A Model of Two-Way Selection System for Human Behavior

Two-way selection is a common phenomenon in nature and society. It appears in the processes like choosing a mate between men and women, making contracts between job hunters and recruiters, and trading between buyers and sellers. In this paper, we propose a model of two-way selection system, and present its analytical solution for the expectation of successful matching total and the regular pattern that the matching rate trends toward an inverse proportion to either the ratio between the two sides or the ratio of the state total to the smaller group''s people number. The proposed model is verified by empirical data of the matchmaking fairs. Results indicate that the model well predicts this typical real-world two-way selection behavior to the bounded error extent, thus it is helpful for understanding the dynamics mechanism of the real-world two-way selection system.     

A method to optimize selection on multiple identified quantitative trait loci

A mathematical approach was developed to model and optimize selection on multiple known quantitative trait loci (QTL) and polygenic estimated breeding values in order to maximize a weighted sum of responses to selection over multiple generations. The model allows for linkage between QTL with multiple alleles and arbitrary genetic effects, including dominance, epistasis, and gametic imprinting. Gametic phase disequilibrium between the QTL and between the QTL and polygenes is modeled but polygenic variance is assumed constant. Breeding programs with discrete generations, differential selection of males and females and random mating of selected parents are modeled. Polygenic EBV obtained from best linear unbiased prediction models can be accommodated. The problem was formulated as a multiple-stage optimal control problem and an iterative approach was developed for its solution. The method can be used to develop and evaluate optimal strategies for selection on multiple QTL for a wide range of situations and genetic models.     

Fisher's lost model of runaway sexual selection

The bizarre elaboration of sexually selected traits such as the peacock's tail was a puzzle to Charles Darwin and his 19th century followers. Ronald A. Fisher crafted an ingenious solution in the 1930s, positing that female preferences would become genetically correlated with preferred traits due to nonrandom mating. These genetic correlations would translate selection for preferred traits into selection for stronger preferences, leading to a self-reinforcing process of ever-elaborating traits and preferences. It is widely believed that Fisher provided only a verbal model of this “runaway” process. However, in correspondence with Charles Galton Darwin, Fisher also laid out a simple mathematical model that purportedly confirms his verbal prediction of runaway sexual selection. Unfortunately, Fisher's model contains inconsistencies that render his quantitative conclusions inaccurate. Here, we correct Fisher's model and show that it contains all the ingredients of a working runaway process. We derive quantitative predictions of his model using numerical techniques that were unavailable in Fisher's time. Depending on parameter values, mean traits and preferences may increase until genetic variance is depleted by selection, exaggerate exponentially while their variances remain stable, or both means and variances may increase super-exponentially. We thus present the earliest mathematical model of runaway sexual selection.     

Perturbation analysis of a two-locus model with directional selection and recombination

A population genetic two-locus model with additive, directional selection and recombination is considered. It is assumed that recombination is weaker than selection; i.e., the recombination parameter r is smaller than the selection coefficients. This assumption is appropriate for describing the effects of two-locus selection at the molecular level. The model is formulated in terms of ordinary differential equations (ODES) for the gamete frequencies x = (x ₁, x ₂, x ₃, x ₄), defined on the simplex S ₄. The ODEs are analyzed using first a regular pertubation technique. However, this approach yields satisfactory results only if r is very small relative to the selection coefficients and if the initial values x(0) are in the interior part of S ₄. To cope with this problem, a novel two-scale perturbation method is proposed which rests on the theory of averaging of vectorfields. It is demonstrated that the zeroth-order solution of this two-scale approach approximates the numerical solution of the model well, even if recombination rate is on the order of the selection coefficients.     

