Search

Population densities of a species measured in different locations are often correlated over time, a phenomenon referred to as synchrony. Synchrony results from dispersal of individuals among locations and spatially correlated environmental variation, among other causes. Synchrony is often measured by a correlation coefficient. However, synchrony can vary with timescale. We demonstrate theoretically and experimentally that the timescale-specificity of environmental correlation affects the overall magnitude and timescale-specificity of synchrony, and that these effects are modified by population dispersal. Our laboratory experiments linked populations of flour beetles by changes in habitat size and dispersal. Linear filter theory, applied to a metapopulation model for the experimental system, predicted the observed timescale-specific effects. The timescales at which environmental covariation occurs can affect the population dynamics of species in fragmented habitats.

Populations fluctuate because of their internal dynamics, which can be nonlinear and stochastic, and in response to environmental variation. Theory predicts how the colour of environmental stochasticity affects population means, variances and correlations with the environment over time. The theory has not been tested for cycling populations, commonly observed in field systems. We applied noise of different colours to cycling laboratory beetle populations, holding other statistical properties of the noise fixed. Theory was largely validated, but failed to predict observations in sufficient detail. The main period of population cycling was shifted up to 33% by the colour of environmental stochasticity. Noise colour affected population means, variances and dominant periodicities differently for populations that cycled in different ways without noise. Our results show that changes in the colour of climatic variability, partly caused by humans, may affect the main periodicity of cycling populations, possibly impacting industry, pest management and conservation.

Stochasticity alters the nonlinear dynamics of inherently cycling populations. The power spectrum can describe and explain the impacts of stochasticity. We fitted models to short observed time series of flour beetle populations in the frequency domain, then used a well fitting stochastic mechanistic model to generate detailed predictions of population spectra. Some predicted spectral peaks represent periodic phenomena induced or modified by stochasticity and were experimentally confirmed. For one experimental treatment, linearization theory explained that these peaks represent overcompensatory decay of deviations from deterministic oscillation. In another treatment, stochasticity caused frequent directional phase shifting around a cyclic attractor. This directional phase shifting was not explained by linearization theory and modified the periodicity of the system. If field systems exhibit directional phase shifting, then changing the intensity of demographic or environmental noise while holding constant the structure of the noise can change the main frequency of population fluctuations.

A scaling rule of ecological theory, accepted but lacking experimental confirmation, is that the magnitude of fluctuations in population densities due to demographic stochasticity scales inversely with the square root of population numbers. This supposition is based on analyses of models exhibiting exponential growth or stable equilibria. Using two quantitative measures, we extend the scaling rule to situations in which population densities fluctuate due to nonlinear deterministic dynamics. These measures are applied to populations of the flour beetle Tribolium castaneum that display chaotic dynamics in both 20-g and 60-g habitats. Populations cultured in the larger habitat exhibit a clarification of the deterministic dynamics, which follows the inverse square root rule. Lattice effects, a deterministic phenomenon caused by the discrete nature of individuals, can cause deviations from the scaling rule when population numbers are small. The scaling rule is robust to the probability distribution used to model demographic variation among individuals.

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that “chaos” should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.

We use the concepts of composite maps, basins of attraction, basin switching, and saddle fly-by's to make the ecological hypothesis of the existence of multiple attractors more accessible to experimental scrutiny. Specifically, in a periodically forced insect population growth model we identify multiple attractors, namely, two locally stable 4-cycles. Using the model-predicted basins of attraction, we examine data time series from a Tribolium experiment for evidence of the multiple attractors. We conclude that the multiple attractor hypothesis together with demographic stochasticity accounts for the experimental observations.

