Stochastic viability in an island model with partial dispersal: Approximation by a diffusion process in the limit of a large number of islands.

In this paper, we investigate a finite population undergoing evolution through an island model with partial dispersal and without mutation, where generations are discrete and non-overlapping. The population is structured into D demes, each containing N individuals of two possible types, A and B, whose viability coefficients, sA and sB, respectively, vary randomly from one generation to the next. We assume that the means, variances and covariance of the viability coefficients are inversely proportional to the number of demes D, while higher-order moments are negligible in comparison to 1/D. We use a discrete-time Markov chain with two timescales to model the evolutionary process, and we demonstrate that as the number of demes D approaches infinity, the accelerated Markov chain converges to a diffusion process for any deme size N≥2. This diffusion process allows us to evaluate the fixation probability of type A following its introduction as a single mutant in a population that was fixed for type B. We explore the impact of increasing the variability in the viability coefficients on this fixation probability. At least when N is large enough, it is shown that increasing this variability for type B or decreasing it for type A leads to an increase in the fixation probability of a single A. The effect of the population-scaled variances, σA2 and σB2, can even cancel the effects of the population-scaled means, µA and µB. We also show that the fixation probability of a single A increases as the deme-scaled migration rate increases. Moreover, this probability is higher for type A than for type B if the population-scaled geometric mean viability coefficient is higher for type A than for type B, which means that µA-σA2/2>µB-σB2/2.

Evolution of cooperation in social dilemmas with assortment in finite populations.

We investigate conditions for the evolution of cooperation in social dilemmas in finite populations with assortment of players by group founders and general payoff functions for cooperation and defection within groups. Using a diffusion approximation in the limit of a large population size that does not depend on the precise updating rule, we show that the first-order effect of selection on the fixation probability of cooperation when represented once can be expressed as the difference between time-averaged payoffs with respect to effective time that cooperators and defectors spend in direct competition in the different group states. Comparing this fixation probability to its value under neutrality and to the corresponding fixation probability for defection, we deduce conditions for the evolution of cooperation. We show that these conditions are generally less stringent as the level of assortment increases under a wide range of assumptions on the payoffs such as additive, synergetic or discounted benefits for cooperation, fixed cost for cooperation and threshold benefit functions. This is not necessarily the case, however, when payoffs in pairwise interactions are multiplicatively compounded within groups.

Conformity and anti-conformity in a finite population.

Conformist and anti-conformist cultural transmission have been studied both empirically, in several species, and theoretically, with population genetic models. Building upon standard, infinite-population models (IPMs) of conformity, we introduce finite-population models (FPMs) and study them via simulation and a diffusion approximation. In previous IPMs of conformity, offspring observe the variants of n adult role models, where n is often three. Numerical simulations show that while the short-term behavior of the FPM with n=3 role models is well approximated by the IPM, stable polymorphic equilibria of the IPM become effective equilibria of the FPM at which the variation persists prior to fixation or loss, and which produce plateaus in curves for fixation probabilities and expected times to absorption. In the FPM with n=5 role models, the population may switch between two effective equilibria, which is not possible in the IPM, or may cycle between frequencies that are not effective equilibria, which is possible in the IPM. In all observed cases of 'equilibrium switching' and 'cycling' in the FPM, model parameters exceed O(1/N), required for the diffusion approximation, resulting in an over-estimation of the actual times to absorption. However, in those cases with n=5 role models that have one effective equilibrium and stable fixation states, even if conformity coefficients exceed O(1/N), the diffusion approximation matches closely the numerical simulations of the FPM. This suggests that the robustness of the diffusion approximation depends not only on the magnitudes of coefficients, but also on the qualitative behavior of the conformity model.

Modelling of the Energy Depletion Process and Battery Depletion Attacks for Battery-Powered Internet of Things (IoT) Devices.

