Population dynamics in variable environments. VI. Cyclical environments

This paper studies the dynamics of an age-structured population which experiences cyclical variation in vital rates. The principal features of population behavior are found to be contained in an explicitly calculable response function. Three distinct regimes of qualitative behavior are described when cycle period is respectively much less than, of the order of, and much greater than the average generation length. These results make explicit the way in which transient properties corresponding to average vital rates determine population response to cycles.     

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.     

Population dynamics in variable environments. III. Evolutionary dynamics of r-selection

The boundary dynamics of a genetic model for an age-structured population in a temporally fluctuating environment are analyzed. The condition for invasion by a new allele identifies the logarithmic growth rate a of each life history phenotype as the fitness measure relevant to “r-selection.” An analytical formula is obtained for fitness a when temporal variance in life history characters is small. This formula reveals the major qualitative and quantitative effects of the average life history, fluctuations, and temporal autocorrelation on fitness. A similar approximation is obtained for the log-variance of population number so that the statistical distribution of population size can be estimated.     

Population dynamics in variable environments. IV. Weak ergodicity in the Lotka equation

The Hilbert projective metric is applied to the continuous-time Lotka equation in demography to establish weak ergodicity: populations with the same time-varying fecundity and mortality schedules ultimately have the same age composition. The analysis displays clearly the dynamic content of Lotka's equation and identifies a contraction operator which forces convergence of birth sequences over time. The relationship between primitivity in the discrete (Leslie) and continuous (Lotka) demographic models is made clear.     

Population and prehistory I: Food-dependent population growth in constant environments

We present a demographic model that describes the feedbacks between food supply, human mortality and fertility rates, and labor availability in expanding populations, where arable land area is not limiting. This model provides a quantitative framework to describe how environment, technology, and culture interact to influence the fates of preindustrial agricultural populations. We present equilibrium conditions and derive approximations for the equilibrium population growth rate, food availability, and other food-dependent measures of population well-being. We examine how the approximations respond to environmental changes and to human choices, and find that the impact of environmental quality depends upon whether it manifests through agricultural yield or maximum (food-independent) survival rates. Human choices can complement or offset environmental effects: greater labor investments increase both population growth and well-being, and therefore can counteract lower agricultural yield, while fertility control decreases the growth rate but can increase or decrease well-being. Finally we establish equilibrium stability criteria, and argue that the potential for loss of local stability at low population growth rates could have important consequences for populations that suffer significant environmental or demographic shocks.     

Age-structured population growth rates in constant and variable environments: a near equilibrium approach

General measures summarizing the shapes of mortality and fecundity schedules are proposed. These measures are derived from moments of probability distributions related to mortality and fecundity schedules. Like moments, these measures form infinite sequences, but the first terms of these sequences are of particular value in approximating the long-term growth rate of an age- structured population that is growing slowly. Higher order terms are needed for approximating faster growing populations. These approximations offer a general nonparametric approach to the study of life-history evolution in both constant and variable environments. These techniques provide simple quantitative representations of the classical findings that, with fixed expected lifetime and net reproductive rate, type I mortality and early peak reproduction increase the absolute magnitude of the population growth rate, while type III mortality and delayed peak reproduction reduce this absolute magnitude.     

Population growth in random environments

This paper reviews some recent advances in single population stochastic differential equation growth models. They are a natural way to model population growth in a randomly varying environment. The question of which calculus, Itô or Stratonovich, is preferable is addressed. The two calculi coincide when the noise term is linear, if we take into account the differences in the interpretation of the parameters. This clarifies, among other things, the controversy on the theory of niche limiting similarity proposed by May and MacArthur. The effects of correlations in the environmental fluctuations and statistical methods for estimating parameters and for prediction based on a single population trajectory are mentioned. Applications to fisheries, wildlife management and particularly to environmental impact assessment are now becoming possible and are proposed in this paper.     

