Persistence of structured populations in random environments

Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and density-dependent interactions influence population dynamics, we provide a general theory for persistence for density-dependent matrix models in random environments. For populations with compensating density dependence, exhibiting “bounded” dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating density-dependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stage-structured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space.     

Regulated growth in random environments

Summary Conditions for extinction, convergence to a stationary distribution and attaining a carrying capacity are given for stochastic versions of the logistic growth process.     

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.     

Habitat dependence and correlations between elasticities of long-term growth rates

In population biology, elasticity is a measure of the importance of a demographic rate on population growth. A relatively small amount of stochasticity can substantially impact the dynamics of a population whose growth is a function of deterministic and stochastic processes. Analyses of natural populations frequently neglect the latter. Even in a population that fluctuates substantially with time, the results of a deterministic perturbation analysis correlated strongly with results of a perturbation analysis of the long-run stochastic growth rate. Population growth was, however, not uniformly sensitive to demographic rates across different environmental conditions. The overall correlation between deterministic and stochastic perturbation analysis may be high, but environmental variability can dramatically alter the contributions of demographic rates in different environmental conditions. This potentially informative detail is neglected by deterministic analysis, yet it highlights one difficulty when extrapolating results from long-term analysis to shorter-term environmental change.     

Extinction and exponential growth in random environments.

The influence of randomly varying environments on unrestricted population growth and extinction is analyzed by means of branching processes with random environments (BPRE). A main theme is the interplay between environmental and sampling (or “demographic”) variability. If the two sources of variationg are of comparable magnitude, the environmental variation will dominate except as regards the event of extinction.A diffusion approximation of BPRE is proposed to study the situation of a large population with small environmental variance and mean offspring size near one.Comments on the ecological literature as well as on the relation of the results to previous work involving stochastic differential equations are also given.     

Response of population size to changing vital rates in random environments

Population size and population growth rate respond to changes in vital rates like survival and fertility. In deterministic environments change in population growth rate alone determines change in population size. In random environments, population size at any time t is a random variable so that change in population size obeys a probability distribution. We analytically show that, in a density-independent population, the proportional change in population size with respect to a small proportional change in a vital rate has an asymptotic normal distribution. Its mean grows linearly at a rate equal to the elasticity of the long-term stochastic growth rate λ _S while the standard deviation scales as $\sqrt t$ . Consequently, a vital rate with a larger elasticity of λ _S may produce a larger mean change in population size compared to one with a smaller elasticity of λ _S. But a given percentage change in population size may be more likely when the vital rate with smaller elasticity is perturbed. Hence, the response of population size to perturbation of a vital rate depends not only on the elasticity of the population growth rate but also on the variance in change in population size. Our results provide a formula to calculate the probability that population size changes by a given percentage that works well even for short time periods.     

Individual growth rates of salps in three populations

One of the fastest growing planktonic species, the salp Thaliademocratica, was analysed for growth rate in three differentoceanic blooms in winter, summer and spring. Each bloom wasmarked by a pair of drogues, and sequentially sampled to obtaina sesies of length frequencies in each population through time.The length frequencies were analysed by an improved method ofmodal analysis. Length-specific growth rate was calculated fromthe progression through time of the resulting series of modes.The blooms showed average growth per individual of 10.4, 14.7and 19.8% in length per hour in winter, summer and spring respectively.These growth rates in the field are compared with growth ratesobtained previously in the laboratory, and the reasons for thevarying estimates of growth rate in the literature are examined.     

Polymorphism in random environments

A two-allele diploid model is described in which the fitnesses of the three genotypes are stationary stochastic processes. It is shown that a stable polymorphism will occur if the geometric mean fitness of the heterozygote exceeds that of both homozygotes. It is possible for the mean fitness of the population to be lower in polymorphic than in the associated monomorphic populations.     