The Internet of Things (IoT) is transforming almost every industry, including agriculture, food processing, health care, oil and gas, environmental protection, transportation and logistics, manufacturing, home automation, and safety. Cost-effective, small-sized batteries are often used to power IoT devices being deployed with limited energy capacity. The limited energy capacity of IoT devices makes them vulnerable to battery depletion attacks designed to exhaust the energy stored in the battery rapidly and eventually shut down the device. In designing and deploying IoT devices, the battery and device specifications should be chosen in such a way as to ensure a long lifetime of the device. This paper proposes diffusion approximation as a mathematical framework for modelling the energy depletion process in IoT batteries. We applied diffusion or Brownian motion processes to model the energy depletion of a battery of an IoT device. We used this model to obtain the probability density function, mean, variance, and probability of the lifetime of an IoT device. Furthermore, we studied the influence of active power consumption, sleep time, and battery capacity on the probability density function, mean, and probability of the lifetime of an IoT device. We modelled ghost energy depletion attacks and their impact on the lifetime of IoT devices. We used numerical examples to study the influence of battery depletion attacks on the distribution of the lifetime of an IoT device. We also introduced an energy threshold after which the device's battery should be replaced in order to ensure that the battery is not completely drained before it is replaced.

Revisiting the number of self-incompatibility alleles in finite populations: From old models to new results.

Under gametophytic self-incompatibility (GSI), plants are heterozygous at the self-incompatibility locus (S-locus) and can only be fertilized by pollen with a different allele at that locus. The last century has seen a heated debate about the correct way of modelling the allele diversity in a GSI population that was never formally resolved. Starting from an individual-based model, we derive the deterministic dynamics as proposed by Fisher (The genetical theory of natural selection - A complete, Variorum edition, Oxford University Press, 1958) and compute the stationary S-allele frequency distribution. We find that the stationary distribution proposed by Wright (Evolution, 18, 609, 1964) is close to our theoretical prediction, in line with earlier numerical confirmation. Additionally, we approximate the invasion probability of a new S-allele, which scales inversely with the number of resident S-alleles. Lastly, we use the stationary allele frequency distribution to estimate the population size of a plant population from an empirically obtained allele frequency spectrum, which complements the existing estimator of the number of S-alleles. Our expression of the stationary distribution resolves the long-standing debate about the correct approximation of the number of S-alleles and paves the way for new statistical developments for the estimation of the plant population size based on S-allele frequencies.

Evolutionary game dynamics with non-uniform interaction rates in finite population.

In this study, we extend evolutionary game dynamics with non-uniform interaction rates to the situation with finite population. Our main goal is to show how the fixation probability is influenced by the non-uniform interaction rates under weak selection. Based on the diffusion approximation of the Moran process and assumption of weak selection, the stochastic dynamic properties of a two-phenotype game with non-uniform interaction rates in a finite population are investigated. By the analysis of some cases, we show that the non-uniform interaction rates may result in the potential evolutionary complexity of game dynamics in finite population.

The effect of the opting-out strategy on conditions for selection to favor the evolution of cooperation in a finite population.

We consider a Prisoner's Dilemma (PD) that is repeated with some probability 1-ρ only between cooperators as a result of an opting-out strategy adopted by all individuals. The population is made of N pairs of individuals and is updated at every time step by a birth-death event according to a Moran model. Assuming an intensity of selection of order 1/N and taking 2N2 birth-death events as unit of time, a diffusion approximation exhibiting two time scales, a fast one for pair frequencies and a slow one for cooperation (C) and defection (D) frequencies, is ascertained in the limit of a large population size. This diffusion approximation is applied to an additive PD game, cooperation by an individual incurring a cost c to the individual but providing a benefit b to the opponent. This is used to obtain the probability of ultimate fixation of C introduced as a single mutant in an all D population under selection, which can be compared to the probability under neutrality, 1/(2N), as well as the corresponding probability for a single D introduced in an all C population under selection. This gives conditions for cooperation to be favored by selection. We show that these conditions are satisfied when the benefit-to-cost ratio, b/c, exceeds some increasing function of ρ that is approximately given by (1+ρ)/(1-ρ). This condition is more stringent, however, than the condition for tit-for-tat (TFT) to be favored against always-defect (AllD) in the absence of opting-out.

Diffusion Model of Preemptive-Resume Priority Systems and Its Application to Performance Evaluation of SDN Switches.

The increasing use of Software-Defined Networks brings the need for their performance analysis and detailed analytical and numerical models of them. The primary element of such research is a model of a SDN switch. This model should take into account non-Poisson traffic and general distributions of service times. Because of frequent changes in SDN flows, it should also analyze transient states of the queues. The method of diffusion approximation can meet these requirements. We present here a diffusion approximation of priority queues and apply it to build a more detailed model of SDN switch where packets returned by the central controller have higher priority than other packets.