Population and prehistory III: Food-dependent demography in variable environments

The population dynamics of preindustrial societies depend intimately on their surroundings, and food is a primary means through which environment influences population size and individual well-being. Food production requires labor; thus, dependence of survival and fertility on food involves dependence of a population’s future on its current state. We use a perturbation approach to analyze the effects of random environmental variation on this nonlinear, age-structured system. We show that in expanding populations, direct environmental effects dominate induced population fluctuations, so environmental variability has little effect on mean hunger levels, although it does decrease population growth. The growth rate determines the time until population is limited by space. This limitation introduces a tradeoff between population density and well-being, so population effects become more important than the direct effects of the environment: environmental fluctuation increases mortality, releasing density dependence and raising average well-being for survivors. We discuss the social implications of these findings for the long-term fate of populations as they transition from expansion into limitation, given that conditions leading to high well-being during growth depress well-being during limitation.     

Age- and density-dependent population dynamics in static and variable environments

We consider a general model of a single-species population with age- and density-dependent per capita birth and death rates. In a static environment we show that if the per capita death rate is independent of age, then the local stability of any stationary state is guaranteed by the requirement that, in the region of the steady state, the density dependence of the birth rate should be negative and that of the death rate positive. In a variable environment we show that, provided the system is locally stable, small environmental fluctuations will give rise to small age structure and population fluctuations which are related to the driving environmental fluctuations by a simple “transfer function.” We illustrate our general theory by examining a model with a per capita death rate which is age and density independent and a per capita birth rate which is zero up to some threshold age a₀, adopts a finite density-dependent value up to a maximum age a_o + α, and is zero thereafter. We conclude from this model that resonance due specifically to single-species age-structure effects will only be of practical importance in populations whose members have a life cycle consisting of a long immature phase followed by a short burst of intense reproductive effort (α a_o).     

Population dynamics of animals in unpredictably-changing tropical environments

We studied population dynamics of a solitary phytophagous beetle,Epilachna viqintioctopunctata and a social stingless bee,Trigona minangkabau, in Sumatra, Indonesia for 5 years from 1981. Population increase ofEpilachna vigintioctopunctata was suppressed in months of normal rainfall (≥300mm) but was released in the 1982–1983 El Nino-Southern. Oscillation when rainfall dropped to 50% of the long-term average. Mechanisms might be direct; rainfall lowered egg hatchability and the time of adult’s residence on host plants. When dry weather continued for more than three generations, theEpilachna vigintioctopunctata population reached a density at which food shortage due to defoliation occurred. Although parasitism of immature stages was high, it was not a population-regulating factor. Thus, there were two types of ecological crunch: competition for food resources at the end of favourable dry periods and high mortality during heavy rainfall periods that usually followed El Nino-Southern Oscillation dry conditions. By an experimental addition of artificial nest sites, colony density ofTrigona minangkabau increased 2.5 times the original density of natural colonies. One-half of artificial nest sites were occupied by arboreal ants and thus competition for nest sites with ants suppressed further increase ofTrigona minangkabau. Intermediate rainfall was favourable forTrigona minangkabau because the rate of colony foundation decreased both during dry El Niño-Southern Oscillation months and months with heavy rain. Colony death was independent from rainfall. Many colonies that survived for 6 months persisted for >2 years and colony density was quite stable.Trigona minangkabau colonies could survive even under unfavourable periods, by hoarding resources in the nest. There was no significant ecological crunch during the study period and colony density almost always tracked the carrying capacity of the habitat, which was basically determined by nest-site abundance. Climatic conditions, especially rainfall, changed with various periodicities, 4–5 years for El Nino-Southern Oscillation, and 2 years for the monsoon and other shorter periods. The contribution of periodicities of 1 and 0.5 years, that were linked to movement of the sun, were weak, indicating that animals could not use seasonal changes of environments,e.g. daylength, to predict environmental changes. We discuss traits adaptive to such unpredictably-changing tropical environments. Separation of predictability of temporal environmental change and synchronous changes among patches improves our understanding. Low oviposition rate and resulting prolonged life-span ofEpilachna vigintioctopunctata, usually associated withK-selected traits of life history, seem to be adaptations for unpredictable environmental changes.     