SOLVING THE PARADOX OF STASIS: SQUASHED STABILIZING SELECTION AND THE LIMITS OF DETECTION

Despite the potential for rapid evolution, stasis is commonly observed over geological timescales—the so‐called “paradox of stasis.” This paradox would be resolved if stabilizing selection were common, but stabilizing selection is infrequently detected in natural populations. We hypothesize a simple solution to this apparent disconnect: stabilizing selection is hard to detect empirically once populations have adapted to a fitness peak. To test this hypothesis, we developed an individual‐based model of a population evolving under an invariant stabilizing fitness function. Stabilizing selection on the population was infrequently detected in an “empirical” sampling protocol, because (1) trait variation was low relative to the fitness peak breadth; (2) nonselective deaths masked selection; (3) populations wandered around the fitness peak; and (4) sample sizes were typically too small. Moreover, the addition of negative frequency‐dependent selection further hindered detection by flattening or even dimpling the fitness peak, a phenomenon we term “squashed stabilizing selection.” Our model demonstrates that stabilizing selection provides a plausible resolution to the paradox of stasis despite its infrequent detection in nature. The key reason is that selection “erases its traces”: once populations have adapted to a fitness peak, they are no longer expected to exhibit detectable stabilizing selection.     

Multiple loci mapping via model-free variable selection

Despite recent flourish of proposals on variable selection, genome-wide multiple loci mapping remains to be challenging. The majority of existing variable selection methods impose a model, and often the homoscedastic linear model, prior to selection. However, the true association between the phenotypical trait and the genetic markers is rarely known a priori, and the presence of epistatic interactions makes the association more complex than a linear relation. Model-free variable selection offers a useful alternative in this context, but the fact that the number of markers p often far exceeds the number of experimental units n renders all the existing model-free solutions that require n > p inapplicable. In this article, we examine a number of model-free variable selection methods for small-n-large-p regressions in the context of genome-wide multiple loci mapping. We propose and advocate a multivariate group-wise adaptive penalization solution, which requires no model prespecification and thus works for complex trait-marker association, and handles one variable at a time so that works for n < p. Effectiveness of the new method is demonstrated through both intensive simulations and a comprehensive real data analysis across 6100 gene expression traits.     

Cumulant dynamics of a population under multiplicative selection, mutation, and drift

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.     

A theory of natural selection incorporating interaction among individuals. VII. Use of groups consisting of one sire and one dam

A previous study, (V), in this series dealt with the problem of interaction among individuals in a population by utilizing a conceptual life-history model that assumed the synchronization of fitness components (viability and fecundity) for members within groups of interacting individuals. The model also assumed that the group members were not differentiated in any way. It was shown that these assumptions of synchronization and homogeneity of group members resulted in overall symmetric fitness values, and that selection operating on these symmetric values produced optimum short- and long-term results.Since the idealized model of paper (V) yields optimum selection results, it is of interest to consider specific biological mechanisms which can be used to fulfill the conceptual assumptions involved. The solution adopted in this study is to, first, require survival to occur at the group level. This forces the viability parameters to be identical for all group members.Second, the group composition is restricted to a single pair mating. This forces the fecundity parameters to be identical for members (sire and dam) of the same group. However, this solution causes a complication. Since each individual is identified as belonging to one of the two sexual types, it is necessary to extend the previous analysis in paper (V), to accommodate differentiated group members. It is shown that this differentiation leads to a non-symmetric fitness matrix. However, it is further shown that the selection results can be obtained by use of a ‘combined’ fitness matrix that is symmetric. Therefore, the important result is demonstrated that selection involving differentiated sire-dam groups possesses all of the optimum short- and long-term properties inherent in the original non-differentiated selection procedure developed in paper (V). It is speculated that these results could be of some significance with regard to the evolution of a monogamous, pair-bonding family structure.     

最优化设计连续的自然保护区

生境破碎是导致生物多样性损失的重要原因之一,避免生境破碎的一个有效方式是建立连续的自然保护区使物种可在保护区内自由移动。不加选择地把大片土地都转为保护区是实现连续的一个途径,但资源是有限的,应当以最优的方式分配。如何最优化设计生态上和经济上都有效的保护区成为生物保护领域一个重要议题。从一组备选地块中选择一部分组成自然保护区,这样的问题主要有两种解法:启发式方法和最优化方法。启发式方法虽然灵活且运算速度快但不能保证最优解因而可能导致稀缺资源的浪费,最优化方法保证得到的解是最优的但建模和运算存在困难。建立一个线性整数规划模型用于设计一个最小的连续保护区,用Dantzig剪切法消除循环确保形成一个连续的树,对应一个连续的保护区,检验了模型的计算效率。结果显示,模型可在合理时间内解决一个包含100个备选地块和30个物种的连续保护区设计问题,计算效率显著优于同类目的的其它方法。以美国伊利诺伊州Cache河流域11种濒危鸟类的保护区设计为例说明了该方法的应用,设计了两种情况下连续的保护区。讨论了模型的局限和数据问题。     

