Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology

Conditions are derived that we conjecture are necessary and sufficient for the existence of stationary densities for a class of two-dimensional diffusion processes. The derivation of the conditions rests on the assumption that a two-dimensional stationary density (which can be viewed as a stable “internal equilibrium”) exists if and only if all “boundary equilibria” are unstable in the sense that small perturbations lead to moving away from the boundaries with high probability. For the models considered, the boundary equilibria are one-dimensional stationary densities and equilibrium points. To demonstrate the usefulness of the conditions, three random environment models are analyzed: a three-allele selection model, a two-species competition model, and a two-locus selection model. Several of the results obtained have been verified by alternate methods.     

SEXUAL SELECTION BY THE HANDICAP MECHANISM

The handicap mechanism of sexual selection by female choice has been strongly criticized because it does not cause sexual selection to reinforce viability selection and it cannot account for the origin of mating preferences. However, several models indicate that the handicap mechanism can have important effects when operating in conjunction with Fisher's mechanism in polygynous populations. These models have been criticized because they require that fitness remains heritable indefinitely. I develop a simple haploid model of the handicap mechanism based on nonheritable variation in paternal investment, thus eliminating the problem of heritable fitness. This model produces the same evolutonary dynamics as both simple and quantitative genetic models of the handicap mechanism based on heritable fitness. If the parameters are such that Fisherian runaway selection does not occur in the null model (i.e., the polymorphic equilibria, which lie along the “Fisher line,” are stable), then the handicap mechanism turns the Fisher line into an evolutionary trajectory upon which all other trajectories converge. This occurs because Fisher's mechanism generates no net selection on female preference when the population is on the Fisher line, so that any additional source of selection (direct or indirect) on female choice causes the population to evolve deterministically along the Fisher line. This change in the evolutionary dynamics has the important consequence of eliminating the potential for rapid population divergence for mating systems via genetic drift along the Fisher line.     

Natural selection under population regulation

The dynamical behavior of multi-allele, one-locus systems is analyzed under population regulation. Weak selection is assumed. It is shown that for sufficiently large times, t, the nth time derivative of the population number N(t) is of order n}+1 in the selection coefficients. These order relations imply there is an asymptotic “quasi-equilibrium” in which population size and mean fitness change slowly relative to changes in gene frequencies. Consistent with the results of other authors, in quasi-equilibrium the mean fitness is second-order in the selection coefficients. In an effort to understand dynamic behavior beyond the immediate neighborhood of equilibrium, the topology of mean fitness surfaces is explored. In general, population regulation leads to regions of decreasing mean fitness in which there are important changes in gene frequencies. To illustrate this and other related phenomena, I analyze models in which there is logarithmic population control, and in which genotypic fitnesses W_i(x) are linear in the allele frequencies x. Exact solutions for mean fitness W(x) are obtained for two- and three-allele systems with symmetric fertilities and mortalities.     

Generalized population models and the nature of genetic drift

The Wright–Fisher model of allele dynamics forms the basis for most theoretical and applied research in population genetics. Our understanding of genetic drift, and its role in suppressing the deterministic forces of Darwinian selection has relied on the specific form of sampling inherent to the Wright–Fisher model and its diffusion limit. Here we introduce and analyze a broad class of forward-time population models that share the same mean and variance as the Wright–Fisher model, but may otherwise differ. The proposed class unifies and further generalizes a number of population-genetic processes of recent interest, including the Λ and Cannings processes. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We characterize in detail the behavior of standard population-genetic quantities across this family of generalized models. Some quantities, such as heterozygosity, remain unchanged; but others, such as neutral absorption times and fixation probabilities under selection, deviate by orders of magnitude from the Wright–Fisher model. We show that generalized population models can produce startling phenomena that differ qualitatively from classical behavior — such as assured fixation of a new mutant despite the presence of genetic drift. We derive the forward-time continuum limits of the generalized processes, analogous to Kimura’s diffusion limit of the Wright–Fisher process, and we discuss their relationships to the Kingman and non-Kingman coalescents. Finally, we demonstrate that some non-diffusive, generalized models are more likely, in certain respects, than the Wright–Fisher model itself, given empirical data from Drosophila populations.     

