The stabilizing effect of a random environment

It is shown that the lottery competition model permits coexistence in a stochastic environment, but not in a constant environment. Conditions for coexistence and competitive exclusion are determined. Analysis of these conditions shows that the essential requirements for coexistence are overlapping generations and fluctuating birth rates which ensure that each species has periods when it is increasing. It is found that a species may persist provided only that it is favored sufficiently by the environment during favorable periods independently of the extent to which the other species is favored during its favorable periods.Coexistence is defined in terms of the stochastic boundedness criterion for species persistence. Using the lottery model as an example this criterion is justified and compared with other persistence criteria. Properties of the stationary distribution of population density are determined for an interesting limiting case of the lottery model and these are related to stochastic boundedness. An attempt is then made to relate stochastic boundedness for infinite population models to the behavior of finite population models.     

Prediction of multi-locus inbreeding coefficients and relation to linkage disequilibrium in random mating populations

An algorithm to predict the level of identity by descent simultaneously at multiple loci is presented, which can in principle be extended to any number of loci. The model assumes a random mating population, with random association of haplotypes. The relationship is shown between coefficients of multi-locus identity or non-identity by descent and moments of multi-locus linkage disequilibrium. Thus, these moments can be computed from the multilocus identity or, using algorithms derived previously to predict the disequilibria moments, vice-versa. The results can be applied to predict multi-locus identity in, for example, gene mapping.     

Diffusion approximations for one-locus multi-allele kin selection,mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size. In memory of Sam Karlin.     

The biennial life strategy in a random environment

A discrete-time population model with two age classes is studied which describes the growth of biennial plants in a randomly varying environment. A fraction of the oldest age class delays its flowering each year. The solution of the model involves products of random matrices. We calculate the exact mean and variance of the long-run geometric growth rate assuming a gamma distribution for the random number of offspring per flowering plant after one year. It is shown, both by analytical calculation and numerical examples, that it is profitable for the population to delay its flowering, in the sense that the average growth rate increases and the extinction probability decreases. The optimal values of the flowering fraction depend upon the environmental and model parameters.     

Equilibrium selection in evolutionary games with random matching of players

We discuss stochastic dynamics of populations of individuals playing games. Our models possess two evolutionarily stable strategies: an efficient one, where a population is in a state with the maximal payoff (fitness) and a risk-dominant one, where players are averse to risks. We assume that individuals play with randomly chosen opponents (they do not play against average strategies as in the standard replicator dynamics). We show that the long-run behavior of a population depends on its size and the mutation level.     

Stability results for delayed-recruitment models in population dynamics

This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.     

The many-demes limit for selection and drift in a subdivided population

A diffusion approximation is obtained for the frequency of a selected allele in a population comprised of many subpopulations or demes. The form of the diffusion is equivalent to that for an unstructured population, except that it occurs on a longer time scale when migration among demes is restricted. This many-demes diffusion limit relies on the collection of demes always being in statistical equilibrium with respect to migration and drift for a given allele frequency in the total population. Selection is assumed to be weak, in inverse proportion to the number of demes, and the results hold for any deme sizes and migration rates greater than zero. The distribution of allele frequencies among demes is also described.     

The biennial life strategy in a random environment

In a previous paper (J. Math. Biol. 26, 199–215 (1988)) we calculated the mean and variance of the long-run geometric growth rate of a discrete-time population model with two age classes in a random environment. The formula which was used in that paper as the starting point for the computation of the variance represents only the contribution of the one-period variances. Here we supplement these results by a calculation of the exact variance. All qualitative conclusions reached before are unaffected.     

