Time-asymptotic convergence rates towards the discrete evolutionary stable distribution

2015 ◽  
Vol 25 (08) ◽  
pp. 1589-1616 ◽  
Author(s):  
Wenli Cai ◽  
Pierre-Emmanuel Jabin ◽  
Hailiang Liu
Keyword(s):  
Stable Distribution ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Continuous Model ◽  
Asymptotic Convergence ◽  
Discrete Dynamics ◽  
Differential Model ◽  
Continuous Trait ◽  
Fully Discrete ◽  
Algebraic Convergence

This paper is concerned with the discrete dynamics of an integro-differential model that describes the evolution of a population structured with respect to a continuous trait. Various time-asymptotic convergence rates towards the discrete evolutionary stable distribution (ESD) are established. For some special ESD satisfying a strict sign condition, the exponential convergence rates are obtained for both semi-discrete and fully discrete schemes. Towards the general ESD, the algebraic convergence rate that we find is consistent with the known result for the continuous model.

Time-asymptotic convergence rates towards discrete steady states of a nonlocal selection-mutation model

2019 ◽  
Vol 29 (11) ◽  
pp. 2063-2087
Author(s):  
Wenli Cai ◽  
Pierre-Emmanuel Jabin ◽  
Hailiang Liu
Keyword(s):  
Large Time ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Continuous Model ◽  
Large Time Behavior ◽  
Asymptotic Convergence ◽  
Time Behavior ◽  
Structured Population ◽  
Mutation Model ◽  
Globally Stable

This paper is concerned with large time behavior of solutions to a semi-discrete model involving nonlinear competition that describes the evolution of a trait-structured population. Under some threshold assumptions, the steady solution is shown unique and strictly positive, and also globally stable. The exponential convergence rate to the steady state is also established. These results are consistent with the results in [P.-E. Jabin, H. L. Liu. Nonlinearity 30 (2017) 4220–4238] for the continuous model.

scholarly journals A continuous model of physiological prion aggregation suggests a role for Orb2 in gating long-term synaptic information

2018 ◽  
Vol 5 (12) ◽  
pp. 180336
Author(s):  
Michele Sanguanini ◽  
Antonino Cattaneo
Keyword(s):  
Rna Binding ◽  
Binding Protein ◽  
Rna Binding Protein ◽  
Mrna Translation ◽  
Continuous Model ◽  
Cellular Level ◽  
Long Term Memory ◽  
Cytoplasmic Polyadenylation ◽  
Differential Model

The regulation of mRNA translation at the level of the synapse is believed to be fundamental in memory and learning at the cellular level. The family of cytoplasmic polyadenylation element binding (CPEB) proteins emerged as an important RNA-binding protein family during development and in adult neurons. Drosophila Orb2 (homologue of mouse CPEB3 protein and of the neural isoform of Aplysia CPEB) has been found to be involved in the translation of plasticity-dependent mRNAs and has been associated with long-term memory. Orb2 protein presents two main isoforms, Orb2A and Orb2B, which form an activity-induced amyloid-like functional aggregate, thought to be the translation-inducing state of the RNA-binding protein. Here we present a first two-states continuous differential model for Orb2A–Orb2B aggregation. This model provides new working hypotheses for studying the role of prion-like CPEB proteins in long-term synaptic plasticity. Moreover, this model can be used as a first step to integrate translation- and protein aggregation-dependent phenomena in synaptic facilitation rules.

scholarly journals EXPONENTIAL CONVERGENCE OF hp-FEM FOR MAXWELL EQUATIONS WITH WEIGHTED REGULARIZATION IN POLYGONAL DOMAINS

2005 ◽  
Vol 15 (04) ◽  
pp. 575-622 ◽  
Author(s):  
MARTIN COSTABEL ◽  
MONIQUE DAUGE ◽  
CHRISTOPH SCHWAB
Keyword(s):  
Finite Element ◽  
Maxwell’S Equations ◽  
Maxwell Equations ◽  
Maxwell's Equations ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Polygonal Domains ◽  
Time Harmonic ◽  
Standard Tool ◽  
Optimal Convergence Rates

The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization converge to wrong solutions in any non-convex polygon. Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h version of finite elements. We prove exponential convergence of hp FEM for the weighted regularization of Maxwell's equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.

scholarly journals FULLY DISCRETE SCHEMES FOR THE SCHRÖDINGER EQUATION: DISPERSIVE PROPERTIES

2007 ◽  
Vol 17 (04) ◽  
pp. 567-591 ◽  
Author(s):  
LIVIU I. IGNAT
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Continuous Model ◽  
Local Smoothing ◽  
Euler Approximation ◽  
One Dimensional ◽  
Fully Discrete ◽  
Backward Euler ◽  
The One

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

Exponential convergence rates for weighted sums in noncommutative probability space

2016 ◽  
Vol 19 (04) ◽  
pp. 1650027 ◽  
Author(s):  
Byoung Jin Choi ◽  
Un Cig Ji
Keyword(s):  
Probability Space ◽  
Von Neumann Algebra ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Large Deviation ◽  
Type Inequality ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Noncommutative Probability ◽  
Von Neumann

We study exponential convergence rates for weighted sums of successive independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. Then we prove a noncommutative extension of the result for the exponential convergence rate by Baum, Katz and Read. As applications, we first study a large deviation type inequality for weighted sums in a noncommutative probability space, and secondly we study exponential convergence rates for weighted free additive convolution sums of probability measures.

scholarly journals Making the most of potential: potential games and genotypic convergence

2021 ◽  
Vol 8 (8) ◽  
pp. 210309
Author(s):  
Omer Edhan ◽  
Ziv Hellman ◽  
Ilan Nehama
Keyword(s):  
Growth Rate ◽  
Sexual Reproduction ◽  
Genetic Linkage ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Basins Of Attraction ◽  
Single Locus ◽  
Potential Game ◽  
Unified Theory ◽  
Varying Environments

We consider genotypic convergence of populations and show that under fixed fitness asexual and haploid sexual populations attain monomorphic convergence (even under genetic linkage between loci) to basins of attraction with locally exponential convergence rates; the same convergence obtains in single locus diploid sexual reproduction but to polymorphic populations. Furthermore, we show that there is a unified theory underlying these convergences: all of them can be interpreted as instantiations of players in a potential game implementing a multiplicative weights updating algorithm to converge to equilibrium, making use of the Baum–Eagon Theorem. To analyse varying environments, we introduce the concept of ‘virtual convergence’, under which, even if fixation is not attained, the population nevertheless achieves the fitness growth rate it would have had under convergence to an optimal genotype. Virtual convergence is attained by asexual, haploid sexual and multi-locus diploid reproducing populations, even if environments vary arbitrarily. We also study conditions for true monomorphic convergence in asexually reproducing populations in varying environments.

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