Population dynamics in variable environments. II. Correlated environments,sensitivity analysis and dynamics

Mean and mean square number are studied for age-structured populations with serially correlated temporally fluctuating vital rates. Results are that (1) Moments of population number can be used effectively to analyse growth rates of the coefficient of variation and an approximate median population number. (2) Analytical approximations to the growth rates of moments reveal dynamic consequences of covarying phenotypic traits and of temporal correlation along environmental sequences. (3) Dynamic properties can be explicitly related to the static sensitivity of an average vital rate matrix. (4) The use of (1), (2) and (3) allows an extension of many applications of static vital rate theory to dynamics with fluctuating rates.     

Stochastic gene expression in fluctuating environments

Stochastic mechanisms can cause a group of isogenic bacteria, each subject to identical environmental conditions, to nevertheless exhibit diverse patterns of gene expression. The resulting phenotypic subpopulations will typically have distinct growth rates. This behavior has been observed in several contexts, including sugar metabolism and pili phase variation. Under fixed environmental conditions, the net growth rate of the population is maximized when all cells are of the fastest growing phenotype, so it is unclear what fitness advantage is conferred by population heterogeneity. However, unlike ideal laboratory conditions, natural environments tend to fluctuate, either periodically or randomly. Here we use a stochastic population model to show that, during growth in such fluctuating environments, a dynamically heterogenous bacterial population can sometimes achieve a higher net growth rate than a homogenous one. By using stochastic mechanisms to sample several distinct phenotypes, the bacteria are able to anticipate and take advantage of sudden changes in their environment. However, this heterogeneity is beneficial only if the bacterial response rate is sufficiently low. Our results could be useful in the design of artificial evolution experiments and in the optimization of fermentation processes.     

Evolution and population dynamics in stochastic environments

Inter-generational temporal variability of the environment is important in the evolution and adaptation of phenotypic traits. We discuss a population-dynamic approach which plays a central role in the analysis of evolutionary processes. The basic principle is that the phenotypes with the greatest long-term average growth rate will dominate the entire population. The calculation of longterm average growth rates for populations under temporal stochasticity can be highly cumbersome. However, for a discrete non-overlapping population, it is identical to the geometric mean of the growth rates (geometric mean fitness), which is usually different from the simple arithmetic mean of growth rates. Evolutionary outcomes based on geometric mean fitness are often very different from the predictions based on the usual arithmetic mean fitness. In this paper we illustrate the concept of geometric mean fitness in a few simple models. We discuss its implications for the adaptive evolution of phenotypes, e.g. foraging under predation risks and clutch size. Next, we present an application: the risk-spreading egg-laying behaviour of the cabbage white butterfly, and develop a two-patch population dynamic model to show how the optimal solution diverges from the ssual arithmetic mean approach. The dynamics of these stochastic models cannot be predicted from the dynamics of simple deterministic models. Thus the inclusion of stochastic factors in the analyses of populations is essential to the understanding of not only population dynamics, but also their evolutionary dynamics.     

The many growth rates and elasticities of populations in random environments

Despite considerable interest in the dynamics of populations subject to temporally varying environments, alternate population growth rates and their sensitivities remain incompletely understood. For a Markovian environment, we compare and contrast the meanings of the stochastic growth rate (lambdaS), the growth rate of average population (lambdaM), the growth rate for average transition rates (lambdaA), and the growth rate of an aggregate represented by a megamatrix (shown here to equal lambdaM). We distinguish these growth rates by the averages that define them. We illustrate our results using data on an understory shrub in a hurricane-disturbed landscape, employing a range of hurricane frequencies. We demonstrate important differences among growth rates: lambdaS lambdaM. We show that stochastic elasticity, ESij, and megamatrix elasticity, EMij, describe a complex perturbation of both means and variances of rates by the same proportion. Megamatrix elasticities respond slightly and stochastic elasticities respond strongly to changing the frequency of disturbance in the habitat (in our example, the frequency of hurricanes). The elasticity EAij of lambdaA does not predict changes in the other elasticities. Because ES, although commonly utilized, is difficult to interpret, we introduce elasticities with a more direct interpretation: ESmu for perturbations of means and ESsigma for variances. We argue that a fundamental tool for studying selection pressures in varying environments is the response of growth rate to vital rates in all habitat states.     

