A stochastic computer model for simulating population growth

A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model's behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka's Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values or r_m and K. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetle Tribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

Fitness of Allozyme Variants in DROSOPHILA PSEUDOOBSCURA. I. Selection at the PGM-1 and ME-2 Loci

We have studied in Drosophila pseudoobscura the effect of allozyme variation on seven fitness components: female fecundity, egg hatchability, egg-to-adult survival under near-optimal and under competitive conditions, rate of development under near-optimal and under competitive conditions, and mating capacity of males. Three genotypes at each of two loci, Pgm-1 and Me-2, have been studied in various combinations. These two loci are highly polymorphic in natural populations of D. pseudoobscura. Statistically significant differences involving one or more genotypes exist for all components of fitness. No single genotype is best for all fitness components; rather the relative fitnesses of genotypes are reversed when different parameters are considered, or when they are studied in different environmental conditions. Also, the average egg-to-adult survival and rate of development are better when different genotypes are reared together than when they occur in pure culture. Four different modes of selection have been uncovered by our experiments. These forms of selection may account for the persistence of the two allozyme polymorphisms in nature, and for previously observed seasonal fluctuations of the allelic frequencies in natural populations.     

Colony-level selection in the social insects: Single locus additive and nonadditive models

Genetic models of colony-level selection applicable to diploids (termites) and haplodiploids (social Hymenoptera) are analysed. In the Additive model colony fitnesses are just the arithmetic average of the contribution of the worker genotypes. In the Nonadditive model the fitness of the heterogenotypic colonies (those comprised of more than one worker genotype) may be altered due to interaction between the different worker genotypes. This is modelled by multiplying the additive fitness by the variable, e_i. With additive selection the same equilibrium gene frequency occurs in diploids and in haplodiploids with both once and twice mated queens. In haplodiploids if selection is nonadditive and strong, up to three polymorphic equilibria can exist; however, only a maximum of two are possible with weak selection. Multiple mating by queens increases the number of equilibria possible. Worker-produced males alter the conditions for the existence of a polymorphic equilibrium, and shift the male and female equilibrium gene frequencies.     

The influence of self‐fertilization and grouping on fitness attributes in the freshwater snail Physa acuta: population and individual inbreeding depression

The genetic structure, selfing rate and inbreeding depression of the hermaphroditic freshwater snail Physa acuta were jointly analysed in a population near Montpellier, France. Allozymic markers revealed moderate gene diversity (0.138), and no heterozygote deficiency. The mean outcrossing rate, estimated by using progeny arrays, was 0.9, with substantial variation among families. This also suggests that the number of fathers among outcrossed offspring of a given mother is low. Inbreeding depression was estimated over more than one generation using 83 first‐laboratory‐generation (G₁) families. The main parameters measured were parental (G₁) fecundity, offspring (G₂) survival and fecundity. Size and growth were also monitored. Parental fecundity was analysed under several conditions (isolation, pair and quadruplet outcrossing). The self‐fertilization depression, including parental fecundity, offspring survival and fecundity, was about 0.9 at the population level. The genetic data obtained in the same population indicate a value of about 0.3 using Ritland’s (1990) technique, suggesting that the depression over the whole life‐cyle might be even higher than 0.9. Grouping affected neither fecundity nor self‐fertilization depression. Substantial variation in depression for survival was detected among individuals, from no survival in some selfed families to better survival than that of outbred families in others. The overall result (outbred population structure, high outcrossing rate and high self‐fertilization depression) is consistent with what is expected in large outcrossing populations in which inbreeding depression is maintained by mutation‐selection balance.     

Estimation of effective generation interval in Drosophila population cages

An approximation method for calculating the effective generation interval in populations with overlapping generations was presented and illustrated with Drosophila melanogaster data where a sustained heterosis (or pseudo-overdominance) was observed for sex-linked genes balanced by an inversion. An equilibrium frequeney was established in replicated 2-bottle population eages which was different for males and females. The fitnesses of the genotypes were thus estimable and the expected gene frequencies with diserete generations were calculated. The effective generation interval was estimated by fitting the observed frequencies to those expected by minimizing the lack of fit chisquare. The best fit resulted if the interval was allowed to increase as the population approached a stable age distribution. The best estimate indicated that there was an average interval of 14.75 days per generation in the first 5 generations and 16.25 days thereafter. This estimate is particular to 2-bottle eages; as the number of bottles increases the expected generation interval is also expected to inerease.Journal Paper Number 79-5-211 of the University of Kentucky Agricultural Experiment Station, and Journal Paper Number 7942 of the Purdue University Agricultural Experiment Station.     