Sexual selection unhandicapped by the Fisher process

A population genetic model of sexual selection is constructed in which, at equilibrium, males signal their quality by developing costly ornaments, and females pay costs to use the ornaments in mate choice. It is shown that the form of the equilibrium is uninfluenced by the Fisher process, that is, by self-reinforcement of female preferences. This is a working model of the handicap principle applied to sexual selection, and places Zahavi's handicap principle on the same logical footing as the Fisher process, in that each can support sexual selection without the presence of the other. A way of measuring the relative importance of the two processes is suggested that can be applied to both theories and facts. A style of modelling that allows simple genetics and complicated biology to be combined is recommended.     

Continuous-genotype models and assortative mating

Feldman and Cavalli-Sforza (1979a,b) have argued that the convergence properties of classical models of assortative mating are not known, and that these models involve arbitrary assumptions which assume rather than derive the achievement of equilibrium. A careful consideration of all models shows that the classical models are well defined and seem to achieve their equilibria. The model used by Feldman and Cavalli-Sforza involves an arbitrary assumption. Consideration of the models of Wright, Fisher, Bulmer, and Lande in the context of assortative mating or of selection versus mutation shows that these models are consistent with each other. The treatment of the balance between mutation and normalizing selection by Cavalli-Sforza and Feldman comes to conclusions sharply different from those of other authors, apparently as a result of this same arbitrary assumption.     

The nearly neutral and selection theories of molecular evolution under the fisher geometrical framework: substitution rate, population size, and complexity

The general theories of molecular evolution depend on relatively arbitrary assumptions about the relative distribution and rate of advantageous, deleterious, neutral, and nearly neutral mutations. The Fisher geometrical model (FGM) has been used to make distributions of mutations biologically interpretable. We explored an FGM-based molecular model to represent molecular evolutionary processes typically studied by nearly neutral and selection models, but in which distributions and relative rates of mutations with different selection coefficients are a consequence of biologically interpretable parameters, such as the average size of the phenotypic effect of mutations and the number of traits (complexity) of organisms. A variant of the FGM-based model that we called the static regime (SR) represents evolution as a nearly neutral process in which substitution rates are determined by a dynamic substitution process in which the population's phenotype remains around a suboptimum equilibrium fitness produced by a balance between slightly deleterious and slightly advantageous compensatory substitutions. As in previous nearly neutral models, the SR predicts a negative relationship between molecular evolutionary rate and population size; however, SR does not have the unrealistic properties of previous nearly neutral models such as the narrow window of selection strengths in which they work. In addition, the SR suggests that compensatory mutations cannot explain the high rate of fixations driven by positive selection currently found in DNA sequences, contrary to what has been previously suggested. We also developed a generalization of SR in which the optimum phenotype can change stochastically due to environmental or physiological shifts, which we called the variable regime (VR). VR models evolution as an interplay between adaptive processes and nearly neutral steady-state processes. When strong environmental fluctuations are incorporated, the process becomes a selection model in which evolutionary rate does not depend on population size, but is critically dependent on the complexity of organisms and mutation size. For SR as well as VR we found that key parameters of molecular evolution are linked by biological factors, and we showed that they cannot be fixed independently by arbitrary criteria, as has usually been assumed in previous molecular evolutionary models.     

Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes

The maintenance of genetic variation in a spatially heterogeneous environment has been one of the main research themes in theoretical population genetics. Despite considerable progress in understanding the consequences of spatially structured environments on genetic variation, many problems remain unsolved. One of them concerns the relationship between the number of demes, the degree of dominance, and the maximum number of alleles that can be maintained by selection in a subdivided population. In this work, we study the potential of maintaining genetic variation in a two-deme model with deme-independent degree of intermediate dominance, which includes absence of G×E interaction as a special case. We present a thorough numerical analysis of a two-deme three-allele model, which allows us to identify dominance and selection patterns that harbor the potential for stable triallelic equilibria. The information gained by this approach is then used to construct an example in which existence and asymptotic stability of a fully polymorphic equilibrium can be proved analytically. Noteworthy, in this example the parameter range in which three alleles can coexist is maximized for intermediate migration rates. Our results can be interpreted in a specialist-generalist context and (among others) show when two specialists can coexist with a generalist in two demes if the degree of dominance is deme independent and intermediate. The dominance relation between the generalist allele and the specialist alleles play a decisive role. We also discuss linear selection on a quantitative trait and show that G×E interaction is not necessary for the maintenance of more than two alleles in two demes.     