Performance Analysis of Packet Aggregation Mechanisms and Their Applications in Access (e.g., IoT, 4G/5G), Core, and Data Centre Networks.

The transmission of massive amounts of small packets generated by access networks through high-speed Internet core networks to other access networks or cloud computing data centres has introduced several challenges such as poor throughput, underutilisation of network resources, and higher energy consumption. Therefore, it is essential to develop strategies to deal with these challenges. One of them is to aggregate smaller packets into a larger payload packet, and these groups of aggregated packets will share the same header, hence increasing throughput, improved resource utilisation, and reduction in energy consumption. This paper presents a review of packet aggregation applications in access networks (e.g., IoT and 4G/5G mobile networks), optical core networks, and cloud computing data centre networks. Then we propose new analytical models based on diffusion approximation for the evaluation of the performance of packet aggregation mechanisms. We demonstrate the use of measured traffic from real networks to evaluate the performance of packet aggregation mechanisms analytically. The use of diffusion approximation allows us to consider time-dependent queueing models with general interarrival and service time distributions. Therefore these models are more general than others presented till now.

Diffusion Model of a Non-Integer Order PIγ Controller with TCP/UDP Streams.

In this article, a way to employ the diffusion approximation to model interplay between TCP and UDP flows is presented. In order to control traffic congestion, an environment of IP routers applying AQM (Active Queue Management) algorithms has been introduced. Furthermore, the impact of the fractional controller PIγ and its parameters on the transport protocols is investigated. The controller has been elaborated in accordance with the control theory. The TCP and UDP flows are transmitted simultaneously and are mutually independent. Only the TCP is controlled by the AQM algorithm. Our diffusion model allows a single TCP or UDP flow to start or end at any time, which distinguishes it from those previously described in the literature.

Randomized matrix games in a finite population: Effect of stochastic fluctuations in the payoffs on the evolution of cooperation.

A diffusion approximation for a randomized 2 × 2-matrix game in a large finite population is ascertained in the case of random payoffs whose expected values, variances and covariances are of order given by the inverse of the population size N. Applying the approximation to a Randomized Prisoner's Dilemma (RPD) with independent payoffs for cooperation and defection in random pairwise interactions, conditions on the variances of the payoffs for selection to favor the evolution of cooperation, favor more the evolution of cooperation than the evolution of defection, and disfavor the evolution of defection are deduced. All these are obtained from probabilities of ultimate fixation of a single mutant. It is shown that the conditions are lessened with an increase in the variances of the payoffs for defection against cooperation and defection and a decrease in the variances of the payoffs for cooperation against cooperation and defection. A RPD game with independent payoffs whose expected values are additive is studied in detail to support the conclusions. Randomized matrix games with non-independent payoffs, namely the RPD game with additive payoffs for cooperation and defection based on random cost and benefit for cooperation and the repeated RPD game with Tit-for-Tat and Always-Defect as strategies in pairwise interactions with a random number of rounds, are studied under the assumption that the population-scaled expected values, variances and covariances of the payoffs are all of the same small enough order. In the first model, the conditions in favor of the evolution of cooperation hold only if the covariance between the cost and the benefit is large enough, while the analysis of the second model extends the results on the effects of the variances of the payoffs for cooperation and defection found for the one-round RPD game.

Diffusion dynamics on the coexistence subspace in a stochastic evolutionary game.

Frequency-dependent selection reflects the interaction between different species as they battle for limited resources in their environment. In a stochastic evolutionary game the species relative fitnesses guides the evolutionary dynamics with fluctuations due to random drift. A selection advantage which depends on a changing environment will introduce additional possibilities for the dynamics. We analyse a simple model in which a random environment allows competing species to coexist for a long time before a fixation of a single species happens. In our analysis we use stability in a linear combination of competing species to approximate the stochastic dynamics of the system by a diffusion on a one dimensional co-existence region. Our method significantly simplifies approximating the probability of first extinction and its expected time, and demonstrates a rigorous model reduction technique for evaluating quasistationary properties of stochastic evolutionary dynamics.

Probing deep tissues with laser-induced thermotherapy using near-infrared light.

