Speeding up Evolutionary Search by Small Fitness Fluctuations

We consider a fixed size population that undergoes an evolutionary adaptation in the weak mutation rate limit, which we model as a biased Langevin process in the genotype space. We show analytically and numerically that, if the fitness landscape has a small highly epistatic (rough) and time-varying component, then the population genotype exhibits a high effective diffusion in the genotype space and is able to escape local fitness minima with a large probability. We argue that our principal finding that even very small time-dependent fluctuations of fitness can substantially speed up evolution is valid for a wide class of models.     

Spatial pattern formation of a ratio-dependent predator-prey model

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator-prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.     

Recombination dramatically speeds up evolution of finite populations

We study the role of recombination, in the form of bacterial transformation, in speeding up Darwinian evolution. This is done by adding a new process to a previously studied Markov model of evolution on a smooth fitness landscape; this new process allows alleles to be exchanged with those in the surrounding medium. Our results, both numerical and analytic, indicate that, for a wide range of intermediate population sizes, recombination dramatically speeds up the rate of evolutionary advance.     

Film flows down a fiber: Modeling and influence of streamwise viscous diffusion

A two-equation model is formulated in terms of two coupled evolution equations for the film thickness h and the local flow rate q within the framework of lubrication theory. Consistency is achieved up to first order in the film parameter epsilon and streamwise diffusion effects are accounted for. The evolution equation obtained by Craster and Matar [1] is recovered in the appropriate limit. Comparisons to the experimental results by [2] and [3] show good agreement in the linear and nonlinear regimes. Second-order viscous diffusion terms are found to potentially enhance the speed and amplitude of nonlinear waves triggered by the Rayleigh-Plateau instability mechanism. Time-dependent computations of the spatial evolution of the film reveal a strong influence of streamwise diffusion on the dynamics of the flow and the wave selection process.     

Global Migration Can Lead to Stronger Spatial Selection than Local Migration

The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection and would therefore favor to a larger extent the evolution of cooperation. To address this issue, we quantify the impacts of these two migration patterns on the evolutionary competition of two strategies in a finite island model. Global migration means that individuals can migrate from any one island to any other island. Local migration means that individuals can only migrate between islands that are nearest neighbors; we study a simple geometry where islands are arranged on a one-dimensional, regular cycle. We derive general results for weak selection and large population size. Our key parameters are: the number of islands, the migration rate and the mutation rate. Surprisingly, our comparative analysis reveals that global migration can lead to stronger spatial selection than local migration for a wide range of parameter conditions. Our work provides useful insights into understanding how different mobility patterns affect evolutionary processes.     

Effect of temperature on rectified diffusion during ultrasound-induced heating

Experimental observations of delayed-onset cavitation during ultrasound insonation have been suggested as being caused by a change in the size distribution of the bubble population due to rectified diffusion. To investigate this hypothesis, a single bubble model is used here to explore the effect of heating and the subsequent elevated temperatures on the rectified diffusion process. Numerical solution of the model, which includes the temperature dependences of seven relevant physical parameters, allows quantification of the change in the pressure threshold for rectified diffusion, as well as the importance of the bulk liquid saturation concentration in determining bubble evolution. Although elevated temperatures and liquid supersaturation reduce the rectified diffusion threshold, it remains coincident with the inertial cavitation thresholds at submicron bubble sizes at all temperatures. This observation suggests that changes in the nucleation environment, rather than bubble growth due to rectified diffusion, is a more likely cause of delayed-onset cavitation events.     

Molecular evolution under fitness fluctuations

Molecular evolution is a stochastic process governed by fitness, mutations, and reproductive fluctuations in a population. Here, we study evolution where fitness itself is stochastic, with random switches in the direction of selection at individual genomic loci. As the correlation time of these fluctuations becomes larger than the diffusion time of mutations within the population, fitness changes from an annealed to a quenched random variable. We show that the rate of evolution has its maximum in the crossover regime, where both time scales are comparable. Adaptive evolution emerges in the quenched fitness regime (evidence for such fitness fluctuations has recently been found in genomic data). The joint statistical theory of reproductive and fitness fluctuations establishes a conceptual connection between evolutionary genetics and statistical physics of disordered systems.     

Probabilistic models of multidimensional piecewise expanding mappings

Strongly chaotic systems (e.g., piecewise expanding mappings) exhibit diffusion-like behavior in the sense of central limit theorems. To find more precise statements about the similarity to probabilistic diffusion, we study how the evolution of probability densities underd-dimensional piecewise expanding mappings can be modeled by Markov processes with smooth transition probabilities (such as diffusion processes). Our results can be viewed as a special type of local limit theorem.     