A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, non- linear ecological interactions, provided such interactions could be correctly identified and modeled. The realization in recent decades that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attri- buting the fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experimentally tested a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. The predictions arose from a system of difference equations (the LPA model) describing the nonlinear life-stage interactions, predominantly cannibalism. We built a stochastic version of the model incorporating demographic vari- ability and obtained conditional least-squares estimates for the model parameters. We gen- erated 2000 ''bootstrapped data sets'' with a time-series bootstrap technique, and for each set we reestimated the model parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distri- butions of the Liapunov exponents for the deterministic portion (the skeleton) of the model as well as for the full stochastic model. Frequency distributions of estimated dynamic behaviors of the skeleton at each experimental treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder of the estimates yielded high-period cycles. The low-dimensional skeleton accounted for at least 90% of the variability in the population abundances and accurately described the responses of populations to experimental demographic manipulations, in- cluding treatments for which the predicted dynamic behavior was chaos. Demographic stochasticity described the remaining noise quite well. We conclude that the fluctuations of experimental flour beetle populations are explained largely by known nonlinear forces involving cannibalistic-stage interactions. Claims of dynamic behavior such as periodic cycles or chaos must be accompanied by a consideration of the reliability of the estimated parameters and a realization that the population fluctuations are a blend of deterministic forces and stochastic events.

Chaos has now been documented in a laboratory population. In controlled laboratory experiments, cultures of flour beetles (Tribolium castaneum) undergo bifurcations in their dynamics as demographic parameters are manipulated. These bifurcations, including a specific route to chaos, are predicted by a well-validated deterministic model called the “LPA model”. The LPA model is based on the nonlinear interactions among the life cycle stages of the beetle (larva, pupa and adult). A stochastic version of the model accounts for the deviations of data from the deterministic model and provides the means for parameterization and rigorous statistical validation. The chaotic attractor of the deterministic LPA model and the stationary distribution of the stochastic LPA model describe the experimental data in phase space with striking accuracy. In addition, model-predicted temporal patterns on the attractor are observed in the data. This paper gives a brief account of the interdisciplinary effort that obtained these results.

Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable 'saddle cycle' separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.

Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identifies boundaries in parameter space where stable equilibria bifurcate to periodic 2-cy- cles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the flour beetle Tribolium yield the time series data. A three-dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.

SIMPLE nonlinear models can generate fixed points, periodic cycles and aperiodic oscillations in population abundance without any external environmental variation. Another familiar theoretical result is that shifts in demographic parameters (such as survival or fecundity) can move a population from one of these behaviours to another1–4. Unfortunately, empirical evidence to support these theoretical possibilities is scarce5–15. We report here a joint theoretical and experimental study to test the hypothesis that changes in demographic parameters cause predictable changes in the nature of population fluctuations. Specifically, we developed a simple model describing population growth in the flour beetle Tribolium16. We then predicted, using standard mathematical techniques to analyse the model, that changes in adult mortality would produce substantial shifts in population dynamic behaviour. Finally, by experimentally manipulating the adult mortality rate we observed changes in the dynamics from stable fixed points to periodic cycles to aperiodic oscillations that corresponded to the transitions forecast by the mathematical model.

Motivated by the genetic hypothesis that natural selection results in the maximization of the equilibrium population size, we quantified this latter equilibrium for laboratory populations of the flour beetle Tribolium castaneum, using the gamma probability density function. Gamma density functions were fitted to adult numbers for each of the experimental treatments that were started with frequencies of the corn oil sensitive (cos) allele in the range 0–1 at intervals of 0.1. The gamma density function adequately described all observed distributions. However, contrary to theory, statistical comparisons of the fitted distribution indicate that the polymorphic populations did not converge to the same identical distribution and that the polymorphic populations are intermediate in population size to the two homozygous groups. The need for a stochastic theory that combines both population size and genetic selection is discussed.Key words: natural selection, Tribolium, gamma distribution, maximum population size, stationary distribution, stochastic differential equation, stochastic population model.

We explore the persistence of corn oil sensitivity in a population of the flour beetle Tribolium castaneum using evolutionary game methods that model population dynamics and changes in the mean strategy of a population over time. The strategy in an evolutionary game is a trait that affects the fitness of the organisms. Corn oil sensitivity represents such a strategy in the flour beetle. We adapt an existing model of the ecological dynamics of T. castaneum into an evolutionary game framework to explore the persistence of corn oil sensitivity in the population. The equilibrium allele frequencies resulting from the evolutionary game are evolutionarily stable strategies and compare favorably with those obtained from the experimental data.