Population dynamics within a microbial consortium during growth on diesel fuel in saline environments

The diversity and dynamics of a bacterial community extracted from an exploited oil field with high natural soil salinity near Comodoro Rivadavia in Patagonia (Argentina) were investigated. Community shifts during long-term incubation with diesel fuel at four salinities between 0 and 20% NaCl were monitored by single-strand conformation polymorphism community fingerprinting of the PCR-amplified V4-V5 region of the 16S rRNA genes. Information obtained by this qualitative approach was extended by flow cytometric analysis to follow quantitatively the dynamics of community structures at different salinities. Dominant and newly developing clusters of individuals visualized via their DNA patterns versus cell sizes were used to identify the subcommunities primarily involved in the degradation process. To determine the most active species, subcommunities were separated physically by high-resolution cell sorting and subsequent phylogenetic identification by 16S rRNA gene sequencing. Reduced salinity favored the dominance of Sphingomonas spp., whereas at elevated salinities, Ralstonia spp. and a number of halophilic genera, including Halomonas, Dietzia, and Alcanivorax, were identified. The combination of cytometric sorting with molecular characterization allowed us to monitor community adaptation and to identify active and proliferating subcommunities.     

Spatial and temporal synchrony in reptile population dynamics in variable environments

Resources are seldom distributed equally across space, but many species exhibit spatially synchronous population dynamics. Such synchrony suggests the operation of large-scale external drivers, such as rainfall or wildfire, or the influence of oasis sites that provide water, shelter, or other resources. However, testing the generality of these factors is not easy, especially in variable environments. Using a long-term dataset (13–22 years) from a large (8000 km²) study region in arid Central Australia, we tested firstly for regional synchrony in annual rainfall and the dynamics of six reptile species across nine widely separated sites. For species that showed synchronous spatial dynamics, we then used multivariate follow a multivariate auto-regressive state–space (MARSS) models to predict that regional rainfall would be positively associated with their populations. For asynchronous species, we used MARSS models to explore four other possible population structures: (1) populations were asynchronous, (2) differed between oasis and non-oasis sites, (3) differed between burnt and unburnt sites, or (4) differed between three sub-regions with different rainfall gradients. Only one species showed evidence of spatial population synchrony and our results provide little evidence that rainfall synchronizes reptile populations. The oasis or the wildfire hypotheses were the best-fitting models for the other five species. Thus, our six study species appear generally to be structured in space into one or two populations across the study region. Our findings suggest that for arid-dwelling reptile populations, spatial and temporal dynamics are structured by abiotic events, but individual responses to covariates at smaller spatial scales are complex and poorly understood.     

Linear analysis solves two puzzles in population dynamics: the route to extinction and extinction in coloured environments

In this paper, we give simple explanations to two unsolved puzzles that have emerged in recent theoretical studies in population dynamics. First, the tendency of some model populations to go extinct from high population densities, and second, the positive effect of autocorrelated environments on extinction risks for some model populations. Both phenomena are given general explanations by simple, linear, sto-chastic models. We emphasize the predictive and explanatory power of such models.     