Best linear prediction of breeding values in a forest tree improvement program

Summary Best Linear Prediction (BLP) was used to predict breeding values for 1,396 parents from progeny test data in an operational slash pine breeding program. BLP rankings of parents were compared to rankings of averaged standard scores, a common approach in forestry. Using BLP rankings, selection of higher ranking parents tends to choose parents in a larger number of more precise progeny tests. The trend is the opposite with standard scores; higher ranking parents tend to be those in fewer, less precise tests. BLP and a related methodology, Best Linear Unbiased Prediction (BLUP), were developed by dairy cattle breeders and have not been used widely outside of animal breeding for predicting breeding values from messy progeny test data. Application of either of these techniques usually requires simplifying assumptions to keep the problem computationally tractable. The more appropriate technique for a given application depends upon which set of assumptions are better for the given problem. An assumption of homogeneous genetic and error variances and covariances, generally made by animal breeders when applying BLUP, was inappropriate for our data. We employed an approach that treated fixed effects as known and treated the same trait measured in different environments as different traits with heterogeneous variance structures. As tree improvement programs become more complex, the ease with which BLP and BLUP handle messy data and incorporate diverse sources of information should make these techniques appealing to forest tree breeders.     

Restricted best linear unbiased prediction and a selection model

Summary An equivalence between restricted best linear unbiased prediction (and thus restricted selection index) and a particular example of a selection model is presented. Specifically, the equivalence is between restricted selection and a model of selection on the residuals of the general mixed linear model. This result illustrates that restricted selection acts by nonrandomly sampling those genes that act pleiotropically in multiple trait genetic models. An expression for a mixed linear model which includes restrictions is also presented.     

Effects of random and non-random errors on phenotypic selection in autotetraploids

Summary A theoretical investigation was made to ascertain the effects of random and non-random deviations, called errors, of phenotypic from genotypic values on population means and on the response to phenotypic recurrent selection. The study was motivated as a selection experiment for disease resistance where there was either variability in the inoculation or environment (the random errors) or where the inoculation was above or below the the optimum rate where genetic differences in resistance are maximized (the non-random errors). The study was limited to the genetics at a diallelic locus (alleles B and b) in an autotetraploid population in random mating equilibrium. The response to selection was measured as the covariance of selection and compared to the exact covariance which was the covariance of selection without errors in phenotype. The random errors were modeled by assuming that a given percentage () of the population was uniformly distributed among the five possible genotype classes independent of their true genotypes. This model was analyzed numerically for a theoretical population with the frequency of the B allele (p) ranging from 0.0 to 1.0 and assumed errors of=0.1 and 0.5 for the following six types of genic action of the B allele: additive, monoplex dominance, partial monoplex dominance, duplex dominance, partial duplex dominance, and recessive. The effect of random error was to consistently reduce the response to selection by a percentage independent of the type of genic action at the locus. The effect on the population mean was an upward bias when p was low and a downward bias when p approached unity. In the non-random error model below optimum inoculations altered the phenotypes by systematically including percentage of susceptible genotypes into one or more other genotype classes with more genetic resistance (a positive shift). With above optimum inoculations, some resistant genotypes are classed with the non-resistant genotypes (a negative shift). The effects on the covariance of selection were found by numerical analysis for the same types of genic action and's as investigated for random error. With a negative shift and a low p, the covariance of selection was always reduced, but for an increasing p the covariance approached and exceeded the exact covariance for all types of genic action except additive. With a positive shift and a low p, response to selection was greatly improved for three types of genic action: duplex dominance, partial duplex dominance, and recessive. The effect of a non-random error on population means was to greatly bias the means upwards for a low p and positive shift, but with increasing p the bias decreased. A relatively slight decrease in the mean occurred with a negative shift. This study indicated check varieties commonly used to monitor selection pressures in screening programs are very responsive to positive non-random shifts, but are relatively unresponsive to negative shifts. The interaction of selection pressure, types of genic action, and genotypes in the class shift models was suggested as a partial explanation for the lack of response to increasing selection pressures observed in some breeding programs.Cooperative investigations of the Alfalfa Production Research Unit, United States Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, Nevada. Paper No. 404 Scientific Journal Series. Nevada Agricultural Experiment Station     