From stochastic environments to life histories and back

Environmental stochasticity is known to play an important role in life-history evolution, but most general theory assumes a constant environment. In this paper, we examine life-history evolution in a variable environment, by decomposing average individual fitness (measured by the long-run stochastic growth rate) into contributions from average vital rates and their temporal variation. We examine how generation time, demographic dispersion (measured by the dispersion of reproductive events across the lifespan), demographic resilience (measured by damping time), within-year variances in vital rates, within-year correlations between vital rates and between-year correlations in vital rates combine to determine average individual fitness of stylized life histories. In a fluctuating environment, we show that there is often a range of cohort generation times at which the fitness is at a maximum. Thus, we expect ‘optimal’ phenotypes in fluctuating environments to differ from optimal phenotypes in constant environments. We show that stochastic growth rates are strongly affected by demographic dispersion, even when deterministic growth rates are not, and that demographic dispersion also determines the response of life-history-specific average fitness to within- and between-year correlations. Serial correlations can have a strong effect on fitness, and, depending on the structure of the life history, may act to increase or decrease fitness. The approach we outline takes a useful first step in developing general life-history theory for non-constant environments.     

Adaptive Host Preference and the Dynamics of Host–Parasitoid Interactions

Models of two independent host populations and a common parasitoid are investigated. The hosts have density-dependent population growth and only interact indirectly by their effects on parasitoid behavior and population dynamics. The parasitoid is assumed to experience a trade-off in its ability to exploit the two hosts. Three alternative types of parasitoid are investigated: (i) fixed generalists whose consumption rates are those that maximize fitness; (ii) “ideal free” parasitoids, which modify their behavior to maximize their rate of finding unparasitized hosts within a generation; and (iii) “evolving” parasitoids, whose capture rates change between generations based on quantitative genetic determination of the relative attack rates on the two hosts. The primary questions addressed are: (1) Do the different types of adaptive processes stabilize or destabilize the population dynamics? (2) Do the adaptive processes tend to equalize or to magnify differences in host densities? The models show that adaptive behavior and evolution frequently destabilize population dynamics and frequently increase the average difference between host densities.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Scaling population responses to spatial environmental variability in advection-dominated systems

We model the spatial dynamics of an open population of organisms that disperse solely through advection in order to understand responses to multiscale environmental variability. We show that the distance over which a population responds to a localized perturbation, called the response length, can be characterized as an organisms average lifetime dispersal distance, unless there is strong density‐dependence in demographic or dispersal rates. Continuous spatial fluctuations in demographic rates at scales smaller than the response length will be largely averaged in the population distribution, whereas those in per capita emigration rates will be strongly tracked. We illustrate these results using a parameterized example to show how responses to environmental variability may differ in streams with different average current velocities. Our model suggests an approach to linking local dynamics dominated by dispersal processes to larger‐scale dynamics dominated by births and deaths.     

Evolution of genetic variability and the advantage of sex and recombination in changing environments.

The role of recombination and sexual reproduction in enhancing adaptation and population persistence in temporally varying environments is investigated on the basis of a quantitative-genetic multilocus model. Populations are finite, subject to density-dependent regulation with a finite growth rate, diploid, and either asexual or randomly mating and sexual with or without recombination. A quantitative trait is determined by a finite number of loci at which mutation generates genetic variability. The trait is under stabilizing selection with an optimum that either changes at a constant rate in one direction, exhibits periodic cycling, or fluctuates randomly. It is shown by Monte Carlo simulations that if the directional-selection component prevails, then freely recombining populations gain a substantial evolutionary advantage over nonrecombining and asexual populations that goes far beyond that recognized in previous studies. The reason is that in such populations, the genetic variance can increase substantially and thus enhance the rate of adaptation. In nonrecombining and asexual populations, no or much less increase of variance occurs. It is explored by simulation and mathematical analysis when, why, and by how much genetic variance increases in response to environmental change. In particular, it is elucidated how this change in genetic variance depends on the reproductive system, the population size, and the selective regime, and what the consequences for population persistence are.     

