Persistence in fluctuating environments for interacting structured populations

Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and species coexistence. To understand these effects, we provide a general theory for coexistence of structured, interacting species living in a stochastic environment. The theory is applicable to nonlinear, multi species matrix models with stochastically varying parameters. The theory relies on long-term growth rates of species corresponding to the dominant Lyapunov exponents of random matrix products. Our coexistence criterion requires that a convex combination of these long-term growth rates is positive with probability one whenever one or more species are at low density. When this condition holds, the community is stochastically persistent: the fraction of time that a species density goes below \(\delta >0\) approaches zero as \(\delta \) approaches zero. Applications to predator-prey interactions in an autocorrelated environment, a stochastic LPA model, and spatial lottery models are provided. These applications demonstrate that positive autocorrelations in temporal fluctuations can disrupt predator-prey coexistence, fluctuations in log-fecundity can facilitate persistence in structured populations, and long-lived, relatively sedentary competing populations are likely to coexist in spatially and temporally heterogenous environments.     

Diffusion analysis and stationary distribution of the two-species lottery competition model

The lottery model is a stochastic population model in which juveniles compete for space. Examples include sedentary organisms such as trees in a forest and members of marine benthic communities. The behavior of this model appears to be characteristic of that found in other sorts of stochastic competition models. In a community with two species, it was previously demonstrated that coexistence of the species is possible if adult death rates are small and environmental variation is large. Environmental variation is incorporated by assuming that the birth rates and death rates are random variables. Complicated conditions for coexistence and competitive exclusion have been derived elsewhere. In this paper, simple and easily interpreted conditions are found by using the technique of diffusion approximation. Formulae are given for the stationary distribution and means and variances of population fluctuations. The shape of the stationary distribution allows the stability of the coexistence to be evaluated.     

The age-structured lottery model

The lottery model of competition between species in a variable environmental has been influential in understanding how coexistence may result from interactions between fluctuating environmental and competitive factors. Of most importance, it has led to the concept of the storage effect as a mechanism of species coexistence. Interactions between environment and competition in the lottery model stem from the life-history assumption that environmental variation and competition affect recruitment to the adult population, but not adult survival. The strong role of life-history attributes in this coexistence mechanism implies that its robustness should be checked for a variety of life-history scenarios. Here, age structure is added to the adult population, and the results are compared with the original lottery model. This investigation uses recently developed shape characteristics for mortality and fecundity schedules to quantify the effects of age structure on the long-term low-density growth rate of a species in competition with its competitor when applying the standard invasibility coexistence criterion. Coexistence conditions are found to be affected to a small degree by the presence of age structure in the adult population: Type III mortality broadens coexistence conditions, and type I mortality makes them narrower. The rates of recovery from low density for coexisting species, and the rates of competitive exclusion in other cases, are modified to a greater degree by age structure. The absolute rates of recovery or decline of a species from low density are increased by type I mortality or early peak reproduction, but reduced by type III mortality or late peak reproduction. Analytical approximations show how the most important effects can be considered as simple modifications of the long-term low-density growth rates for the original lottery model.     

Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments

This paper presents a study of a nonlinear reaction–diffusion population model in fragmented environments. The model is set on , with periodic heterogeneous coefficients obtained using stochastic processes. Using a criterion of species persistence based on the notion of principal eigenvalue of an elliptic operator, we provided a precise numerical analysis of the interactions between habitat fragmentation and species persistence. The obtained results clearly indicated that species persistence strongly tends to decrease with habitat fragmentation. Moreover, comparing two stochastic models of landscape pattern generation, we observed that in addition to local fragmentation, a more global effect of the position of the habitat patches also influenced species persistence.      

Survival Analysis of Stochastic Competitive Models in a Polluted Environment and Stochastic Competitive Exclusion Principle

Stochastic competitive models with pollution and without pollution are proposed and studied. For the first system with pollution, sufficient criteria for extinction, nonpersistence in the mean, weak persistence in the mean, strong persistence in the mean, and stochastic permanence are established. The threshold between weak persistence in the mean and extinction for each population is obtained. It is found that stochastic disturbance is favorable for the survival of one species and is unfavorable for the survival of the other species. For the second system with pollution, sufficient conditions for extinction and weak persistence are obtained. For the model without pollution, a partial stochastic competitive exclusion principle is derived.     