Timescales of population rarity and commonness in random environments

This is a mathematical study of the interactions between non-linear feedback (density dependence) and uncorrelated random noise in the dynamics of unstructured populations. The stochastic non-linear dynamics are generally complex, even when the deterministic skeleton possesses a stable equilibrium. There are three critical factors of the stochastic non-linear dynamics; whether the intrinsic population growth rate (lambda) is smaller than, equal to, or greater than 1; the pattern of density dependence at very low and very high densities; and whether the noise distribution has exponential moments or not. If lambda < 1, the population process is generally transient with escape towards extinction. When lambda > or = 1, our quantitative analysis of stochastic non-linear dynamics focuses on characterizing the time spent by the population at very low density (rarity), or at high abundance (commonness), or in extreme states (rarity or commonness). When lambda >1 and density dependence is strong at high density, the population process is recurrent: any range of density is reached (almost surely) in finite time. The law of time to escape from extremes has a heavy, polynomial tail that we compute precisely, which contrasts with the thin tail of the laws of rarity and commonness. Thus, even when lambda is close to one, the population will persistently experience wide fluctuations between states of rarity and commonness. When lambda = 1 and density dependence is weak at low density, rarity follows a universal power law with exponent -3/2. We provide some mathematical support for the numerical conjecture [Ferriere, R., Cazelles, B., 1999. Universal power laws govern intermittent rarity in communities of interacting species. Ecology 80, 1505-1521.] that the -3/2 power law generally approximates the law of rarity of 'weakly invading' species with lambda values close to one. Some preliminary results for the dynamics of multispecific systems are presented.     

Expected utility maximization and selection of stable plant cultivars

Summary In most plant breeding programs, selection of the best commercially suitable cultivars for a target group of environments is based on information obtained from evaluation trials cultivated in a sample of environments. Information on the performance of cultivars collected in a sample of environments can only be approximate and, consequently, selection of the best cultivar involves choosing among cultivars that respond uncertainly in many environments. The agronomic and/or economic value of a cultivar across environments may be considered the general or overall utility of the cultivar. Data from a sample of environments therefore provides only an estimate of any cultivar's overall utility, with the overall goal of selection among all cultivars being the maximization of the expected utility. Within this frame-work, expected utility maximization, an approach to decision making that has been well developed in the disciplines of economics and statistics, can assist the plant breeder in making such decisions. This research was initiated (1) to determine how expected utility maximization might be used to develop indices that are useful for selecting broadly adapted plant cultivars, and (2) to determine how the breeder's preferences might affect choice of the best cultivar. The data used in this research were from USDA Regional Soybean Tests. The results indicated that expected utility maximization, which explicitly incorporates into the selection rule the plant breeder's preferences regarding stability, can be a useful aid in the selection of stable plant cultivars.     

Efficiency of including first-generation information in second-generation ranking and selection: results of computer simulation

Using computer simulation, we evaluated the impact of using first-generation information to increase selection efficiency in a second-generation breeding program. Selection efficiency was compared in terms of increase in rank correlation between estimated and true breeding values (i.e., ranking accuracy), reduction in coefficient of variation of correlation coefficients (i.e., ranking reliability), and increase in realized gain, with best linear unbiased prediction (BLUP). The test populations were generated with varying parameters: selection strategy (forward vs backward selection of parents); number of parents (24∼96); number of crosses per parent (1∼8); heritability (0.05∼0.35); ratio of dominance to additive variance (0∼3); ratio of additive-by-site to additive variance (0∼3); and ratio of dominance-by-site to additive variance (0∼3). The two selection strategies gave distinct results. When parents of the second-generation crosses had been selected via backward selection, adding first-generation information markedly increased selection efficiency. Conversely, when parents had been selected via forward selection, first-generation information provided little increase in efficiency. The amount of increase depended more on heritabilities in both generations and less on dominance and genotype–by–environment effects. Including first-generation information helped more when there were many parents and few crosses per parent in the second generation. Only in the case of extremely low first-generation heritabilities was there no benefit to adding first-generation information in terms of improved ranking reliability and accuracy.     