Sex in random environments

Based on the theory of natural selection it is not obvious why sexual reproduction should evolve in Mendelian populations. Sexually reproducing organisms incur a “cost of meiosis”: an asexual lineage would grow at twice the rate of a comparable sexual lineage. A plausible and popular explanation for the widespread occurrence of sexual reproduction is that it adapts a lineage to temporal uncertainty in the environment. Computer simulation of a model introduced and partially analyzed in a companion paper (Hines & Moore,1981) suggests that under some of the hypothetical conditions, sexuality is advantageous, but the conditions are very restricted if only one or a few loci are selected. In the companion paper, to make analytical progress, it was necessary to assume small environmental effects or that the fitnesses of the homozygotes at each locus were identical in each generation, although fluctuating between generations. No such assumptions were made here. In addition the effect of an absorbing barrier was studied in the simulations.The computer model envisages from 1–4 loci, each with two alleles, selected independently. In each generation, each locus experiences one of three selection regimes chosen at random; each genotype is favored by one of the three selection regimes. The fitness of a multi-locus genotype is the product of the fitnesses of the independent loci. The sexual species produce genetically varied offspring according to Mendel's laws; the recombination frequency between all loci is 0–5. Members of the asexual species produce offspring that are genetic replicates of themselves. It is important to note that the model represents segregation and independent assortment of genes but not linkage disequilibrium.Computer simulation results were consistent with analytical results, suggesting that inferences can be extrapolated from the analysis without danger of serious error. Both the analysis and simulations reveal a dilemma for the hypothesis that sex is an adaptation to temporal uncertainty; viz., the conditions that are most favorable for sexually are somewhat antithetical (but not prohibitive) to the maintenance of genetic polymorphism in the sexual species whereas sex is useless in a monomorphic population. The dilemma is particularly apparent when only one or a few loci are selected; however, as the number of selected loci increases, the disadvantage in sexuality diminishes. Thus, environmental uncertainty may explain the adaptive significance of sex provided many loci are selected in the prescribed manner.     

Effects of random motility on growth of bacterial populations

A spatially distributed mathematical model is developed to elucidate the effects of chemical diffusion and cell motility as well as cell growth, death, and substrate uptake on steady-state bacterial population growth in a finite, one-dimensional, nonmixed region. The situation considered is growth limited by a diffusing substrate from an adjacent phase not accessible to the bacteria. Chemotactic movement is not considered in this paper; we consider only randomwalk-type random motility behavior here. The following important general concepts are suggested by the results of our theoretical analysis: (a) The significance of random motility effects depends on the magnitude of the ratio/kL ², where is the bacterial random motility coefficient,k is the growth rate constant, andL is the linear dimension of the confined growth region. (b) In steady-state growth in a confined region, the bacterial population size decreases as increases. (c) The effect of on population size can be great; in fact, sometimes relative population sizes of two species can be governed primarily by the relative values of rather than by the relative values ofk.     

Population dynamics in variable environments I. Long-run growth rates and extinction

A model of population growth is studied in which the Leslie matrix for each time interval is chosen according to a Markov process. It is shown analytically that the distribution of total population number is lognormal at long times. Measures of population growth are compared and it is shown that a mean logarithmic growth rate and a logarithmic variance effectively describe growth and extinction at long times. Numerical simulations are used to explore the convergence to lognormality and the effects of environmental variance and autocorrelation. The results given apply to other geometric growth models which involve nonnegative growth matrices.     

The rapid determination in controlled environments of parameters for predicting seedling growth rates in natural conditions

Parameters which characterise the response of seedling growth rate to temperature and mean daily radiant exposure were determined for cultivars of eight vegetable species and two ornamental bedding plant species using controlled environments. All species were grown for approximately four weeks in a series of four controlled environments consisting of factorial combinations of 20°C or 15°C with 7.78 or 0.98 MJ m^-2 radiant exposure per day to photosynthetically active radiation. Four of the vegetable cultivars had been used previously in a study of growth in the glasshouse at different seasons. The parameters derived from controlled environments were used, with temperature and light measurements from the glasshouse, to predict seedling shoot growth in the glasshouse. These predictions accounted for 92% of the observed variance in log shoot dry weight in a series of glasshouse experiments involving thirteen combinations of species and sowing date, each grown at four plant densities and sampled on four occasions. Equivalent parameters derived previously from the glasshouse data themselves accounted for 92% of the variance in the same predictive exercise. Differences in base temperature for growth, the responsiveness of relative growth rate to daily radiant exposure (light), and the potential relative growth rate were apparent between species and were quantitatively characterised by the parameters. To determine these parameters for a cultivar requires the equivalent of one square metre of controlled environment growing area for approximately sixteen days. Such parameters could be applied for scheduling and management in crop production, particularly in transplant production.     