A nonlinear demographic model was used to predict the population dynamics of the flour beetle Tribolium under laboratory conditions and to establish the experimental protocol that would reveal chaotic behavior. With the adult mortality rate experimentally set high, the dynamics of animal abundance changed from equilibrium to quasiperiodic cycles to chaos as adult-stage recruitment rates were experimentally manipulated. These transitions in dynamics corresponded to those predicted by the mathematical model. Phase-space graphs of the data together with the deterministic model attractors provide convincing evidence of transitions to chaos.

This chapter summarizes the concepts and models associated with chaos and bifurcations. It reviews the modeling methodology, nonlinear phenomenon, and dynamics of biological populations. Model assessment is important to distinguish between procedures to parameterize (calibrate) a model and those to validate a model. Part of the success of the deterministic LPA model can be attributed to the model's design around the dominant mechanism known to drive the dynamics of the species. Another aspect of the mechanistic modeling approach that is considered is the inclusion of stochasticity. The connection of the deterministic model to real data is accomplished by modeling the expected deviations of data from model predictions in terms of probabilistic assumptions concerning environmental and demographic sources of noise. One conclusion drawn from the studies is that the deterministic LPA model provides a remarkably accurate description of the flour beetle cultures under various laboratory conditions. It made accurate a priori predictions of the long-term dynamics of beetle populations when demographic parameters of the populations are changed by direct manipulation. One of the features of nonlinearity is that responses are not (necessarily) proportional to disturbances; they might be very unexpected and nonintuitive. Animal numbers or densities come from a finite set of discrete numbers called lattice. The LPA model provides an accurate explanation of patterns. The chapter provides insights into nonlinear population dynamics and a modest step toward the hardening of ecological science.

When observation and theory collide, scientists turn to carefully designed experiments for resolution. Their motivation is especially high in the case of biological systems, which are typically far too complex to be grasped by observation and theory alone. The best procedure, as in the rest of science, is first to simplify the system, then to hold it more or less constant while varying the important parameters one or two at a time to see what happens. —Edward O. Wilson (2002)

This chapter highlights one of the chaotic treatments in the route-to-chaos experiment. A study of complex dynamics—in particular, chaotic dynamics—requires a sufficiently long time series of data. In this experiment, chaotic treatment is considered in more detail by examining some temporal patterns predicted by the chaotic attractor. Sensitivity to initial conditions is the subject of a follow-up experiment designed to test hallmark property of chaos. The model-predicted consequences of small perturbations to population numbers near the most sensitive region of the chaotic attractor—the hot spot—are dramatically borne out by the experimental data. In addition to corroborating the influence of the chaotic attractor on the beetle populations, the hot-spot experiment also serves as an interesting demonstration of controlling chaos. Temporal patterns in chaos are complicated. The investigation of temporal patterns in the chaos treatment leads to another insights into modeling the dynamics of biological populations. Recurrent temporal pattern that is observed in the experimental data accounts for the lattice effect, a pattern arising from the fact that animals come in whole numbers and data necessarily lie on a discrete lattice of points in state space.

Lattice effects in ecological time-series are patterns that arise because of the inherent discreteness of animal numbers. In this paper, we suggest a systematic approach for predicting lattice effects. We also show that an explanation of all the patterns in a population time-series may require more than one deterministic model, especially when the dynamics are complex.

Animals and many plants are counted in discrete units. The collection of possible values (state space) of population numbers is thus a nonnegative integer lattice. Despite this fact, many mathematical population models assume a continuum of system states. The complex dynamics, such as chaos, often displayed by such continuous-state models have stimulated much ecological research; yet discrete-state models with bounded population size can display only cyclic behavior. Motivated by data from a population experiment, we compared the predictions of discrete-state and continuous-state population models. Neither the discrete- nor continuous-state models completely account for the data. Rather, the observed dynamics are explained by a stochastic blending of the chaotic dynamics predicted by the continuous-state model and the cyclic dynamics predicted by the discrete-state models. We suggest that such lattice effects could be an important component of natural population fluctuations.

