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.     

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.     

Variable effort fishing models in random environments

We study the growth of populations in a random environment subjected to variable effort fishing policies. The models used are stochastic differential equations and the environmental fluctuations may either affect an intrinsic growth parameter or be of the additive noise type. Density-dependent natural growth and fishing policies are of very general form so that our results will be model independent. We obtain conditions on the fishing policies for non-extinction and for non-fixation at the carrying capacity that are very similar to the conditions obtained for the corresponding deterministic model. We also obtain conditions for the existence of stationary distributions (as well as expressions for such distributions) very similar to conditions for the existence of an equilibrium in the corresponding deterministic model. The results obtained provide minimal requirements for the choice of a wise density-dependent fishing policy.     

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.     

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.     

Invariant distributions for multi-population models in random environments

We obtain the existence of a solution and invariant distribution for systems of stochastic differential equations which represent populations in random environments. The method used is a stochastic Lyapunov function, based on a theorem of Kushner. The method is applied to a system of two populations exchainging individuals through migration, and to a generalized n-dimensional Lotka-Volterra system.     

An uncertain life: demography in random environments

This paper concisely reviews the demography of populations with random vital rates, highlights examples and techniques which yield insight into population dynamics, summarizes the state of significant applications of the theory, and points to open problems. The central picture in this theory is of a time-varying but statistically stationary equilibrium for population, sharply distinct from the notions of classical demography. The deepest biological insights from the theory reveal the temporal structure of life histories to be a rich arena for natural selection.     

Timescales of population rarity and commonness in random environments

This is a mathematical study of the interactions between non-linear feedback (density dependence) and uncorrelated random noise in the dynamics of unstructured populations. The stochastic non-linear dynamics are generally complex, even when the deterministic skeleton possesses a stable equilibrium. There are three critical factors of the stochastic non-linear dynamics; whether the intrinsic population growth rate (lambda) is smaller than, equal to, or greater than 1; the pattern of density dependence at very low and very high densities; and whether the noise distribution has exponential moments or not. If lambda < 1, the population process is generally transient with escape towards extinction. When lambda > or = 1, our quantitative analysis of stochastic non-linear dynamics focuses on characterizing the time spent by the population at very low density (rarity), or at high abundance (commonness), or in extreme states (rarity or commonness). When lambda >1 and density dependence is strong at high density, the population process is recurrent: any range of density is reached (almost surely) in finite time. The law of time to escape from extremes has a heavy, polynomial tail that we compute precisely, which contrasts with the thin tail of the laws of rarity and commonness. Thus, even when lambda is close to one, the population will persistently experience wide fluctuations between states of rarity and commonness. When lambda = 1 and density dependence is weak at low density, rarity follows a universal power law with exponent -3/2. We provide some mathematical support for the numerical conjecture [Ferriere, R., Cazelles, B., 1999. Universal power laws govern intermittent rarity in communities of interacting species. Ecology 80, 1505-1521.] that the -3/2 power law generally approximates the law of rarity of 'weakly invading' species with lambda values close to one. Some preliminary results for the dynamics of multispecific systems are presented.     

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.     

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.     

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.     

Predictive performance of random forest on the identification of mangrove species in arid environments

Machine learning methods are the most popular approaches for carrying out classification in remote sensing studies. Of the methods available, random forest (RF) is the one most often used due to its high predictive performance. The objective of this study was to assess the predictive performance of RF in identifying (classifying) mangrove species in an arid environment using two cameras: one conventional (visible part of the light, RGB), the other specialized (Green, Red, Near-infrared, GRN). The RGB and GRN bands were used with derived vegetation indexes (for each camera), the canopy height model (derived from photogrammetry), and distance to water (derived from raster analysis) to classify the study area in eight classes (including three mangrove species) using RF. Results suggest only slight differences in predictive performance (validation) between the products derived from the GRN and RGB cameras, the accuracy values ranged from 0.58 to 0.77 and from 0.53 to 0.72 for RGB and GRN, respectively. The most important variables were the distance to water and canopy height model for both cameras, followed by specific bands and vegetation indices. The study concludes that conventional cameras mounted in commercial drones can be used efficiently to identify mangrove species in arid environments when the classification model uses physical variables of the species (tree height) and the system (distance to water). Results of this study can be applied to describe spatial distributions by species in small or large patches of mangroves in arid environments, thus improving our ecological knowledge of this ecosystem.     