Evidence for lottery recruitment in Mediterranean old fields

Abstract. The hypothesis of lottery establishment ( Sale 1977 ) explains coexistence of species with similar niches through processes of stochastic recruitment. This initial idea forms the basis for a variety of mathematical models, but has not been tested empirically. This study is a field investigation of lottery establishment for plants with a seed bank, using Canonical Correspondence Analysis to compare the compositions of the vegetation and the seed bank according to different hypotheses on the mechanisms of establishment. This method was used for a data set from old fields from southern France. The weighted lottery (i.e. a random draw from the seed pool, weighted by the frequencies of each species) appeared as the best suited hypothesis to explain the high degree of similarity between the vegetation and the seed bank and the relative spatial distributions of the species. Several mechanisms are probably interacting, depending on the life histories of the species. Modelling and experimental approaches are needed to further test the hypothesis of lottery recruitment.     

Coexistence in a fluctuating environment by the effect of relative nonlinearity: A minimal model

The minimal model of the “relative nonlinearity” type fluctuation-maintained coexistence is investigated. The competing populations are affected by an environmental white noise. With quadratic density dependence, the long-term growth rates of the populations are determined by the average and the variance of the (fluctuating) total density. At most two species can coexist on these two “regulating” variables; competitive exclusion would ensue in a constant environment. A numerical study of the expected time until extinction of any of the two species reveals that the criterion of mutual invasibility predicts the parameter range of long-term coexistence correctly in the limit of zero extinction threshold. However, any extinction threshold consistent with a realistic population size will allow only short-term coexistence. Therefore, our simulations question the biological relevance of mutual invasibility, as a sufficient condition of coexistence, for large density fluctuations. We calculate the average and the variance of the fluctuating density of the coexisting populations analytically via the moment-closure approximation; the results are reasonably close to the simulated behavior. Based on this treatment, robustness of coexistence is studied in the limit of infinite population size. We interpret the results of this analysis in the context of necessity of niche segregation with respect to the regulating variables using a framework theory published earlier.     

Effects of temporal variability on rare plant persistence in annual systems

Traditional conservation biology regards environmental fluctuations as detrimental to persistence, reducing long-term average growth rates and increasing the probability of extinction. By contrast, coexistence models from community ecology suggest that for species with dormancy, environmental fluctuations may be essential for persistence in competitive communities. We used models based on California grasslands to examine the influence of interannual fluctuations in the environment on the persistence of rare forbs competing with exotic grasses. Despite grasses and forbs independently possessing high fecundity in the same types of years, interspecific differences in germination biology and dormancy caused the rare forb to benefit from variation in the environment. Owing to the buildup of grass competitors, consecutive favorable years proved highly detrimental to forb persistence. Consequently, negative temporal autocorrelation, a low probability of a favorable year, and high variation in year quality all benefited the forb. In addition, the litter produced by grasses in a previously favorable year benefited forb persistence by inhibiting its germination into highly competitive grass environments. We conclude that contrary to conventional predictions of conservation and population biology, yearly fluctuations in climate may be essential for the persistence of rare species in invaded habitats.     

A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability

A discrete-time Markov chain model, a continuous-time Markov chain model, and a stochastic differential equation model are compared for a population experiencing demographic and environmental variability. It is assumed that the environment produces random changes in the per capita birth and death rates, which are independent from the inherent random (demographic) variations in the number of births and deaths for any time interval. An existence and uniqueness result is proved for the stochastic differential equation system. Similarities between the models are demonstrated analytically and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models satisfy certain consistency conditions.     

Invasibility and stochastic boundedness in monotonic competition models

We give necessary and sufficient conditions for stochastically bounded coexistence in a class of models for two species competing in a randomly varying environment. Coexistence is implied by mutual invasibility, as conjectured by Turelli. In the absence of invasibility, a species converges to extinction with large probability if its initial population is small, and extinction of one species must occur with probability one regardless of the initial population sizes. These results are applied to a general symmetric competition model to find conditions under which environmental fluctuations imply coexistence or competitive exclusion.     

A comparison of three different stochastic population models with regard to persistence time

Results are summarized from the literature on three commonly used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.     

Stochastic instability and Liapunov stability

Summary The relationship between the deterministic stability of nonlinear ecological models and the properties of the stochastic model obtained by adding weak random perturbations is studied. It is shown that the expected escape time for the stochastic model from a bounded region with nonsingular boundary is determined by a Liapunov function for the nonlinear deterministic model. This connection between stochastic and deterministic models brings together various notions of persistence and vulnerability of ecosystems as defined for deterministically perturbed or randomly perturbed models.     