The study of population dynamics has opened the way to new explanations of data, to new procedures for data analyses, and will surely lead to a new phase of research in which experiments are focused directly on phenomena such as stable equilibria, bifurcations, multiple attractors, limit cycles, life not lived, and stationary distributions. In this final chapter we speculate on the emerging discipline of population dynamics and the role of Tribolium as an experimental animal model.

In the deterministic and stochastic models of the previous chapters, population size was treated as a continuous variable. But animals come in discrete units. For large populations, a continuous variable may be a reasonable approximation; a model which predicts a population size of 11,924.38 is not much different from one that predicts a population of 11,924. But in small populations the difference may be important; a prediction of 5 animals is more reasonable than a prediction of 5.42. This is especially true when one considers the question of extinction, where the dynamics of small populations are of fundamental interest. A more realistic model should consider population size as a discrete variable restricted to the nonnegative integers.

Even under carefully controlled laboratory conditions, uncertainty is an inherent feature of Tribolium populations. Replicate populations which are initiated with the same initial age structures and maintained in the same incubator will exhibit variation in the densities of individuals at any given time. The densities in a single population will also fluctuate over time in an apparently random way. Deterministic models are incapable of describing this type of behavior. Variability must be described using a stochastic model.

It would be hard to imagine a much simpler experimental system. Place some flour and yeast in a glass container with a few graminivorous beetles. Keep the container in a dark incubator at a constant temperature and humidity. At regular intervals, count the beetles and change the flour. An incubator, a scale, and a few plates and sifters are all that is needed. Yet despite its simplicity, this experimental system has the potential for producing a variety of interesting population phenomena including stable equilibria, Hopf bifurcations, multiple attractors, and stable limit cycles. How the physiology and behavior of flour beetles of the genus Tribolium can lead to these topics is the subject of this chapter and the next.

1. We experimentally set adult mortality rates, μ a , in laboratory cultures of the flour beetle Tribolium at values predicted by a biologically based, nonlinear mathematical model to place the cultures in regions of different asymptotic dynamics. 2. Analyses of time-series residuals indicated that the stochastic stage-structured model described the data quite well. Using the model and maximum-likelihood parameter estimates, stability boundaries and bifurcation diagrams were calculated for two genetic strains. 3. The predicted transitions in dynamics were observed in the experimental cultures. The parameter estimates placed the control and μ a = 0.04 treatments in the region of stable equilibria. As adult mortality was increased, there was a transition in the dynamics. At μ a = 0.27 and 0.50 the populations were located in the two-cycle region. With μ a = 0.73 one genetic strain was close to a two-cycle boundary while the other strain underwent another transition and was in a region of equilibrium. In the μ a = 0.96 treatment both strains were close to the boundary at which a bifurcation to aperiodicities occurs; one strain was just outside this boundary, the other just inside the boundary. 4. The rigorous statistical verification of the predicted shifts in dynamical behaviour provides convincing evidence for the relevance of nonlinear mathematics in population biology.

Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.

The viability pattern of the unsaturated fatty acid sensitive mutant (cos/cos) of Tribolium castaneum was examined with reference to the normal genotype from egg hatching through adult emergence. The beetles were cultured on diets with 0, 3, or 5% corn oil in growth chambers maintained at 33 +/- 1 degree C and either 42 +/- 6% or 75 +/- 3% relative humidity. The pattern of the mutant was altered during the larval stage and was dependent on both the concentration of dietary fatty acid and the relative humidity of the culture chamber. Values of the viability component of fitness were assigned to the cos/cos, +/-/cos and +/-/+/- genotypes for multiple environments based on the number of eggs surviving to pupation. In large, random mating populations segregating at this locus, a stable genetic polymorphism is forecast. Predicted rates of change of allele frequency are dependent upon the culture conditions.

A new macroparameter characterizing the movement of a non-equilibrium initial population structure toward the equilibrium structure under the natural selection process was suggested in a previous paper (Ginzburg, 1977). This parameter, named selective delay, is directly measurable from the population size growth curve without any knowledge of the frequencies of the different genotypes in the population. An equation was proposed describing the relationship between selective delay, average population fitness at the equilibrium point and the entropy distance between initial and equilibrium states of the population. The purpose of this paper is an experimental check of this equation. Experimental results with populations of flour beetle Tribolium castaneum segregating at the unsaturated fatty acid sensitive locus have shown a good correspondence to the theory.