Modelling animal growth in random environments: An application using nonparametric estimation

We study a stochastic differential equation growth model to describe individual growth in random environments. In particular, in this paper, we discuss the estimation of the drift and the diffusion coefficients using nonparametric methods for the case of nonequidistant data for several trajectories. We illustrate the methodology by using bovine growth data. Our goal is to assess: (i) if the parametric models (with specific functional forms for the drift and the diffusion coefficients) previously used by us to describe the evolution of bovine weight were adequate choices; (ii) whether some alternative specific parameterized functional forms of these coefficients might be suggested for further parametric analysis of this data.     

Effect of plant growth on dehydration rates and microbial populations in sewage biosolids

The high water content of sewage biosolids makes them expensive to transport. Two experiments were done to see if it was practical to use transpiration by plants as a low cost method to dehydrate biosolids. (i) To assess biosolids as a growth-medium for plants, maize, beans, pumpkin, forage oats and an annual ryegrass were grown in pots of fresh biosolids. Plant growth varied between species and potassium deficiency was found to be a limiting factor for barley. Roots were slow to penetrate anoxic biosolids in the bottom of the pots. (ii) Dehydration rates were measured in 30 litre boxes of biosolids from two different sources. Boxes were planted with maize or beans, or kept fallow. Despite the high (73-83%) initial water content of the biosolids, plants were susceptible to wilting on warm days which suggested that a significant proportion of the water in biosolids is not freely available to plants. Shrinkage caused a major reduction in biosolids volume in both experiments. When change in volume of the biosolids residue was accounted for, there was a 56% average water loss in planted treatments and 34% water loss in the fallow treatment. This indicates that planting may have some potential as a technique to accelerate dehydration of biosolids. Water contents were not reduced sufficiently to influence biosolids microbial populations.     

The physiological homeostasis of human populations in variable environments

A close evolutionary relationship of physiology and ecology of organisms determines the dynamic dependence of the population physiological status in man on ecological factors. For the explanation of the stability and variability of population physiological status the concept of physiological homeostasis was applied. The investigation of physiological status in several populations of Middle Asia, Kazakhstan, North-East Asia and Khakassia has shown that reversible changes in the environment may temporarily destabilize the equilibrium in the "population-environment" system and prolonged stresses may cause a state of disadaptation. For estimation of population physiological homeostasis dependent on the environment, generalized dispersion, correlation and factor analyses are very informative. They not only mark the violation of population homeostasis, but also indicate changes in the environment.     

Variation in juvenile growth rates among and within latitudinal populations of the medaka

In ectotherms, lower temperatures at high latitudes would theoretically reduce annual growth rates of individuals. If slower growth and resulting smaller body size reduce fitness, individuals at high latitudes may evolve compensatory growth. This study compares individual growth rates among and within 12 latitudinal populations of the medaka (Oryzias latipes). Growth rates during juvenile stage were measured in a common, temperature-controlled (28°C) environment. The results revealed that juvenile growth rates differed significantly among the populations. Growth rates were, moreover, significantly correlated with latitudes of source populations, such that higher-latitude individuals grew faster. Significant variation in growth rates among full-sib families within populations was also demonstrated. The results strongly suggest that higher-latitude O. latipes have acquired a greater capacity for growth as an adaptation to shorter growing seasons (which would reduce annual growth rates), thus refuting probability processes, i.e., genetic drift, founder, or bottleneck effects, as a cause of the among-population variation.     

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.