A minimum stochastic model evaluating the interplay between population density and drift for species coexistence

Despite the general acknowledgment of the role of niche and stochastic process in community dynamics, the role of species relative abundances according to both perspectives may have different effects regarding coexistence patterns. In this study, we explore a minimum probabilistic stochastic model to determine the relationship of populations relative and total abundances with species chances to outcompete each other and their persistence in time (i.e., unstable coexistence). Our model is focused on the effects drift (i.e., random sampling of recruitment) under different scenarios of selection (i.e., fitness differences between species). Our results show that taking into account the stochasticity in demographic properties and conservation of individuals in closed communities (zero-sum assumption), initial population abundance can strongly influence species chances to outcompete each other, despite fitness inequalities between populations, and also, influence the period of coexistence of these species in a particular time interval. Systems carrying capacity can have an important role in species coexistence by exacerbating fitness inequalities and affecting the size of the period of coexistence. Overall, the simple stochastic formulation used in this study demonstrated that populations initial abundances could act as an equalizing mechanism, reducing fitness inequalities, which can favor species coexistence and even make less fitted species to be more likely to outcompete better-fitted species, and thus to dominate ecological communities in the absence of niche mechanisms. Although our model is restricted to a pair of interacting species, and overall conclusions are already predicted by the Neutral Theory of Biodiversity, our main objective was to derive a model that can explicitly show the functional relationship between population densities and community mono-dominance odds. Overall, our study provides a straightforward understanding of how a stochastic process (i.e., drift) may affect the expected outcome based on species selection (i.e., fitness inequalities among species) and the resulting outcome regarding unstable coexistence among species.     

Lottery coexistence models extended to plants with disjoint generations

Abstract. Neither conventional niche theory nor current lottery models offer a satisfactory theoretical scope for modelling coexistence of species with disjoint generations. South-African fynbos and Australian kwongan include many species which are killed by, and recruit only after, fire. We propose a density-dependent lottery model which accommodates the unusual demographics of these species. We show that coexistence requires density dependence in recruitment. The result applies to a wider class of populations than the one considered here. It is applied to non-resprouting species in fynbos and kwongan. We show that the lottery assumption of recruitment in proportion to propagules is often satisfied, while the production of such propagules is often density-dependent, and we discuss some evidence of mechanisms whereby this may occur.     

General theory of competitive coexistence in spatially-varying environments

A general model of competitive and apparent competitive interactions in a spatially-variable environment is developed and analyzed to extend findings on coexistence in a temporally-variable environment to the spatial case and to elucidate new principles. In particular, coexistence mechanisms are divided into variation-dependent and variation-independent mechanisms with variation-dependent mechanisms including spatial generalizations of relative nonlinearity and the storage effect. Although directly analogous to the corresponding temporal mechanisms, these spatial mechanisms involve different life history traits which suggest that the spatial storage effect should arise more commonly than the temporal storage effect and spatial relative nonlinearity should arise less commonly than temporal relative nonlinearity. Additional mechanisms occur in the spatial case due to spatial covariance between the finite rate of increase of a local population and its local abundance, which has no clear temporal analogue. A limited analysis of these additional mechanisms shows that they have similar properties to the storage effect and relative nonlinearity and potentially may be considered as enlargements of the earlier mechanisms. The rate of increase of a species perturbed to low density is used to quantify coexistence. A general quadratic approximation, which is exact in some important cases, divides this rate of increase into contributions from the various mechanisms above and admits no other mechanisms, suggesting that opportunities for coexistence in a spatially-variable environment are fully characterized by these mechanisms within this general model. Three spatially-implicit models are analyzed as illustrations of the general findings and of techniques using small variance approximations. The contributions to coexistence of the various mechanisms are expressed in terms of simple interpretable formulae. These spatially-implicit models include a model of an annual plant community, a spatial multispecies version of the lottery model, and a multispecies model of an insect community competing for spatially-patchy and ephemeral food.     

Carrying capacity and demographic stochasticity: scaling behavior of the stochastic logistic model

The stochastic logistic model is the simplest model that combines individual-level demography with density dependence. It explicitly or implicitly underlies many models of biodiversity of competing species, as well as non-spatial or metapopulation models of persistence of individual species. The model has also been used to study persistence in simple disease models. The stochastic logistic model has direct relevance for questions of limiting similarity in ecological systems. This paper uses a biased random walk heuristic to derive a scaling relationship for the persistence of a population under this model, and discusses its implications for models of biodiversity and persistence. Time to extinction of a species under the stochastic logistic model is approximated by the exponential of the scaling quantity U=(R-1)(2) N/R(R+1), where N is the habitat size and R is the basic reproductive number.     