We develop a model of larval cannibalism of Tribolium eggs that is both analytically and numerically tractable and that is consistent with available data on Tribolium dynamics. The model consists of a system of nonlinear integral equations amenable to a bifurcation analysis. We are thus able to predict the presence of multiple attractors for the same parameter values in the model. Our model lets us determine the effect of varying cannibalism rates, fecundity, survival, and lengths of egg and larval stages on stability. In particular, intermediate lengths for larval stages are least stable, and increasing the length of the egg stage is always destabilizing. The model thus provides a conceptual framework for further studies of Tribolium dynamics.

A stochastic differential equation model for the rate of change of adult numbers was used to analyze data from two genetically differentiated groups of laboratory populations of the flour beetle, Tribolium castaneum. The first group of populations was homozygous for the corn oil sensitive allele and the other group was polymorphic at this genetic locus. Previous work has shown that mean larval viability is increased in the polymorphic populations. This suggests that the average number of potential recruits was higher in the polymorphic populations when compared to the homozygous ones. The stochastic model was used to derive predictions for the stationary distributions and the serial correlations at demographic equilibrium. These predictions were evaluated using 66 weeks of census data from both groups of populations while adult numbers were fluctuating in the region of their steady states. The data supported the following theoretical predictions: (i) the fluctuations in adult numbers can be described using a gamma probability density function; (ii) increased recruitment in the polymorphic populations results in a larger mean and variance for adult numbers; (iii) the autocorrelation of adult numbers decays exponentially with time; and (iv) increased recruitment in the polymorphic populations results in a faster rate of decay in the autocorrelations. These results suggest that genetically based fitness differences among populations can be reflected in the stationary stochastic dynamics of population size.Key words: Tribolium, population dynamics, stochastic models, stationary distributions, gamma distribution.

A biological counterpart to mathematical stability analysis was demonstrated using the Tribolium model. The responses to deliberate demographic and genetic perturbations of T. castaneum populations, initially homozygous for the corn oil sensitive (cos) allele and near demographic equilibrium, were examined experimentally. An equilibrium was said to be ecologically stable if it was locally stable to perturbations in population numbers alone and genetically stable if it was locally stable to changes in allele frequencies alone. A qualitative stability analysis of the within population age-class interactions indentified several conditions for stability, such as, the product of the rate of egg cannibalism by larvae times the rate of pupal cannibalism by adults must be greater than the product of the "net" fecundity of adults times the overall larval viability. The populations which were subjected to the addition or removal of genetically similar adults or immature life stages were ecologically stable. The cultures altered by the introduction of the + allele were genetically unstable to this perturbation and converged to an average polymorphic cos equilibrium allele frequency of 0.42. The incorporation of the + allele into populations initially homozygous for the cos allele was examined in terms of natural selection in a density-regulated population. The experimental data supported the theoretical prediction that selection is expected to move an ecologically stable system toward an evolutionarily stable strategy resulting in the maximization of adult numbers.

The growth and distribution of single- and mixed-species cultures of Tribolium castaneum and T. brevicornis maintained in fine-grained habitats defined by the proportion of standard and corn oil media were examined. The growth patterns of the single species cultures were qualitatively different. For the corn oil-sensitive strain of T. castaneum, the initial adult-pupa coordinates were drastically modified by corn oil medium, which appeared to stabilize the approach to demographic equilibrium. For T. brevicornis, the presence of corn oil patches extended the duration of the oscillations in these coordinates. Although niche breadth and overlap measures revealed different species utilization of the media types, T. castaneum excluded T. brevicornis in all test habitats. These single and mixed species observations were interpreted in the context of the usual cannibalism model for the regulation of population numbers with the addition of a newly identified adult-larval interaction in T. brevicornis. The primary...