Optically tunable gold nanoparticles have been widely used in research with near-infrared light as a means to enhance laser-induced thermal therapy since it capitalizes on nanoparticles' plasmonic heating properties. There have been several studies published on numerical models replicating this therapy in such conditions. However, there are several limitations on some of the models which can render the model unfaithful to therapy simulations. In this paper, two techniques of simulating laser-induced thermal therapy with a high-absorbing localized region of interest inside a phantom are compared. To validate these models, we conducted an experiment of an agar-agar phantom with an inclusion reproducing it with both models. The phantom was optically characterized by absorption and total attenuation. The first model is based on the macroperspective solution of the radiative transfer equation given by the diffusion equation, which is then coupled with the Pennes bioheat equation to obtain the temperature. The second is a Monte Carlo model that considers a stochastic solution of the same equation and is also considered as input to the Pennes bioheat transfer equation which is then computed. The Monte Carlo is in good agreement with the experimental data having an average percentage difference of 4.5% and a correlation factor of 0.98, while the diffusion method comparison with experimental data is 61% and 0.95 respectively. The optical characterization of the phantom and its inclusion were also validated indirectly since the Monte Carlo, which used those parameters, was also validated. While knowing the temperature in all points inside a body during photothermal therapy is important, one has to be mindful of the model which fits the conditions and properties. There are several reasons to justify the discrepancy of the diffusion method: low-scattering conditions, absorption, and reduced scattering are comparable. The error bars that are normally associated when characterizing an optical phantom can justify also a part of that uncertainty. For low-size tumors in depth, one may have to increase the light dosage in photothermal therapies to have a more effective treatment.

Stationary distribution of the linkage disequilibrium coefficient r2.

The linkage disequilibrium coefficient r2 is a measure of statistical dependence of the alleles possessed by an individual at different genetic loci. It is widely used in association studies to search for the locations of disease-causing genes on chromosomes. Most studies to date treat r2 as a fixed property of two loci in a finite population, and investigate the sampling distribution of estimators due to the statistical sampling of individuals from the population. Here, we instead consider the distribution of r2 itself under a process of genetic sampling through the generations. Using a classical two-locus model for genetic drift, mutation, and recombination, we investigate the probability density function of r2 at stationarity. This density function provides a tool for inference on evolutionary parameters such as mutation and recombination rates. We reconstruct the approximate stationary density of r2 by calculating a finite sequence of the distribution's moments and applying the maximum entropy principle. Our approach is based on the diffusion approximation, under which we demonstrate that for certain models in population genetics, moments of the stationary distribution can be obtained without knowing the probability distribution itself. To illustrate our approach, we show how the stationary probability density of r2 can be used in a maximum likelihood framework to estimate mutation and recombination rates from sample data of r2.

Diffusion approximation for an age-class-structured population under viability and fertility selection with application to fixation probability of an advantageous mutant.

In this paper, we ascertain the validity of a diffusion approximation for the frequencies of different types under recurrent mutation and frequency-dependent viability and fertility selection in a haploid population with a fixed age-class structure in the limit of a large population size. The approximation is used to study, and explain in terms of selection coefficients, reproductive values and population-structure coefficients, the differences in the effects of viability versus fertility selection on the fixation probability of an advantageous mutant.

Haploids, polymorphisms and fluctuating selection.

I analyze the joint impact of directional and fluctuating selection with reversible mutation in finite bi-allelic haploid populations using diffusion approximations of the Moran and chemostat models. Results differ dramatically from those of the classic Wright-Fisher diffusion. There, a strong dispersive effect attributable to fluctuating selection dissipates nascent polymorphisms promoted by a relatively weak emergent frequency dependent selective effect. The dispersive effect in the Moran diffusion with fluctuations every birth-death event is trivial. The same frequency dependent selective effect now dominates and polymorphism is promoted. The dispersive effect in the chemostat diffusion with fluctuations every generation is identical to that in the Wright-Fisher diffusion. Nevertheless, polymorphism is again promoted because the emergent frequency dependent effect is doubled, an effect attributable to geometric reproduction within generations. Fluctuating selection in the Moran and chemostat diffusions can also promote bi-allelic polymorphisms when one allele confers a net benefit. Rapid fluctuations within generations are highly effective at promoting polymorphism in large populations. The bi-allelic distribution is approximately Gaussian but becomes uniform and then U-shaped as the frequency of environmental fluctuations decreases to once a generation and then once every multiple generations. Trade-offs (negative correlations in fitness) help promote polymorphisms but are not essential. In all three models the frequency dependent effect raises the probability of ultimate fixation of new alleles, but less effectively in the Wright-Fisher diffusion. Individual-based forward simulations confirm the calculations.