The determination of microbial hydrocarbon metabolism in natural environments

Summary An investigation was made of the hydrocarbon-oxidising microorganisms in an upland moorland soil and the underlying shale band. The results obtained were in agreement with an earlier survey, although the methods used were not the same. The procedures used involve the assessment of numbers and activities of hydrocarbon oxidisers. The population studies were done, on this occasion, using the buried slide technique which, although simple to perform, did not always reveal the trends known to occur at the sites investigated. The metabolic activity of the micro-organisms was measured either respirometrically or by following the disappearance of the substrate. The respirometric methods were much less cumbersome and more sensitive to changes in the metabolic pattern in the samples. Of the two approaches the measurement of activity would appear to be the more useful since population studies yield no information on the rates at which micro-organisms are capable of degrading and thus removing contaminating oil from any given environment.     

Molecular diversity after a range expansion in heterogeneous environments

Recent range expansions have probably occurred in many species, as they often happen after speciation events, after ice ages, or after the introduction of invasive species. While it has been shown that range expansions lead to patterns of molecular diversity distinct from those of a pure demographic expansion, the fact that many species do live in heterogeneous environments has not been taken into account. We develop here a model of range expansion with a spatial heterogeneity of the environment, which is modeled as a gamma distribution of the carrying capacities of the demes. By allowing temporal variation of these carrying capacities, our model becomes a new metapopulation model linking ecological parameters to molecular diversity. We show by extensive simulations that environmental heterogeneity induces a loss of genetic diversity within demes and increases the degree of population differentiation. We find that metapopulations with low average densities are much more affected by environmental heterogeneity than metapopulations with high average densities, which are relatively insensitive to spatial and temporal variations of the environment. Spatial heterogeneity is shown to have a larger impact on genetic diversity than temporal heterogeneity. Overall, temporal heterogeneity and local extinctions are not found to leave any specific signature on molecular diversity that cannot be produced by spatial heterogeneity.     

中国沙棘克隆生长对灌水强度的响应

为了解中国沙棘克隆生长调节对土壤水分资源供应水平的响应规律并寻找最佳灌水强度,研究了种群生长量、生物量、子株数量、克隆器官延伸能力、克隆器官分枝级数等种群参数与同灌水强度的关系。结果表明：（1）种群参数增幅与灌水强度之间呈二次抛物线关系,即种群参数增幅随着灌水强度的增加先升后降。小于最佳灌水强度,种群参数增幅随着灌水强度的增加而上升;大于最佳灌水强度,种群参数增幅随着灌水强度的增加而下降。（2）不同种群参数增幅最大时的最佳灌水强度具有一定差异,种群及其构件生物量的最佳灌水强度为每月4.0次、相当于本地年均降水量的2.0倍,子株数量、克隆器官延伸能力、克隆器官分枝级数增幅的最佳灌水强度为每月3.0次、相当于本地年均降水量的1.5倍。（3）适当灌水能够促进中国沙棘种群生长、提高克隆繁殖能力,但在一定灌水强度下生长和繁殖之间存在权衡关系,而种群生长和生物量积累之间、克隆子株数量与克隆器官延伸能力和克隆器官分枝级数之间则有协同作用。（4）在不同土壤水分资源供应水平下,中国沙棘主要通过个体大小、种群生物量分配、子株数量、克隆器官延伸能力、克隆器官分枝强度等调节做出响应,这些调节直接影响种群的稳定性和克隆的持久性。     

Population and prehistory II: space-limited human populations in constant environments

We present a population model to examine the forces that determined the quality and quantity of human life in early agricultural societies where cultivable area is limited. The model is driven by the non-linear and interdependent relationships between the age distribution of a population, its behavior and technology, and the nature of its environment. The common currency in the model is the production of food, on which age-specific rates of birth and death depend. There is a single non-trivial equilibrium population at which productivity balances caloric needs. One of the most powerful controls on equilibrium hunger level is fertility control. Gains against hunger are accompanied by decreases in population size. Increasing worker productivity does increase equilibrium population size but does not improve welfare at equilibrium. As a case study we apply the model to the population of a Polynesian valley before European contact.     