Dietary Effects on Fitness Components at the PGM-1 Locus of TRIBOLIUM CASTANEUM

The pattern of genetic differentiation among experimental populations of the flour beetle Tribolium castaneum suggested the hypothesis that relative fitness of three genotypes at the PGM-1 locus (or other linked loci) depends directly on diet. This hypothesis was tested by measuring several fitness components (developmental time, survival, fecundity, rate of egg cannibalism) on groups of individuals differing at the PGM-1 locus that were reared on three types of flour (wheat, corn and a mixture of wheat, corn, barley and rye). Flour type had large effects on all traits except larval survival to 3 weeks of age. Relative fitnesses of the three genotypes differed significantly for fecundity. Diet was found to significantly influence the relative developmental times of the three genotypes.     

Evaluation of Spanish Arundo scale Rhizaspidiotus donacis (Hemiptera; Diaspididae) survival and fecundity on three new world genotypes of Arundo donax (Poaceae; Arundinoideae)

A pre-release evaluation of survival and fecundity of the arundo scale, Rhizaspidiotus donacis, was conducted on three invasive genotypes of the riparian weed, Arundo donax. The three A. donax genotypes were collected from Laredo, Austin and Balmorhea, TX, which represented the majority of the genotypic diversity found in Texas watersheds. Although R. donacis developed on all three genotypes of the plant, the Austin A. donax genotype, followed by the Laredo genotype, were the most suitable in terms of the size of first-generation immature and adult scale populations that developed after crawler release. Both the Laredo and Austin genotypes of A. donax are likely to be of Spanish origin and are close genetic matches with scale's original host plant genotype in Alicante, Spain. In comparison, survival was lowest on the phylogenetically distant genotype of A. donax from Balmorhea, TX. Although the population size of settled, immature second-generation scales varied in a manner similar to that of the first generation, the fecundity of isolated first-generation females was not significantly different across the three plant genotypes, suggesting that R. donacis is not a genotype specialist in terms of nutrient assimilation for reproduction. Rather, differences in genotype suitability affect rates of success of crawler settling. These results indicate that selection of scale genotype from the native range may have a moderate influence on the success of R. donacis and ultimately the biological control programme.     

Population Dynamics of Evolutionary Change: Demographic Parameters as Indicators of Fitness

I evaluated demographic parameters as indicators of fitness by calculating the net reproductive rate (R₀), exponential rate of change (r), lifetime reproductive success (LRS), and Malthusian parameter (m) for nine genotypes and four phenotypes (two alleles at each of two independent loci) of an age-structured population. The given starting conditions included age-specific survival rates of males and females and age-specific fecundity of females for each genotype (to simplify the problem I presumed no differences in survivorship or fecundity of genotypes with the same phenotype) and the same age structure for each genotype. The prevailing genotype had the greatestm, but it did not have the greatestr,R₀, or LRS, or even the greatest survivorship of either juveniles or adults, or the greatest fecundity. This result indicates thatmis the only correct measure of fitness (i.e., as a predictor of which genotype should prevail from among a group of genotypes) and that comparisons ofr,R₀, LRS, juvenile or adult survival rates, or fecundity may be misleading indicators of which genotype should prevail (i.e., be most “fit”) over time (i.e., be selected for).     

Proportions of different habitat types are critical to the fate of a resistance allele

We describe a simple deterministic theoretical framework for analysing the gene frequency evolution of two alternative alleles at a single genetic locus in a habitat comprising two environments in which the genotypes have different relative fitnesses. We illustrate this for adaptation of pest insects, where one allele (resistance to toxins expressed in transgenic crops) is favoured in one environment (transgenic plants) and the other allele (susceptibility to toxins) is favoured in the other environment (‘refuges’ of non-transgenic plants). The evolution of allele frequencies depends on selection pressure because of relative sizes of the environments and relative fitnesses of the genotypes in each environment. We demonstrate that there are critical threshold proportions for habitat division that determine equilibrium allele frequencies. The stability of the system depends on relationships between the relative genotype fitnesses. In some cases, the division of the habitat in exactly the threshold proportions removes selection pressure and maintains polymorphism at all allele frequencies.     