The dynamical attainability of ESS in evolutionary games

In this paper, the attainability of ESS of the evolutionary game among n players under the frequency-independent selection is studied by means of a mathematical model describing the dynamical development and a concept of stability (strongly determined stability). It is assumed that natural selection and small mutations cause the phenotype to change gradually in the direction of fitness increasing. It is shown that (1) the ESS solution is not always evolutionarily attainable in the evolutionary dynamics, (2) in the game where the interaction between two species is completely competitive, the Nash solution is always attainable, and (3) one of two species may attain the state of minimum fitness as a result of evolution. The attainability of ESS is also examined in two game models on the sex ratio of wasps and aphids in light of our criterion of the attainability of ESS.     

Modelling publication bias and p-hacking

Publication bias and p-hacking are two well-known phenomena that strongly affect the scientific literature and cause severe problems in meta-analyses. Due to these phenomena, the assumptions of meta-analyses are seriously violated and the results of the studies cannot be trusted. While publication bias is very often captured well by the weighting function selection model, p-hacking is much harder to model and no definitive solution has been found yet. In this paper, we advocate the selection model approach to model publication bias and propose a mixture model for p-hacking. We derive some properties for these models, and we compare them formally and through simulations. Finally, two real data examples are used to show how the models work in practice.     

Designing connected nature reserve networks using a graph theory approach

Habitat fragmentation has been cited as one of the critical reasons for biodiversity loss. Establishing connected nature reserve networks is an effective way to reduce habit fragmentation. However, the resources devoted to nature reserves have always been scarce. Therefore it is important to allocate our scarce resources in an optimal way. The optimal design of a reserve network which is effective both ecologically and economically has become an important research topic in the reserve design literature. The problem of optimal selection of a subset from a larger group of potential habitat sites is solved using either heuristic or formal optimization methods. The heuristic methods, although flexible and computationally fast, can not guarantee the solution is optimal therefore may lead to scarce resources being used in an ineffective way. The formal optimization methods, on the other hand, guarantees the solution is optimal, but it has been argued that it would be difficult to model site selection process using optimization models, especially when spatial attributes of the reserve have to be taken into account. This paper presents a linear integer programming model for the design of a minimal connected reserve network using a graph theory approach. A connected tree is determined corresponding to a connected reserve. Computational performance of the model is tested using datasets randomly generated by the software GAMS. Results show that the model can solve a connected reserve design problem which includes 100 potential sites and 30 species in a reasonable period of time. As an empirical application, the model is applied to the protection of endangered and threatened bird species in the Cache River basin area in Illinois, US. Two connected reserve networks are determined for 13 bird species.     

A genetic model with mutation and selection

Two deterministic models of a multiallele population in which mutation and selection both operate are considered, and formulae for the gene frequencies are obtained. Both models are of a diploid population in which selection is additive and mutation is general; generations are discrete and nonoverlapping. In the first model, the stationary solution of the discrete equations is found. In the second, the discrete time process is approximated by a continuous time one, and the resulting differential equations are solved. The transient case for two alleles is solved explicitly, and the results are graphed. An application is given to sequences of sites.     

Morphological Integration, Evolutionary Constraints, and Extinction: A Computer Simulation-Based Study

It is well known that there is a strong relationship among the environment, selection, and extinction, but the underlying role of genetics and genetic constraints in contributing to extinction is less appreciated. Integration of characters may enhance survivability for species, providing that selective pressure is parallel with the patterns of morphological integration. However, we hypothesize that, if the direction of selection shifts, integration may also prevent populations from responding quickly enough to the new directions of selection. This would lead to the inability to find a successful adaptive solution, causing downward pressure on the population, and ultimately, extinction. We test this model with a computer simulation, using an adaptive landscape model. We generate populations of varying levels of multivariate integration and generate selection pressures to test the ability of the populations to respond to selection both parallel and orthogonal to the axis of maximum variation. In these simulations, more highly integrated populations survived longer when selection was in the direction of maximum variation. However, when selection was closer to orthogonal to the axis of maximum variation, extinction was more rapid in highly integrated populations. These results suggest that integration may play a strong role in both survivability and extinction. Tightly integrated populations are highly persistent when selection pressure is close to the axis of maximum variation, which is expected to frequently be the case since integration is likely often a product of selection. However, these highly integrated taxa are more susceptible to extinction when the direction of selection shifts, and is closer to orthogonal to the axis of maximum variation.     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Numerical and exact solutions for continuum of alleles models