THE LANDE–KIRKPATRICK MECHANISM IS THE NULL MODEL OF EVOLUTION BY INTERSEXUAL SELECTION: IMPLICATIONS FOR MEANING,HONESTY, AND DESIGN IN INTERSEXUAL SIGNALS

The Fisher‐inspired, arbitrary intersexual selection models of Lande (1981) and Kirkpatrick (1982) , including both stable and unstable equilibrium conditions, provide the appropriate null model for the evolution of traits and preferences by intersexual selection. Like the Hardy–Weinberg equilibrium, the Lande–Kirkpatrick (LK) mechanism arises as an intrinsic consequence of genetic variation in trait and preference in the absence of other evolutionary forces. The LK mechanism is equivalent to other intersexual selection mechanisms in the absence of additional selection on preference and with additional trait‐viability and preference‐viability correlations equal to zero. The LK null model predicts the evolution of arbitrary display traits that are neither honest nor dishonest, indicate nothing other than mating availability, and lack any meaning or design other than their potential to correspond to mating preferences. The current standard for demonstrating an arbitrary trait is impossible to meet because it requires proof of the null hypothesis. The LK null model makes distinct predictions about the evolvability of traits and preferences. Examples of recent intersexual selection research document the confirmationist pitfalls of lacking a null model. Incorporation of the LK null into intersexual selection will contribute to serious examination of the extent to which natural selection on preferences shapes signals.     

The free energy method and the Wright–Fisher model with 2 alleles

We systematically investigate the Wright–Fisher model of population genetics with the free energy functional formalism of statistical mechanics and in the light of recent mathematical work on the connection between Fokker–Planck equations and free energy functionals. In statistical physics, entropy increases, or equivalently, free energy decreases, and the asymptotic state is given by a Gibbs-type distribution. This also works for the Wright–Fisher model when rewritten in divergence to identify the correct free energy functional. We not only recover the known results about the stationary distribution, that is, the asymptotic equilibrium state of the model, in the presence of positive mutation rates and possibly also selection, but can also provide detailed formulae for the rate of convergence towards that stationary distribution. In the present paper, the method is illustrated for the simplest case only, that of two alleles.     

The effects of intraspecific competition and stabilizing selection on a polygenic trait

The equilibrium properties of an additive multilocus model of a quantitative trait under frequency- and density-dependent selection are investigated. Two opposing evolutionary forces are assumed to act: (i) stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and (ii) intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the equilibrium structure, in particular, number, degree of polymorphism, and genetic variance of stable equilibria, is affected by the strength of frequency dependence, and what role the number of loci, the amount of recombination, and the demographic parameters play. To this end, we employ a statistical and numerical approach, complemented by analytical results, and explore how the equilibrium properties averaged over a large number of genetic systems with a given number of loci and average amount of recombination depend on the ecological and demographic parameters. We identify two parameter regions with a transitory region in between, in which the equilibrium properties of genetic systems are distinctively different. These regions depend on the strength of frequency dependence relative to pure stabilizing selection and on the demographic parameters, but not on the number of loci or the amount of recombination. We further study the shape of the fitness function observed at equilibrium and the extent to which the dynamics in this model are adaptive, and we present examples of equilibrium distributions of genotypic values under strong frequency dependence. Consequences for the maintenance of genetic variation, the detection of disruptive selection, and models of sympatric speciation are discussed.     