Fixation probability in spatially changing environments.

The fixation probability of a mutant in a subdivided population with spatially varying environments is investigated using a finite island model. This probability is different from that in a panmictic population if selection is intermediate to strong and migration is weak. An approximation is used to compute the fixation probability when migration among subpopulations is very weak. By numerically solving the two-dimensional partial differential equation for the fixation probability in the two subpopulation case, the approximation was shown to give fairly accurate values. With this approximation, we show in the case of two subpopulations that the fixation probability in subdivided populations is greater than that in panmictic populations mostly. The increase is most pronounced when the mutant is selected for in one subpopulation and is selected against in the other subpopulation. Also it is shown that when there are two types of environments, further subdivision of subpopulations does not cause much change of the fixation probability in the no dominance case unless the product of the selection coefficient and the local population size is less than one. With dominance, the effect of subdivision becomes more complex.     

Hybridization between Felis silvestris silvestris and Felis silvestris catus in two contrasted environments in France

European wildcat (Felis silvestris silvestris) populations are fragmented throughout most of the whole range of the subspecies and may be threatened by hybridization with the domestic cat F.s. catus. The underlying ecological processes promoting hybridization remain largely unknown. In France, wildcats are mainly present in the northeast and signs of their presence in the Pyrenees have been recently provided. However, no studies have been carried out in the French Pyrenees to assess their exposure to hybridization. We compared two local populations of wildcats, one living in a continuous forest habitat in the French Pyrenees, the other living in a highly fragmented forest‐agricultural landscape in northeastern France to get insights into the variability of hybridization rates. Strong evidence of hybridization was detected in northeastern France and not in the Pyrenees. Close kin in the Pyrenees were not found in the same geographic location contrary to what was previously reported for females in the northeastern wildcat population. The two wildcat populations were significantly differentiated (F_ST = 0.072) to an extent close to what has been reported (F_ST = 0.103) between the Iberian population, from which the Pyrenean population may originate, and the German population, which is connected to the northeastern population. The genetic diversity of the Pyrenean wildcats was lower than that of northeastern wildcat populations in France and in other parts of Europe. The lower hybridization in the Pyrenees may result from the continuity of natural forest habitats. Further investigations should focus on linking landscape features to hybridization rates working on local populations.     

Search and navigation in dynamic environments – from individual behaviors to population distributions

Animal movement receives widespread attention within ecology and behavior. However, much research is restricted within isolated sub-disciplines focusing on single phenomena such as navigation (e.g. homing behavior), search strategies (e.g. Levy flights) or theoretical considerations of optimal population dispersion (e.g. ideal free distribution). To help synthesize existing research, we outline a unifying conceptual framework that integrates individual-level behaviors and population-level spatial distributions with respect to spatio-temporal resource dynamics. We distinguish among (1) non-oriented movements based on diffusion and kinesis in response to proximate stimuli, (2) oriented movements utilizing perceptual cues of distant targets, and (3) memory mechanisms that assume prior knowledge of a target's location. Species' use of these mechanisms depends on life-history traits and resource dynamics, which together shape population-level patterns. Resources with little spatial variability should facilitate sedentary ranges, whereas resources with predictable seasonal variation in spatial distributions should generate migratory patterns. A third pattern, 'nomadism', should emerge when resource distributions are unpredictable in both space and time. We summarize recent advances in analyses of animal trajectories and outline three major components on which future studies should focus: (1) integration across alternative movement mechanisms involving links between state variables and specific mechanisms, (2) consideration of dynamics in resource landscapes or environments that include resource gradients in predictability, variability, scale, and abundance, and finally (3) quantitative methods to distinguish among population distributions. We suggest that combining techniques such as evolutionary programming and pattern oriented modeling will help to build strong links between underlying movement mechanisms and broad-scale population distributions.     