We propose the use of a stationary probability distribution for the analysis of data on population size. Predicting this long term population property from short term individual events is accomplished by the use of the asymptotic theory of stochastic processes. A WKB approximation to the stationary density is obtained and then applied to observations on the flour beetleTribolium.

Demographic and genetic data of continuously growing populations of Tribolium castaneum initiated with identical age structures but with different frequencies of the unsaturated fatty acid-sensitive allele were collected for 68 weeks.—The life history data provided the following insights: Genotypic differences for total number of offspring were due primarily to larval viability. The total lifetime offspring production of the genotypes predicted a stable polymorphic genetic equilibrium. The genotypic reproductive functions forecast that a stable age structure was not a prerequisite for genetic equilibrium.—Those cultures initially segregating for the unsaturated fatty acid-sensitive allele converged to an equilibrium allele frequency of 0.25 and a genotypic array composed of equal numbers of +/+ and +/cos individuals.—The numbers of larvae, pupae and adults during the first six weeks were quadratic functions of the initial frequency of the sensitive allele. Qualitative age structure changes that followed were similar in all cultures and demographic equilibrium was realized at week 50. The overall demographic pattern during the 68-week study was interpreted in terms of the interactions among the numbers of small larvae, large larvae plus pupae, and adults.

Genetic analysis of a population of Tribolium. IV. Gene expression modified by corn oil and relative humidity 1

Genetic analysis of a population of Tribolium III. Fatty acid composition of unsaturated fatty acid sensitive mutant 2

I N quantitative genetic theory a population is specified by genotypic frequencies and genotypic values. The genotypic values are numbers assigned to the genotypes under discussion and the subsequent analysis of the population examines the variation of these values. Since the concept of genotypic value is central to quantitative theory, it is the objective of this communication to express the genotypic values as a function of developmental time and hence present a more comprehensive, workable system of population dynamics. Genotypic value in the classical sense (FISHER 1918) is a number that reflects the cumulative effects of a genotype at the time in the life cycle it is assigned. Inferences concerning the genetic structure of the population are necessarily restricted to the time the genotypic values were assigned or experimentally when the genotypes were measured (see ROBERTSON 1967 for recent comment). To remove this limitation oine must describe the association between the value of a genotype and time. There is no universal mathematical function for all organisms that can fulfill this requirement; however, this need not deter one from exploring the concept of developmental time as it influences the genetic structure of populations. A preliminary report of this work was given by COSTANTINO (1967a).

The entropy H(po,p*) of a population with the initial allele frequency po given the equilibrium polymorphic frequency p* has been proposed as a measure of natural selection. In the present paper, we have extended this concept to include a particular aspect of density-dependent selection. We compared size trajectory of a population initially at genetic equilibrium, N(t), with the size trajectories of populations not initially at p*,N(t), but which do eventually converge to a common equilibrium allele frequency and equilibrium density, N*. The following experimentally testable hyopthesis was established. The total area defined by the difference between the trajectories of N(t) and N(t) as they converge to N* is directly proportional to the fitness entropy when population size is transformed using the density-dependent fitness value. Two properties of this relationship were noted. First, it is independent of the magnitude of natural selection and, secondly, it does not depend upon the initial population density as long as the equilibrium and nonequilibrium populations have the same initial numbers. This hypothesis was evaluated with experimental data on the flour beetle Tribolium castaneum.

TAYLOR (1960), after a comprehensive search of the literature, observed that “of all the structures of the egg, the yolk seems to be the least influenced by hereditary factors.” Current interest in cholesterol as related to human health has stimulated the study of hereditary and other factors which influence yolk cholesterol level in chicken eggs. Harris and Wilcox (1958) found significant differences in egg yolk cholesterol among dams in a Randombred population of Leghorns. Edwards et al. (1960) observed significant strain or breed differences in cholesterol content of the fat of eggs. The phenotypic correlation between yolk cholesterol and egg production in a randombred population from October through December was estimated by Harris and Wilcox (1963a,b). The correlation coefficient (r) for November cholesterol and for March cholesterol was the same, −0.17. Correlation coefficients between body weight in March and yolk cholesterol in November and in March were +0.15 and +0.13, . . .