Limit theorems for sums determined by branching and other exponentially growing processes

A branching process counted by a random characteristic has been defined as a process which at time t is the superposition of individual stochastic processes evaluated at the actual ages of the individuals of a branching population. Now characteristics which may depend not only on age but also on absolute time are considered. For supercritical processes a distributional limit theorem is proved, which implies that classical limit theorems for sums of characteristics evaluated at a fixed age point transfer into limit theorems for branching processes counted by these characteristics. A point is that, though characteristics of different individuals should be independent, the characteristics of an individual may well interplay with the reproduction of the latter. The result requires a sort of L^p-continuity for some 1 ? p ? 2. Its proof turns out to be valid for a wider class of processes than branching ones.For the case p = 1 a number of Poisson type limits follow and for p = 2 some normality approximations are concluded. For example results are obtained for processes for rare events, the age of the oldest individual, and the error of population predictions.This work has been supported by a grant from the Swedish Natural Science Research Council.     

Periodic solutions and oscillation of discrete non-linear delay population dynamics model with external force

^** Email: emelabbasy{at}mans.edu.eg^*** Email: shsaker{at}mans.edu.eg In this paper, we consider the discrete non-linear delay populationdynamics model [graphic: see PDF] where m is a positive integer, p(n), Q(n) and (n) are positiveperiodic sequences of period . By the method that involves theapplication of the Gaines and Mawhins coincidence degree theory,we prove that there exists a positive -periodic solution (n). We prove that every positive solutionof (*) which does not oscillate about (n)satisfies lim_t[y(n)–(n)]=0.We establish some necessary and sufficient conditions for theoscillation of every positive solution about (n), and finally, we establish the lower and upperbounds of the oscillatory solutions.     

Limit theorems for the integral of a population process with immigration

A limit theorem is proven for the integral of a general class of population processes possessing independent immigration components. For the special case of the Bellman-Harris process with immigration, further results are obtained.     

PERMANENCE OF A NONAUTOMONOUS POPULATION MODEL

1.IntroductionPersistenceofpopulationsplaysanimportantroleinmathematicalecology.Inrecentyears)muchattentionhasbeengiventothepersistenceofnonautomonouspopulationmodelsll--6].[1]proposedtheconceptofpersistenceinthemeanofpopulations.Thisisofimportancebecausenotonlyitprovidesawaytocharacterizethepersistenceofpopula-nons,butalsothereareonlythresholdsbetweenpersistenceinthemeanandtheextinctionofpopulationsforgeneralnonautomonouspopulationmodels.In[2]thethresholdwasestablishedforone-dimensionalnonaut…     

Evolution on a Smooth Landscape: The Role of Bias

The role of mutational bias in evolution on a smooth landscape is investigated. We consider both a finite-length genome where the bias increases linearly with the fitness, and an infinite genome with a fixed bias. We present simulations of finite populations in a waiting time model, showing both the nonequilibrium dynamics and the equilibrium fitness distributions that are reached. We compute the equilibrium analytically in several cases, using approximate direct solution of the master equations and truncated hierarchies.     

频率啁啾激光场中锂原子的激发与态囚禁

采用含时多态展开方法,对不同频率啁啾激光场中里德堡锂原子布居数的相干迁移特性进行了计算研究。结果表明：布居数跃迁几率对激光脉冲形状、激光场强度、啁啾率等参数非常敏感,在合适的激光参数下,可以实现布居数在四个量子态之间的完全迁移和量子态的囚禁。     

Phase and amplitude control of the group velocity in a two-level system with shelving state

We study the propagation of two weak light pulses with the same temporal shape, symmetrically detuned from a stronger cw pump, in a two-level system which decays via a shelving state. We show that the group velocity of the weak pulses can be controlled, with small distortion, by modifying either the relative amplitude of the weak pulses or the total phase Φ. The changes in the local group velocity on propagation are due to pump absorption, and to changes in the total phase on propagation which occur for all cases apart from the case where the amplitudes of the weak fields are equal and or π. Slow light occurs when the initial total phase but not when ; when one pulse is less intense than the other, it experiences the greater slowing down, and the effect is greater than in the case of equal intensities.     

Modelling and stability analysis in human population genetics with selection and mutation

Population genetics is a scientific discipline that has extensively benefitted from mathematical modelling; since the Hardy‐Weinberg law (1908) to date, many mathematical models have been designed to describe the genotype frequencies evolution in a population. Existing models differ in adopted hypothesis on evolutionary forces (such as, for example, mutation, selection, and migration) acting in the population. Mathematical analysis of population genetics models help to understand if the genetic population admits an equilibrium, ie, genotype frequencies that will not change over time. Nevertheless, the existence of an equilibrium is only an aspect of a more complex issue concerning the conditions that would allow or prevent populations to reach the equilibrium. This latter matter, much more complex, has been only partially investigated in population genetics studies. We here propose a new mathematical model to analyse the genotype frequencies distribution in a population over time and under two major evolutionary forces, namely, mutation and selection; the model allows for both infinite and finite populations. In this paper, we present our model and we analyse its convergence properties to the equilibrium genotype frequency; we also derive conditions allowing convergence. Moreover, we show that our model is a generalisation of the Hardy‐Weinberg law and of subsequent models that allow for selection or mutation. Some examples of applications are reported at the end of the paper, and the code that simulates our model is available online at https://www.ding.unisannio.it/persone/docenti/del-vecchio for free use and testing.