An asymptotic theory for model selection inference in general semiparametric problems

Hjort & Claeskens (2003) developed an asymptotic theoryfor model selection, model averaging and subsequent inferenceusing likelihood methods in parametric models, along with associatedconfidence statements. In this article, we consider a semiparametricversion of this problem, wherein the likelihood depends on parametersand an unknown function, and model selection/averaging is tobe applied to the parametric parts of the model. We show thatall the results of Hjort & Claeskens hold in the semiparametriccontext, if the Fisher information matrix for parametric modelsis replaced by the semiparametric information bound for semiparametricmodels, and if maximum likelihood estimators for parametricmodels are replaced by semiparametric efficient profile estimators.Our methods of proof employ Le Cam's contiguity lemmas, leadingto transparent results. The results also describe the behaviourof semiparametric model estimators when the parametric componentis misspecified, and also have implications for pointwise-consistentmodel selectors.     

Optimal experiment selection for parameter estimation in biological differential equation models

ABSTRACT: BACKGROUND: Parameter estimation in biological models is a common yet challenging problem. In this work we explore the problem for gene regulatory networks modeled by differential equations with unknown parameters, such as decay rates, reaction rates, Michaelis-Menten constants, and Hill coefficients. We explore the question to what extent parameters can be efficiently estimated by appropriate experimental selection. RESULTS: A minimization formulation is used to find the parameter values that best fit the experiment data. When the data is insufficient, the minimization problem often has many local minima that fit the data reasonably well. We show that selecting a new experiment based on the local Fisher Information of one local minimum generates additional data that allows one to successfully discriminate among the many local minima. The parameters can be estimated to high accuracy by iteratively performing minimization and experiment selection. We show that the experiment choices are roughly independent of which local minima is used to calculate the local Fisher Information. CONCLUSIONS: We show that by an appropriate choice of experiments, one can, in principle, efficiently and accurately estimate all the parameters of gene regulatory network. In addition, we demonstrate that appropriate experiment selection can also allow one to restrict model predictions without constraining the parameters using many fewer experiments. We suggest that predicting model behaviors and inferring parameters represent two different approaches to model calibration with different requirements on data and experimental cost.     

Controllability of selection-mutation systems

The well-known Fisher type selection-mutation model is studied from the point of view of mathematical systems theory. Mutation rates are considered as control functions. Based on a general sufficient condition for local controllability of non-linear systems with invariant manifold, a method is proposed to guarantee the controllability of the considered population into a polymorphic equilibrium.     

Population Genetics Inference for Longitudinally-Sampled Mutants Under Strong Selection

Longitudinal allele frequency data are becoming increasingly prevalent. Such samples permit statistical inference of the population genetics parameters that influence the fate of mutant variants. To infer these parameters by maximum likelihood, the mutant frequency is often assumed to evolve according to the Wright–Fisher model. For computational reasons, this discrete model is commonly approximated by a diffusion process that requires the assumption that the forces of natural selection and mutation are weak. This assumption is not always appropriate. For example, mutations that impart drug resistance in pathogens may evolve under strong selective pressure. Here, we present an alternative approximation to the mutant-frequency distribution that does not make any assumptions about the magnitude of selection or mutation and is much more computationally efficient than the standard diffusion approximation. Simulation studies are used to compare the performance of our method to that of the Wright–Fisher and Gaussian diffusion approximations. For large populations, our method is found to provide a much better approximation to the mutant-frequency distribution when selection is strong, while all three methods perform comparably when selection is weak. Importantly, maximum-likelihood estimates of the selection coefficient are severely attenuated when selection is strong under the two diffusion models, but not when our method is used. This is further demonstrated with an application to mutant-frequency data from an experimental study of bacteriophage evolution. We therefore recommend our method for estimating the selection coefficient when the effective population size is too large to utilize the discrete Wright–Fisher model.     

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.     