Inference from the stationary distribution of allele frequencies in a family of Wright-Fisher models with two levels of genetic variability.

The distribution of allele frequencies obtained from diffusion approximations to Wright-Fisher models is useful in developing intuition about the population level effects of evolutionary processes. The statistical properties of the stationary distributions of K-allele models have been extensively studied under neutrality or under selection. Here, we introduce a new family of Wright-Fisher models in which there are two hierarchical levels of genetic variability. The genotypes composed of alleles differing from each other at the selected level have fitness differences with respect to each other and evolve under selection. The genotypes composed of alleles differing from each other only at the neutral level have the same fitness and evolve under neutrality. We show that with an appropriate scaling of the mutation parameter with respect to the number of alleles at each level, the frequencies of alleles at the selected and the neutral level are conditionally independent of each other, conditional on knowing the number of alleles at all levels. This conditional independence allows us to simulate from the joint stationary distribution of the allele frequencies. We use these simulated frequencies to perform inference on parameters of the model with two levels of genetic variability using Approximate Bayesian Computation.

Mathematical approach to nonlocal interactions using a reaction-diffusion system.

In recent years, spatial long range interactions during developmental processes have been introduced as a result of the integration of microscopic information, such as molecular events and signaling networks. They are often called nonlocal interactions. If the profile of a nonlocal interaction is determined by experiments, we can easily investigate how patterns generate by numerical simulations without detailed microscopic events. Thus, nonlocal interactions are useful tools to understand complex biosystems. However, nonlocal interactions are often inconvenient for observing specific mechanisms because of the integration of information. Accordingly, we proposed a new method that could convert nonlocal interactions into a reaction-diffusion system with auxiliary unknown variables. In this review, by introducing biological and mathematical studies related to nonlocal interactions, we will present the heuristic understanding of nonlocal interactions using a reaction-diffusion system.

Evolutionary control: Targeted change of allele frequencies in natural populations using externally directed evolution.

Random processes in biology, in particular random genetic drift, often make it difficult to predict the fate of a particular mutation in a population. Using principles of theoretical population genetics, we present a form of biological control that ensures a focal allele's frequency, at a given locus, achieves a prescribed probability distribution at a given time. This control is in the form of an additional evolutionary force that acts on a population. We provide the mathematical framework that determines the additional force. Our analysis indicates that generally the additional force depends on the frequency of the focal allele, and it may also depend on the time. We argue that translating this additional force into an externally controlled process, which has the possibility of being implemented in a number of different ways corresponding to selection, migration, mutation, or a combination of these, may provide a flexible instrument for targeted change of traits of interest in natural populations. This framework may be applied, or used as an informed form of guidance, in a variety of different biological scenarios including: yield and pesticide optimisation in crop production, biofermentation, the local regulation of human-associated natural populations, such as parasitic animals, or bacterial communities in hospitals.

Hybrid Markov chain models of S-I-R disease dynamics.

Deterministic epidemic models are attractive due to their compact nature, allowing substantial complexity with computational efficiency. This partly explains their dominance in epidemic modelling. However, the small numbers of infectious individuals at early and late stages of an epidemic, in combination with the stochastic nature of transmission and recovery events, are critically important to understanding disease dynamics. This motivates the use of a stochastic model, with continuous-time Markov chains being a popular choice. Unfortunately, even the simplest Markovian S-I-R model-the so-called general stochastic epidemic-has a state space of order [Formula: see text], where N is the number of individuals in the population, and hence computational limits are quickly reached. Here we introduce a hybrid Markov chain epidemic model, which maintains the stochastic and discrete dynamics of the Markov chain in regions of the state space where they are of most importance, and uses an approximate model-namely a deterministic or a diffusion model-in the remainder of the state space. We discuss the evaluation, efficiency and accuracy of this hybrid model when approximating the distribution of the duration of the epidemic and the distribution of the final size of the epidemic. We demonstrate that the computational complexity is [Formula: see text] and that under suitable conditions our approximations are highly accurate.