On the origin of the phase–space diffusion limit in (dis)ordered protein aggregation

Derivation of a phase–space diffusion limit (D-L) allows to obtain a useful formula for a characteristic width of the macroion-channeling filter, controlling model (dis)ordered protein aggregations in a non-ideal aqueous solution. The channel’s width is estimated at the order of an inner half-width of the Stern-type double layer circumventing the growing object and depends in turn on an interplay of the local thermal and electrostatic conditions. The interfacial channeling effect manifests at the edge of biomolecular hydration-duration dependent (non)Markovianity of the system. The interface vs. solution aggregation late-time dynamics are discussed in such local (non)isothermal context with the aim to suggest their experimental assessment.     

Evolutionary game theory in growing populations

Existing theoretical models of evolution focus on the relative fitness advantages of different mutants in a population while the dynamic behavior of the population size is mostly left unconsidered. We present here a generic stochastic model which combines the growth dynamics of the population and its internal evolution. Our model thereby accounts for the fact that both evolutionary and growth dynamics are based on individual reproduction events and hence are highly coupled and stochastic in nature. We exemplify our approach by studying the dilemma of cooperation in growing populations and show that genuinely stochastic events can ease the dilemma by leading to a transient but robust increase in cooperation.     

Anderson Localization, Non-linearity and Stable Genetic Diversity

In many models of genotypic evolution, the vector of genotype populations satisfies a system of linear ordinary differential equations. This system of equations models a competition between differential replication rates (fitness) and mutation. Mutation operates as a generalized diffusion process on genotype space. In the large time asymptotics, the replication term tends to produce a single dominant quasi-species, unless the mutation rate is too high, in which case the asymptotic population becomes de-localized. We introduce a more macroscopic picture of genotypic evolution wherein a random fitness term in the linear model produces features analogous to Anderson localization. When coupled with density dependent non-linearities, which limit the population of any given genotype, we obtain a model whose large time asymptotics display stable genotypic diversity.Research partially supported by DARPA under the FUNBIO program and the Francis J. Carey Term Chair.     

Adaptive ultrafast nano-optics in a tight focus

We theoretically demonstrate the control of electromagnetic field properties on a sub-diffraction length scale, by polarization shaping of tightly focused femtosecond laser pulses. The field distribution in a tight focus is represented as a superposition of plane waves. The near-field of a model nanostructure is then obtained as a sum of the near-field distributions induced by the planar waves components. A self-consistent solution of Maxwell’s equations in the frequency domain yields the near-field distributions for planar wave illumination. Adaptive optimization of the incident polarization pulse shape using an evolutionary algorithm allows controlling of a number of observables, such as local nonlinear flux, simultaneous spatial and temporal control of the intensity evolution, and control of the local spectrum. The tight focusing reduces the controllability of the flux distribution in comparison to plane wave illumination. However, it is still possible to control the spatial and temporal field evolution for particular locations in the vicinity of the nanostructure. PACS 42.65.Re; 78.47.+p; 78.67.-n; 82.53.-k     

Some Analytical Properties of the Model for Stochastic Evolutionary Games in Finite Populations with Non-uniform Interaction Rate

Traditional evolutionary games assume uniform interaction rate, which means that the rate at which individuals meet and interact is independent of their strategies. But in some systems, especially biological systems, the players interact with each other discriminately. Taylor and Nowak (2006) were the first to establish the corresponding non-uniform interaction rate model by allowing the interaction rates to depend on strategies. Their model is based on replicator dynamics which assumes an infinite size population. But in reality, the number of individuals in the population is always finite, and there will be some random interference in the individuals' strategy selection process. Therefore, it is more practical to establish the corresponding stochastic evolutionary model in finite populations. In fact, the analysis of evolutionary games in a finite size population is more difficult. Just as Taylor and Nowak said in the outlook section of their paper, "The analysis of non-uniform interaction rates should be extended to stochastic game dynamics of finite populations." In this paper, we are exactly doing this work. We extend Taylor and Nowak's model from infinite to finite case, especially focusing on the influence of non-uniform connection characteristics on the evolutionary stable state of the system. We model the strategy evolutionary process of the population by a continuous ergodic Markov process. Based on the limit distribution of the process, we can give the evolutionary stable state of the system. We make a complete classification of the symmetric 2×2 games. For each case game, the corresponding limit distribution of the Markov-based process is given when noise intensity is small enough. In contrast with most literatures in evolutionary games using the simulation method, all our results obtained are analytical. Especially, in the dominant-case game, coexistence of the two strategies may become evolutionary stable states in our model. This result can be used to explain the emergence of cooperation in the Prisoner is Dilemma Games to some extent. Some specific examples are given to illustrate our results.     