Niche partitioning and stochastic processes shape community structure following whitefly invasions

One of the most detrimental impacts of invasive species is the exclusion of native species, which reduces biodiversity and can alter community structure. Coexistence between invaders and native species across large scales, however, might be promoted by niche partitioning and/or stochastic processes, even when one species is excluded in some habitats. Here, we examined the effects of species traits, stochastic processes, and niche partitioning on coexistence of two morphocryptic whitefly species in the Bemisia tabaci complex: the invasive Mediterranean (MED) species and the native Middle East-Asia Minor 1 (MEAM1) species. These species engage in intense reproductive interference, which can result in the exclusion of one species or the other in shared habitats. Both species, however, have coexisted in sympatry in Israel for many years, where MED is invasive and MEAM1 is native. Using a spatially explicit model, we show that both stochastic processes and niche partitioning can promote coexistence between MEAM1 and MED, although predicted community structure differs drastically in each scenario. Comparison of field observations with model results indicated that variation in habitat use leading to niche partitioning was a primary factor driving coexistence between MEAM1 and MED across landscapes, although stochastic processes affected the establishment of rare species within habitats. In many systems, combining models with field surveys can be used to isolate and test mechanisms underlying patterns of community structure following invasions.     

The effects of serotiny and rainfall-cued dispersal on fitness: bet-hedging in the threatened cactus <Emphasis Type="Italic">Mammillaria pectinifera</Emphasis>

Serotiny—the retention of seeds in the mother plant for over a year—in unpredictable environments may increase the probability that at least some seeds are dispersed during favorable periods. Propagules may be expelled when environmental cues announcing favorable conditions occur, or be gradually released into the environment. This could be a bet-hedging strategy increasing the long-term fitness by reducing interannual variability in reproduction. However, the impact of seed retention on the population dynamics of serotinous species and its contribution to fitness has been barely explored under field conditions. We assessed these issues in the threatened Mammillaria pectinifera, a small globose cactus that gets established only in exceptionally rainy years. This species expels some seeds actively during unusually rainy periods, while dispersing others passively over several years. Dynamics of the seeds in the mother plant over two very contrasting years in terms of precipitation was incorporated into a stochastic matrix model. Seed retention was found to increase significantly the probability that some of the seeds retained in any given year are dispersed within a subsequent rainy period. Active seed-expulsion raises this probability even further. As expected in bet hedgers, seed retention increased fitness in the presence of temporal variability. Active fruit expulsion did not affect fitness, but reduced demographic stochasticity. The incomplete serotiny and fruit expulsion observed is the evolutionary outcome expected for the environment and life-history attributes of the species.     

Contributions to nonstationary community theory

ABSTRACT

The study of the role of environmental variation in community dynamics has traditionally assumed that the environment is a stationary stochastic process or a periodic deterministic process. However, the physical environment in nature is nonstationary. Moreover, anthropogenically driven climate change provides a new challenge emphasizing a persistent but frequently ignored problem: how to make predictions about the dynamics of communities when the nonstationarity of the physical environment is recognized. Recent work is providing a path to conclusions with none of the traditional assumptions of environmental stationarity or periodicity. Traditional assumptions about convergence of long-term averages of functions of environmental states can be replaced by assumptions about temporal sums, allowing convergence and persistence of population processes to be demonstrated in general nonstationary environments. These tools are further developed and illustrated here with some simple models of nonstationary community dynamics, including the Beverton-Holt model, the threshold exponential and the lottery model.     

Divide and conquer? Persistence of infectious agents in spatial metapopulations of hosts

Persistence of an infectious agent in a population is an important issue in epidemiology. It is assumed that spatially fragmenting a population of hosts increases the probability of persistence of an infectious agent and that movement of hosts between the patches is vital for that. The influence of migration on persistence is however often studied in mean-field models, whereas in reality the actual distance travelled can be limited and influence the movement dynamics. We use a stochastic model, where within- and between-patch dynamics are coupled and movement is modelled explicitly, to show that explicit consideration of movement distance makes the relation between persistence of infectious agents and the metapopulation structure of its hosts less straightforward than previously thought. We show that the probability of persistence is largest at an intermediate movement distance of the host and that spatially fragmenting a population of hosts is not necessarily beneficial for persistence.