 Two results are presented for problems involving alleles with a continuous range of effects. The first result is a simple yet highly accurate numerical method that determines the equilibrium distribution of allelic effects, moments of this distribution, and the mutational load. The numerical method is explicitly applied to the mutation-selection balance problem of stabilising selection. The second result is an exact solution for the distribution of allelic effects under weak stabilising selection for a particular distribution of mutant effects. The exact solution is shown to yield a distribution of allelic effects that, depending on the mutation rate, interpolates between the ``House of Cards' approximation and the Gaussian approximation. The exact solution is also used to test the accuracy of the numerical method. Received: 7 November 2001 / Revised version: 5 September 2002 / Published online: 18 December 2002 Key words or phrases: Continuum of alleles – Numerical solution – Exact solution – Mutation selection balance – Stabilising selection     

Designing connected nature reserve networks using a graph theory approach

Habitat fragmentation has been cited as one of the critical reasons for biodiversity loss. Establishing connected nature reserve networks is an effective way to reduce habit fragmentation. However, the resources devoted to nature reserves have always been scarce. Therefore it is important to allocate our scarce resources in an optimal way. The optimal design of a reserve network which is effective both ecologically and economically has become an important research topic in the reserve design literature. The problem of optimal selection of a subset from a larger group of potential habitat sites is solved using either heuristic or formal optimization methods. The heuristic methods, although flexible and computationally fast, can not guarantee the solution is optimal therefore may lead to scarce resources being used in an ineffective way. The formal optimization methods, on the other hand, guarantees the solution is optimal, but it has been argued that it would be difficult to model site selection process using optimization models, especially when spatial attributes of the reserve have to be taken into account. This paper presents a linear integer programming model for the design of a minimal connected reserve network using a graph theory approach. A connected tree is determined corresponding to a connected reserve. Computational performance of the model is tested using datasets randomly generated by the software GAMS. Results show that the model can solve a connected reserve design problem which includes 100 potential sites and 30 species in a reasonable period of time. As an empirical application, the model is applied to the protection of endangered and threatened bird species in the Cache River basin area in Illinois, US. Two connected reserve networks are determined for 13 bird species.     

Biodynamic modeling and simulation of multistage cell mutations

The aim of this study is to present a mathematical computer simulation model for multistage carcinogenesis. The population genetic model is developed based on the reaction diffusion, logistic behavior, and Hollings Type II interactions between normal, benign, and premalignant cells. The simple form of the Fisher-Haldane-Wright equation of the genetic model of tumor suppressor gene and oncogenes is used to describe this type of interaction. Through computer simulation, we observe the behavior, stability, and traveling wave solution of the premalignant stage mutation as well as its survival under natural selection pressure. As a simple case of this model, the interaction between normal and tumor cells with one or two stages of mutations is analyzed.     

An introduction to the mathematical structure of the Wright–Fisher model of population genetics

In this paper, we develop the mathematical structure of the Wright–Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker–Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.     

A model for the minimum cost configuration problem in flexible manufacturing systems

This paper presents a mathematical programming model to help select equipment for a flexible manufacturing system, i.e., the selection of the types and numbers of CNC machines, washing stations, load/unload stations, transportation vehicles, and pallets. The objective is to minimize equipment costs and work-in-process inventory cost, while fulfilling production requirements for an average period. Queueing aspects and part flow interactions are considered with the help of a Jacksonian-type closed queueing network model in order to evaluate the system's performance. Since the related decision problem of our model can be shown to be NP-complete, the proposed solution procedure is based on implicit enumeration. Four bounds are provided, two lower and two upper bounds. A tight lower bound is obtained by linearizing the model through the application of asymptotic bound analysis. Furthermore, asymptotic bound analysis allows the calculation of a lower bound for the number of pallets in the system. The first upper bound is given by the best feasible solution and the second is based on the anti-starshaped form of the throughput function.