High-resolution DOSY NMR with spins in different chemical surroundings: influence of particle exchange

The effect of chemical exchange in the diffusion-ordered (DOSY) spectra of a two-site system in the slow-exchange limit with respect to the chemical shift is studied. The problem is addressed both theoretically and experimentally. The relationship between diffusion time (t) and mean lifetimes (tau) is studied by the simulation of the magnetization attenuations as a function of the gradient strength, under PFG conditions. The influence of the difference in populations and diffusion coefficients of the two sites is also considered. In analogy to the limiting cases of fast- and slow-exchange with respect to the chemical shift, limiting cases with respect to the diffusion dimension are defined. The slow-exchange limit in diffusion corresponds to the relation of t and tau that allows us to observe the two spins in exchange associated with the individual diffusion coefficients of the two sites when no exchange is present. The fast-exchange limit in diffusion is reached when the relation of t and tau is such that the two spins present the same apparent diffusion coefficient. By using a model system consisting of water/t-butanol it is shown that by recording several DOSY experiments with increasing diffusion times it is possible to estimate the value of the exchange rate.     

SEMIDISCRETE INTEGRABLE NONLINEAR SYSTEMS GENERATED BY THE NEW FOURTH-ORDER SPECTRAL OPERATOR: LOCAL CONSERVATION LAWS

Starting with the semidiscrete integrable nonlinear Schrödinger system on a zigzag-runged ladder lattice we have presented the generalization and an essentially off-diagonal enlargement of its spectral operator which in the framework of zero-curvature equation allows to generate at least two new types of semidiscrete integrable nonlinear systems. The two types of evolutionary operators consistent with the extended spectral operator are proposed. In order to fix arbitrary sampling functions in each type of evolution operators we have to rely upon a restricted collection of lowest local conservation laws whose local densities are independent on the type of admissible evolution operators. For this purpose the modified procedure of seeking the infinite hierarchy of local conservation laws based upon several distinct generating functions has been developed and some lowest local conservation laws have been explicitly obtained.     

Wavelength selection in Rayleigh-Bénard convection

Rayleigh-Bénard convection is considered in the presence of a weakly nonuniform temperature distribution and weakly nonuniform layer height with their gradients perpendicular to the (parallel) convection rolls. In the infinite Prandtl number limit the phase equation that describes the slow spatial and temporal evolution of the local wave number is derived. Evaluating the equation near threshold and for stress-free boundary conditions we find nonuniversal wavenumber selection, forced phase diffusion with moving patterns, and other measurable effects.     

Effects of migration on the evolutionary game dynamics in finite populations with community structures

We investigate the impacts of migration on the evolutionary game dynamics in finite populations with community structures in the framework of evolutionary game theory. In contrast to deterministic dynamics, our model incorporates stochastic factors induced by the finite population size. Based on the analysis of the stationary distribution of the evolutionary process in the limit of rare mutations, we prove that it is most likely to find the population in the community where all individuals have the lower migration rate. Furthermore, we show that reducing the difference between the migration rates of distinct communities can increase the first hitting time to the homogeneous absorbing state and can prolong the coexistence time of different species, promoting the conservation of biodiversity.     

The geometrical aspect of the gauge fields and the possibility of a general description of local and nonlocal interaction in QED

It is shown that the use of analogues of the QCD gauge-invariant structures (type of “strings” and “stars”) in QED enables one to consider the generalized gauge-invariant amplitude, which satisfies the requirements of the quantum theory of gauge fields and allows one to describe electromagnetic (EM) interactions both with local and nonlocal charged matter fields beyond the scope of the Lagrange approach. This causes no negative consequences for the theory as a whole. The invariant character of the amplitude structure in relation to hierarchical evolution of the structural forces and structural elements of the nonlocal field allows its use in an unchanged form to describe EM interaction processes in different scales of matter structure. The generalized amplitude features a continuous limit in transition from nonlocal to local fields.     

Restricted diffusion and exchange of water in porous media: average structure determination and size distribution resolved from the effect of local field gradients on the proton NMR spectrum

NMR Pulsed field gradient measurements of the restrained diffusion of confined fluids constitute an efficient method to probe the local geometry in porous media. In most practical cases, the diffusion decay, when limited to its principal part, can be considered as Gaussian leading to an apparent diffusion coefficient. The evolution of the latter as a function of the diffusion interval yields average information on the surface/volume ratio of porosities and on the tortuosity of the network. In this paper, we investigate porous model systems of packed spheres (polystyrene and glass) with known mean diameter and polydispersity, and, in addition, a real porous polystyrene material. Applying an Inverse Laplace Transformation in the second dimension reveals an evolution of the apparent diffusion coefficient as a function of the resonance frequency. This evolution is related to a similar evolution of the transverse relaxation time T2. These results clearly show that each resonance frequency in the water proton spectrum corresponds to a particular magnetic environment produced by a given pore geometry in the porous media. This is due to the presence of local field gradients induced by magnetic susceptibility differences at the liquid/solid interface and to slow exchange rates between different pores as compared to the frequency differences in the spectrum. This interpretation is nicely confirmed by a series of two-dimensional exchange experiments.