Population growth and structure in a variable environment

When a population experiences temporal changes in the vital rates due to environmental or biotic variation, change is not only expected in the rate of population growth but also in the structure of the population. In this study we present a method for transforming observed patterns (notably how vital rates change with temperature) into functions that can be used in population growth models and analysis of population structure. The method is exemplified by applying it to cohort studies in different constant temperatures of four species of aphids, Lipaphis erysimi (K.), Metopolophium dirhodum (Wlk.), Rhopaliosiphum padi and Macrosiphum avenae (F.). We use piece-wise linear functions to transform the vital rates of the cohort studies. The lifespans are divided into phases, each phase having linear rates. A projection matrix is formulated, where the elements are temperature dependent fecundities, survivorships and developmental rates. The major result is, contrary to what theory predicts as reasonable (Caswell 1989), that population structure of these aphid species will become almost fixed although the temperature varies. This result is consistent with findings of earlier field studies (Wiktelius 1982). A fixed population structure implies that it is possible to calculate the population growth rate on the basis of intrinsic rates of increase. By simulating different temperature regimes we also show that initial oscillations in the population structure dampen out after a few days. After initial oscillations, calculations of population growth using intrinsic rates of increase are consistent with calculations made by a matrix model.     

Population dynamics in tumor growth modeling

A new approach based on local interaction between cancer and tissue cells was applied to the problem of the onset and growth of solid tumors in homogeneous tissues and effects associated with dramatic changes in tumor growth after crossing the boundary between different tissues. The characteristic sizes and growth rates of spherical tumors, the points of the beginning and the end of spherical growth, and the further development of complex structures from the spherical ones (rough interface between the tumor and the host tissue, elongate outgrowths, dendritic structures, and metastases) were inferred assuming that the reproduction rate of a population of cancer cells is a nonmonotone function of their local concentration and thus of the local curvature of the tumor surface.     

Population dynamics in modeling tumor growth

A new approach based on local interaction between cancer and tissue cells is applied to the problem of the onset and growth of solid tumors in homogeneous tissues and effects associated with dramatic changes in tumor growth after crossing the boundary between different tissues. The characteristic sizes and growth rates of spherical tumors, the points of the beginning and the end of spherical growth, and further development of complex structures from the spherical ones (rough interface between the tumor and the host tissue, elongate outgrowths, dendritic structures, and metastases) are inferred assuming that the reproduction rate of a population of cancer cells is a nonmonotonous function of their local concentration and thus of the local curvature of the tumor surface. The growth behavior changes dramatically when the tumor crosses a boundary between two tissues.     

Population growth rates: issues and an application

Current issues in population dynamics are discussed in the context of The Royal Society Discussion Meeting 'Population growth rate: determining factors and role in population regulation'. In particular, different views on the centrality of population growth rates to the study of population dynamics and the role of experiments and theory are explored. Major themes emerging include the role of modern statistical techniques in bringing together experimental and theoretical studies, the importance of long-term experimentation and the need for ecology to have model systems, and the value of population growth rate as a means of understanding and predicting population change. The last point is illustrated by the application of a recently introduced technique, integral projection modelling, to study the population growth rate of a monocarpic perennial plant, its elasticities to different life-history components and the evolution of an evolutionarily stable strategy size at flowering.     

Population dynamics with age- and time-dependent birth and death rates

A continous, deterministic mathematical model is used to predict population distributions by age at any time, given the initial distribution and the variation of birth and death rates with age and time. Solutions are obtained on a computer using a semi-discretization algorithm in which time derivatives in the partial differential equations are replaced by finite-difference expressions. The resulting sets of ordinary differential equations are solved by a predictor-corrector method. Graphical results are shown for some examples.     

Community extinction patterns in coloured environments

Understanding community responses to environmental variation is a fundamental aspect of ecological research, with direct ecological, conservation and economic implications. Here, we examined the role of the magnitude, correlation and autocorrelation structures of environmental variation on species' extinction risk (ER), and the probability of actual extinction events in model competitive communities. Both ER and probability increased with increasing positive autocorrelation when species responded independently to the environment, yet both decreased with a strong correlation between species-specific responses. These results are framed in terms of the synchrony between--and magnitude of variation within--species population sizes and are explained in terms of differences in noise amplification under different conditions. The simulation results are robust to changes in the strength of interspecific density dependence, and whether noise affects density-independent or density-dependent population processes. Similar patterns arose under different ranges of noise severity when these different model assumptions were examined. We compared our results with those from an analytically derived solution, which failed to capture many features of the simulation results.