Convergence of multilocus systems under weak epistasis or weak selection

 The convergence of multilocus systems under viability selection with constant fitnesses is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. It is proved that if epistasis or selection is sufficiently weak (and satisfies a certain nondegeneracy assumption whose genericity we establish), then there is always convergence to some equilibrium point. In particular, cycling cannot occur. The behavior of the mean fitness and some other aspects of the dynamics are also analyzed. Received: 15 November 1997 / Revised version: 25 May 1998     

Genetic changes in a barley population analyzed in terms of some life cycle components of selection

A bulk-hybrid population of barley (CC-XXI) was investigated for changes over a period of six generations, in (1) generation means and variances for three quantitative characters (2) phenotypic and genotypic frequencies at several marker loci and (3) the components of selection at several life cycle stages (germination and seedling survival, fertility, fecundity) for the dominant-recessive phenotypic classes at three of the above loci. The changes in mean and variance for the quantitative characters appeared to be due to both directional and stabilizing types of selection. The estimates of selective values from frequency data suggested heterozygote advantage along with unequal homozygote fitnesses at locuss and heterozygote disadvantage at locibl, r andv, from computations based on higher outcrossing rates in the presence of male-steriles.Of the components studied, fertility and fecundity appeared to account for a larger portion of the total selective differential than the pre-adult stages (germination and seedling establishment). Net selective values based on component analysis approximated the selective values obtained from the frequency data for locusr. For locis, andv, however, the two estimates were often quite different, although they generally agreed in direction. The discrepancies between the two sets of estimates were discussed in terms of the complexity of overall measures of fitness.     

Fitness of allozyme variants in Drosophila pseudoobscura III. Possible factors contributing to the maintenance of polymorphisms in nature

Experiments have been performed to investigate the mechanisms maintaining enzyme polymorphisms in natural populations. We have measured effects on fitness of genotypic variants at three loci, Est-5, Odh, and Mdh-2, in D. pseudoobscura. Significant differences exist among the genotypes in the rate of development from egg-to-adult; there is also indication of differences in larval survival. In a population segregating for allelic variants at all three loci, there is indication that segregation distortion at meiosis or some form of gametic selection might be involved. The relative fitnesses of alternative genotypes are reversed when either different fitness components are considered, or the genotypic frequencies are changed, or the larval density is increased. These fitness reversals may contribute to the maintenance of the polymorphisms, and may account for cyclical oscillations of allozyme frequencies observed in natural populations.Research supported by U.S. Public Health Service Research Fellowship (1F05 TWO 1991-01) to D.M. and by contract AT(04-3)34 with the U.S. Atomic Energy Commission. Adress reprint requests from Europe to D.M.; from elsewhere to F.J.A.     

Fitness relationships under different temperature regimes in Drosophila melanogaster: the eyeless/shaven-naked genetic system

Genotypic fitnesses were estimated over the temperature range 15°C to 29°C for genotypes of the eyeless/shaven-naked system. Total fitness was determined directly from estimates of mating ability, fecundity and egg-to-adult development time and viability, by gene frequency changes in discrete generation populations and in a single generation population experiment involving culture on a rotational basis at 29°C and 15°C. Genotypic differences were detected for mating ability and egg to adult development time and survival. Heterozygote advantage was observed for total fitness and this effect was greatest at 15°C and for culture on a rotational basis at 29°C and 15°C. There was evidence for genetic associations among some fitness components. The tendency for heterozygote advantage in extreme environments supports the general observation of high expressed genetic variation for fitness under extreme stresses. The results suggest an approach to the understanding of the genetic basis of fitness variation in natural populations based upon direct assessments of environmental stresses of ecological importance.     

An experimental analysis of the relationships between fitness components and malic enzyme genotypes inTribolium confusum

The hypothesis that natural selection is capable of maintaining allozyme variation in natural populations was tested using a species of flour beetles,Tribolium confusum. We selected a polymorphic locus (a locus encoding variation for malic enzyme) in an experimental population ofT. confusum and scored the genotypes at this locus for a series of fitness components on different flour types. Measurements included survival rate, development time, fecundity, and rate of egg cannibalism. Flour type had significant effects on most traits. Significant differences among genotypes for fecundity and rates of egg cannibalism and the presence of genotype × flour type interactions for development time were demonstrated. Thus, changes in allele frequencies at the malic enzyme locus could in part be under the influence of natural selection. The existence of genotype × flour type interactions suggests that environmental heterogeneity could maintain allozyme variation at the malic enzyme locus.     