Haploids adapt faster than diploids across a range of environments

Despite a great deal of theoretical attention, we have limited empirical data about how ploidy influences the rate of adaptation. We evolved isogenic haploid and diploid populations of Saccharomyces cerevisiae for 200 generations in seven different environments. We measured the competitive fitness of all ancestral and evolved lines against a common competitor and find that in all seven environments, haploid lines adapted faster than diploids, significantly so in three environments. We apply theory that relates the rates of adaptation and measured effective population sizes to the properties of beneficial mutations. We obtained rough estimates of the average selection coefficients in haploids between 2% and 10% for these first selected mutations. Results were consistent with semi-dominant to dominant mutations in four environments and recessive to additive mutations in two other environments. These results are consistent with theory that predicts haploids should evolve faster than diploids at large population sizes.     

Stochastic gene expression in switching environments

Organisms are known to adapt to regularly varying environments. However, in most cases, the fluctuations of the environment are irregular and stochastic, alternating between favorable and unfavorable regimes, so that cells must cope with an uncertain future. A possible response is population diversification. We assume here that the cell population is divided into two groups, corresponding to two phenotypes, having distinct growth rates, and that cells can switch randomly their phenotypes. In static environments, the net growth rate is maximized when the population is homogeneously composed of cells having the largest growth rate. In random environments, growth rates fluctuate and observations reveal that sometimes heterogeneous populations have a larger net growth rate than homogeneous ones, a fact illustrated recently through Monte-Carlo simulations based on a birth and migration process in a random environment. We study this process mathematically by focusing on the proportion f(t) of cells having the largest growth rate at time t, and give explicitly the related steady state distribution π. We also prove the convergence of empirical averages along trajectories to the first moment , and provide efficient numerical methods for computing .      

ESS germination strategies in randomly varying environments. I. Logistic-type models

ESS germination strategies are studied in a model of annual plant population dynamics in a randomly varying environment. The possible strategies are different values of the annual germination fraction G, either constant over time or varying in response to a "cue" correlated with upcoming environmental conditions. The model generalizes D. Cohen's model (1966, J. Theor. Biol. 12, 119-129; 1968, J. Ecol. 56, 219-228) by allowing density-dependent per capita seed yields. ESSs are characterized in terms of the resulting harmonic mean growth rate of population density. The ESS criterion cannot be solved analytically, but qualitative relationships between the value of the ESS and other population parameters are obtained, and environments in which 100% germination is an ESS are identified. Some explicit predictions of the theory are summarized and compared with ideas of M. Westoby (1981, Amer. Nat. 118, 882-885). The results of this study are compared with those of Cohen (op. cit.) in a companion paper.     

Life-history evolution in guppies. VII. The comparative ecology of high- and low-predation environments

Prior research has demonstrated a strong association between the species of predators that co-occur with guppies and the evolution of guppy life histories. The evolution of these differences in life histories has been attributed to the higher mortality rates experienced by guppies in high-predation environments. Here, we evaluate whether there might be indirect effects of predation on the evolution of life-history patterns and whether there are environmental differences that are correlated with predation. To do so, we quantified features of the physical and chemical environment and the population biology of guppies from seven high- and low-predation localities. We found that high-predation environments tend to be larger streams with higher light levels and higher primary productivity, which should enhance food availability for guppies. We also found that guppy populations from high-predation environments have many more small individuals and fewer large individuals than those from low-predation environments, which is caused by their higher birth rates and death rates. Because of these differences in size distribution, guppies from high-predation environments have only one-fourth of the biomass per unit area, which should also enhance food availability for guppies in these localities. Guppies from high-predation sites allocate more resources to reproduction, grow faster, and attain larger asymptotic sizes, all of which are consistent with higher levels of resource availability. We conclude that guppies from high-predation environments experience higher levels of resource availability in part because of correlated differences in the environment (light levels, primary productivity) and in part as an indirect consequence of predation (death rates and biomass density). These differences in resource availability can, in turn, augment the effect of predator-induced mortality as factors that shape the evolution of guppy life-history patterns. We found no differences in the invertebrate communities from high- and low-predation localities, so we conclude that there do not appear to be multitrophic, indirect effects associated with these differences in predation.