Niche overlap and invasion of competitors in random environments I. Models without demographic stochasticity

The relationship between persistent, small to moderate levels of random environmental fluctuations and limits to the similarity of competing species is studied. The analytical theory hinges on deriving conditions under which a rare invading species will tend to increase when faced with an array of resident competitors in a fluctuating environment. A general approximation scheme predicts that the effects of low levels of stochasticity will typically be small. The technique is applied explicitly to a class of symmetric, discrete-time stochastic analogs of the Lotka-Volterra equations that incorporate cross-correlation but no autocorrelation. The random environment limits to similarity are always very close to the corresponding constant environment limits. However, stochasticity can either facilitate or hinder invasion. The exact limits to similarity are extremely model-dependent. In addition to the symmetric models, an analytically tractable class of models is presented that incorporates both auto- and cross-correlation and no symmetry assumptions. For all of the models investigated, the analytical theory predicts that small-scale stochasticity does little, if anything, to limit similarity. Extensive Monte Carlo results are presented that confirm the analytical results whenever the dynamics of the discretetime models are biologically reasonable in the sense that trajectories do not exhibit unrealistic crashes. Interestingly, the class of stochastic models that is well behaved in this sense includes models whose deterministic analogs are chaotic. The qualitative conclusion, supported by both the analytical and simulation results, is that for competitive guilds adequately modeled by Lotka-Volterra equations including small to moderate levels of random fluctuations, practical limits to similarity can be obtained by ignoring the stochastic terms and performing a deterministic analysis. The mathematical and biological robustness of this conclusion is discussed.     

Pairing statistics and melting of random DNA oligomers: Finding your partner in superdiverse environments

Understanding of the pairing statistics in solutions populated by a large number of distinct solute species with mutual interactions is a challenging topic, relevant in modeling the complexity of real biological systems. Here we describe, both experimentally and theoretically, the formation of duplexes in a solution of random-sequence DNA (rsDNA) oligomers of length L = 8, 12, 20 nucleotides. rsDNA solutions are formed by 4^L distinct molecular species, leading to a variety of pairing motifs that depend on sequence complementarity and range from strongly bound, fully paired defectless helices to weakly interacting mismatched duplexes. Experiments and theory coherently combine revealing a hybridization statistics characterized by a prevalence of partially defected duplexes, with a distribution of type and number of pairing errors that depends on temperature. We find that despite the enormous multitude of inter-strand interactions, defectless duplexes are formed, involving a fraction up to 15% of the rsDNA chains at the lowest temperatures. Experiments and theory are limited here to equilibrium conditions.     

Polymorphism maintenance in populations with mixed random mating and apomixis subjected to stabilizing and cyclical selection

We analysed a diploid population model with a mixed breeding system that includes panmixia and apomixis. Each individual produces a part (ss) of its progeny by random mating, the remainder (1-ss) being a result of precise copying (vegetative reproduction or apomixis) of the parental genotype. Both constant and periodically varying selection regimes were considered. In the main model, the selected trait was controlled by two diallelic additive or semidominant loci, A/a and B/b, whereas the parameter of breeding system (ss) was genotype-independent. A numerical iteration of the evolutionary equations were used to evaluate the proportion (V) of population trajectories converging to internal (polymorphic) fixed points. The results were the following. (a) A complex pattern of dependence of polymorphism stability on interaction among the breeding system, recombination rate, and the genetic architecture of the selected trait emerged. (b) The recombination provided some advantage to sex at intermediate period lengths and strong-to-moderate selection intensities. (c) The complex limiting behavior (CLB) was quite compatible with sexual reproduction, at least within the framework of pure genetic (not including variations in population density) models of multilocus varying selection.     

Polymorphism in heterogeneous environments,evolution of habitat selection and sympatric speciation: Soft and hard selection models

Summary The adaptation to a variable environment has been studied within soft and hard selection frameworks. It is shown that an epistatically determined habitat preference, following a Markovian process, always leads to the maintenance of an adaptive polymorphism, in a soft selection context. Although local mating does not alter the conditions for polymorphism maintenance, it is shown that, in that case, habitat selection also leads to the evolution of isolated reproductive units within each available habitat. Habitat selection, however, cannot evolve in the total absence of adaptive polymorphism. This represents a theoretical problem for all models assuming habitat selection to be an initially fixed trait, and means that within a soft selection framework, all the available habitats will be exploited, even the less favourable ones.On the other hand, polymorphism cannot be maintained when selection is hard, even when all individuals select their habitat. Here, the evolution of habitat selection does not need any prerequisite polymorphism, and always leads to the exploitation of only one habitat by the most specialized genotype. It appears then that hard selection can account for the existence of empty habitat and for an easier evolution of habitat specialization.     

Statistical properties of estimators for continuous time Markov branching processes with varying and random environments

Asymptotic properties are established for estimators of time dependent intensities in Markov branching processes with varying and random environments. For the varying environment model, the estimators are shown to be uniformly strongly consistent on bounded intervals as the initial population size X₀ → ∞, and, when considered as empirical stochastic processes, to converge weakly to Gaussian processes with independent increments. For random environments, the estimators are shown to be asymptotically normal as t → ∞, where t is the time parameter.