The maintenance of gynodioecy and androdioecy in angiosperms

Algebraic models of gynodioecy show that the effects on the equilibrium sex ratio of the relative survival and seed production of the sexes and of inbreeding of male-fertile plants are identical for all genic modes of inheritance, provided that different genotypes among male-fertile plants (or among females) do not differ in average fitness. The effects of three modes of inbreeding on equilibrium sex ratios are examined. If there is competition between self- and cross-fertilization of male-fertile individuals, a stable sexual dimorphism can be maintained by an outbreeding advantage of females if both the proportion of cross-fertilized seeds among those borne on male-fertile individuals,t, and the inbreeding depression (fitness inbred/outbred seeds),i, are less than one half. A lower frequency of females is obtained for the same values oft andi if self-fertilization precedes cross-fertilization. If self-fertilization follows cross-fertilization, gynodioecy cannot be maintained by an outbreeding advantage of females. When the sex phenotypes of gynodioecious populations are determined by cytoplasmic inheritance, females need only a slight advantage over males in survival, ovule production or outbreeding to persist at equilibrium. When determined by nuclear genes, androdioecy can be maintained by greater fecundity or a higher survival rate of males than of female-fertile plants, but not by an outbreeding advantage. Androdioecy cannot be maintained with cytoplasmic inheritance of sex. The models suggest explanations for the more frequent occurrence of gynodioecy than of andrdioecy, the high frequency of gynodioecy in Hawaii and New Zealand, and the origin of gynodioecy from hermaphrodite but not from monoecious ancestors.     

Fecundity,reproductive rate,longevity, and intrinsic rate of increase of an aphidiid parasitoidLysiphlebia mirzai

A life-table was constructed for a little known aphidiid waspLysiphlebia mirzai, a parasitoid of cereal aphid,Rhopalosiphum maidis. The female parasitoid survived 6.4 ± 1.17 (SD) days and oviposited intensively 4.0 ± 0.47 days. The total fecundity rate, R_t, was 169.2 ± 6.94 mummies/female and net reproductive rate, R_o, was 92.70 female offspring/female. The intrinsic total fecundity rate, r_t, and intrinsic rate of natural increase, r_m, the finite rate of total fecundity, λ^t, and finite rate of increase, λ^m, was 0.27048, 0.24155, 1.31059 and 1.27322 respectively. The mean generation time (18.75 days) and doubling time of the population (2.87 days) was slightly higher than other aphidiids studied so far. The proportion of female progenies decreased significantly on the successive oviposition days.      

Evolutionary game dynamics in a Wright-Fisher process

Evolutionary game dynamics in finite populations can be described by a frequency dependent, stochastic Wright-Fisher process. We consider a symmetric game between two strategies, A and B. There are discrete generations. In each generation, individuals produce offspring proportional to their payoff. The next generation is sampled randomly from this pool of offspring. The total population size is constant. The resulting Markov process has two absorbing states corresponding to homogeneous populations of all A or all B. We quantify frequency dependent selection by comparing the absorption probabilities to the corresponding probabilities under random drift. We derive conditions for selection to favor one strategy or the other by using the concept of total positivity. In the limit of weak selection, we obtain the 1/3 law: if A and B are strict Nash equilibria then selection favors replacement of B by A, if the unstable equilibrium occurs at a frequency of A which is less than 1/3.     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

One selectionist's perspective

Following introductory comments expressing doubts about the validity of genetic load and Haldane's "cost of natural selection," the role of selection (expressed as the average number of adult daughters per female) on gene frequencies in populations has been partitioned into population and time arenas. The population arena (a geometric plane) deals with the fitnesses of different genotypes under the many situations encountered by individual members of the population in a single generation; average fitnesses of carriers of various genotypes are obtained by calculating across these many situations. The population arena includes the point signifying that, on the average, each mother leaves one daughter as her replacement within the population. It is the plane within which evolutionarily significant norms of reaction exist. The time arena is also a (geometric) plane, one that is composed of the edge-on limit (average fitness) of each successive population arena. It does not include the effects of individual situations on relative fitnesses within each population arena; it encompasses only the temporal sequence of average relative fitnesses. Amino acid substitutions in proteins and base-pair substitutions in DNA are events of concern in the time arena; within the population arena, however, gene action (not merely gene structure) is a matter of considerable concern. Thus, the discussions of the 1950s and 1960s regarding genetic variation which were reasonable within the population arena seem less so within the time arena where structural, rather than functional, variation is stressed. The function-structure dichotomy is entangled with the neutralist-selectionist controversy.     

Male-female interactions in Drosophila melanogaster: model with one-locus and three-alleles

We develop a fertility model of fitness that is general in that it does not assume that the fitnesses of the mating combinations are symmetrical or that they are additive or multilicative (i. e., that they can be inferred from fitnesses of the two genotypes involved in a mating). %he model considers one locus with three alleles. An experimental test with Drosophila rnelanogaster confirms that the fitnesses of the mating types depart from both additivity (or multiplicativity) and symmetry although this last property is of no consequence for the development of analytical models). urnerical simulations yield the same, or very nearly the same, equilibrium freuencies as the analytical model, independently of whether or not Hardy-Weinberg equilibrium Trequencies are assumed at the beginning of each selection cycle.