Extinction in relation to demographic and environmental stochasticity in age-structured models

The demographic variance of an age-structured population is defined. This parameter is further split into components generated by demographic stochasticity in each vital rate. The applicability of these parameters are investigated by checking how an age-structured population process can be approximated by a diffusion with only three parameters. These are the deterministic growth rate computed from the expected projection matrix and the environmental and demographic variances. We also consider age-structured populations where the fecundity at any stage is either zero or one, and there is neither environmental stochasticity nor dependence between individual fecundity and survival. In this case the demographic variance is uniquely determined by the vital rates defining the projection matrix. The demographic variance for a long-lived bird species, the wandering albatross in the southwestern part of the Indian Ocean, is estimated. We also compute estimates of the age-specific contributions to the total demographic variance from survival, fecundity and the covariance between survival and fecundity.     

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population with random mating

Global convergence to equilibria is proved for the model in the title under the assumption that death rates (and fertilities) are constant and fertilities are additive and positive. The phase portraits are determined. There is at most one polymorphic equilibrium or there are a continuum of equilibria.     

Evaluation of parents and selection for heat tolerance in the early generations of a potato (Solanum tuberosum L.) breeding program

Summary Potato cultivars (Solanum tuberosum L.) were evaluated for their capacity to form tubers under high temperature conditions (heat tolerance). In an experiment conducted in the hot summer season in the field and in experiments conducted under controlled environments in glasshouses, differences in tolerance to heat were noted among the various genotypes. Although heat tolerance tended to be associated with earliness in maturation, differences in the response to high temperatures were observed in genotypes of the same maturity class. Progeny of the cultivars Desiree and Cara, and hybrid progeny of Cara x Desiree and of Blanka x Cara were screened for heat tolerance in controlled environments in glasshouses. Long photoperiod delayed tuberization but high temperatures strongly inhibited tuber formation, which was 4% in the progeny of Desiree, <2% in the progeny of Cara and of Cara x Desiree, and ca. 16% in progeny of Blanka x Cara. For seedlings that did not form tubers in the hot glasshouse but were subsequently transferred to the cool glasshouse, tuberization was much enhanced and clones selected through this dual procedure exhibited tolerance to heat in the first conal year. The results of this study indicate that: (a) tolerance to heat is associated with earliness; however, differences in the response to heat exist among late-maturing cultivars; (b) tolerance to heat, as assessed by the capacity to form tubers under high temperatures, is genetically controlled, and the timing of tuberization might be independent of the rate of tuberization in hybrid progeny; (c) the incidence of heat tolerance in progeny population is correlated with the relative tolerance of the parents; and (d) the dual selection procedure is an efficient approach for the selection of heat-tolerant clones.     

Coexistence and error propagation in pre-biotic vesicle models: a group selection approach

Compartmentalization of unlinked, competing templates is widely accepted as a necessary step towards the evolution of complex organisms. However, preservation of information by templates confined to isolated vesicles of finite size faces much harder obstacles than by free templates: random drift allied to mutation pressure wipe out any template that does not replicate perfectly, no matter how small the error probability might be. In addition, drift alone hinders the coexistence of distinct templates in a same compartment. Here, we investigate the conditions for group selection to prevail over drift and mutation and hence to guarantee the maintenance and coexistence of distinct templates in a vesicle. Group selection is implemented through a vesicle survival probability that depends on the template composition. By considering the limit case of an infinite number of vesicles, each one carrying a finite number of templates, we were able to derive a set of recursion equations for the frequencies of vesicles with different template compositions. Numerical iteration of these recursions allows the exact characterization of the steady state of the vesicle population-a quasispecies of vesicles-thus revealing the values of the mutation and group selection intensities for which template coexistence is possible. Within the main assumption of the model-a fixed, finite or infinite, number of vesicles-we find no fundamental impediment to the coexistence of an arbitrary number of template types with the same replication rate inside a vesicle, except of course for the vesicle capacity. Group selection in the form of vesicle selection is a must for compartmentalized primordial genetic systems even in the absence of intra-genomic competition of different templates.