Models of general frequency-dependent selection and mating-interaction effects and the analysis of selection patterns in Drosophila inversion polymorphisms

We investigate mechanisms of balancing selection by extending two deterministic models of selection in a one-locus two-allele genetic system to allow for frequency-dependent fitnesses. Specifically we extend models of constant selection to allow for general frequency-dependent fitness functions for sex-dependent viabilities and multiplicative fertilities, while non-multiplicative mating-dependent components remain constant. We compute protected polymorphism conditions that take the form of harmonic means involving both the frequency- and the mating-dependent parameters. This allows for a direct comparison of the equilibrium properties of the frequency-dependent models with those of the constant models and for an analysis of equilibrium of the general model of constant fertility. We then apply the theory to analyze the maintenance of inversion polymorphisms in Drosophila subobscura and D. pseudoobscura, for which data on empirical fitness component estimates are available in the literature. Regression on fitness estimates obtained at different starting frequencies enables us to implement explicit fitness functions in the models and therefore to perform complete studies of equilibrium and stability for particular sets of data. The results point to frequency dependence of fitness components as the main mechanism responsible for the maintenance of the inversion polymorphisms considered, particularly in relation to heterosis, although we also discuss the contribution of other selection mechanisms.     

Mate‐sampling costs and sexy sons

Costly female mating preferences for purely Fisherian male traits (i.e. sexual ornaments that are genetically uncorrelated with inherent viability) are not expected to persist at equilibrium. The indirect benefit of producing ‘sexy sons’ (Fisher process) disappears: in some models, the male trait becomes fixed; in others, a range of male trait values persist, but a larger trait confers no net fitness advantage because it lowers survival. Insufficient indirect selection to counter the direct cost of producing fewer offspring means that preferences are lost. The only well‐cited exception assumes biased mutation on male traits. The above findings generally assume constant direct selection against female preferences (i.e. fixed costs). We show that if mate‐sampling costs are instead derived based on an explicit account of how females acquire mates, an initially costly mating preference can coevolve with a male trait so that both persist in the presence or absence of biased mutation. Our models predict that empirically detecting selection at equilibrium will be difficult, even if selection was responsible for the location of the current equilibrium. In general, it appears useful to integrate mate sampling theory with models of genetic consequences of mating preferences: being explicit about the process by which individuals select mates can alter equilibria.     

Stripes,spots, or reversed spots in two-dimensional Turing systems

Two-dimensional Turing models can generate stationary striped patterns or spotted patterns, and are used to explain the body pattern formation of animals. We studied the effects of the choice of reaction terms on pattern selection, i.e., which pattern is likely to be formed. We examined in detail a model with linear reaction terms and additional constraint terms that confine two variables within a finite range. In the one-dimensional model, a periodic stationary pattern can be formed only when the activator level is constrained both from below and from above. In the two-dimensional model, the relative distance of the equilibrium level of the activator between the upper and lower limitations determines the pattern selection. Striped patterns are produced when the equilibrium is equally distant from the upper and the lower limitations, but spotted patterns are produced when the equilibrium is clearly closer to one than to the other of two limitations. We then examined models with nonlinear reaction terms, including both activator-inhibitor and activator-depletion substrate type models; we attempted to explain the pattern selection of these nonlinear models based on the results of linear models with constraints. The distribution of the activator level is skewed positively and negatively for spotted patterns and reversed spotted patterns, respectively. In contrast, the skew of the distribution of the activator level was close to zero in the case of striped patterns. This observation provides a heuristic argument of how the location of the equilibrium between the constraints leads to pattern selection.     

Stable long-period cycling and complex dynamics in a single-locus fertility model with genomic imprinting

Although long-period population size cycles and chaotic fluctuations in abundance are common in ecological models, such dynamics are uncommon in simple population-genetic models where convergence to a fixed equilibrium is most typical. When genotype-frequency cycling does occur, it is most often due to frequency-dependent selection that results from individual or species interactions. In this paper, we demonstrate that fertility selection and genomic imprinting are sufficient to generate a Hopf bifurcation and complex genotype-frequency cycling in a single-locus population-genetic model. Previous studies have shown that on its own, fertility selection can yield stable two-cycles but not long-period cycling characteristic of a Hopf bifurcation. Genomic imprinting, a molecular mechanism by which the expression of an allele depends on the sex of the donating parent, allows fitness matrices to be nonsymmetric, and this additional flexibility is crucial to the complex dynamics we observe in this fertility selection model. Additionally, we find under certain conditions that stable oscillations and a stable equilibrium point can coexist. These dynamics are characteristic of a Chenciner (generalized Hopf) bifurcation. We believe this model to be the simplest population-genetic model with such dynamics.