differential Models from Population Genetics
TLDR
A mathematical analysis of an integro-differential model arising in population genetics that describes the dynamics of fitness distribution in an asexual population under the effect of mutation and selection and derives asymptotic estimates of the distribution as the fitness tends to $\pm \infty$.
Evolutionary Branching via Replicator–Mutator Equations
We consider a class of non-local reaction–diffusion problems, referred to as replicator–mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write
Dynamics of fitness distributions in the presence of a phenotypic optimum: an integro-differential approach
TLDR
An integro-differential description of the dynamics of the fitness distribution in an asexual population under mutation and selection, in the presence of a phenotype optimum, is proposed and it is proved that the equation admits a unique time-global solution.
Dynamics of adaptation in an anisotropic phenotype-fitness landscape
TLDR
It is proved here that the equation admits a unique solution, which is interpreted as the phenotype distribution, and a new and general framework is proposed to the study of the quantitative behavior of this solution, and it is shown that the anisotropic model leads to a very good fit of Escherichia coli long-term evolution experiment.
Explicit Solutions for Replicator-Mutator Equations: Extinction versus Acceleration
TLDR
It is proved that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time.
Replicator-mutator equations with quadratic fitness
This work completes our previous analysis on models arising in evolutionary genetics. We consider the so-called replicator-mutator equation, when the fitness is quadratic. This equation is a heat
STATIONARY DISTRIBUTIONS UNDER MUTATION- SELECTION BALANCE: STRUCTURE AND PROPERTIES
A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume
From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution
We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the
Derivation of replicator–mutator equations from a model in population genetics
We introduce a Markov chain model to study evolution of a continuous trait based on population genetics. It corresponds to individual-based model which includes frequency dependent selection caused
On the maintenance of genetic variation: global analysis of Kimura's continuum-of-alleles model
  • R. Bürger
  • Biology, Medicine
    Journal of mathematical biology
  • 1986
TLDR
The present analysis provides the first proof that in Kimura's model an equilibrium in fact exists and, moreover, that it is globally stable and shows that continuum-of-alleles models may be excellent approximations to multiallele models, if analysed appropriately.
...
1
2
3
4
...