Turbulent transport of suspended particles and dispersing benthic organisms: the hitting-distance problem for the local exchange model

The local exchange model developed by McNair et al. (1997) provides a stochastic diffusion approximation to the random-like motion of fine particles suspended in turbulent water. Based on this model, McNair (2000) derived equations governing the probability distribution and moments of the hitting time, which is the time until a particle hits the bottom for the first time from a given initial elevation. In the present paper, we derive the corresponding equations for the probability distribution and moments of the hitting distance, which is the longitudinal distance a particle has traveled when it hits the bottom for the first time. We study the dependence of the distribution and moments on a particle's initial elevation and on two dimensionless parameters: an inverse Reynolds number M (a measure of the importance of viscous mixing compared to turbulent mixing of water) and the Rouse number ?(a measure of the importance of deterministic gravitational sinking compared to stochastic turbulent mixing in governing the vertical motion of a particle). We also compute predicted hitting-distance distributions for two published data sets. The results show that for fine particles suspended in moderately to highly turbulent water, the hitting-distance distribution is strongly skewed to the right, with mode相似文献   

Stochastic demography and population dynamics in the red kangaroo Macropus rufus

1.  Many organisms inhabit strongly fluctuating environments but their demography and population dynamics are often analysed using deterministic models and elasticity analysis, where elasticity is defined as the proportional change in population growth rate caused by a proportional change in a vital rate. Deterministic analyses may not necessarily be informative because large variation in a vital rate with a small deterministic elasticity may affect the population growth rate more than a small change in a less variable vital rate having high deterministic elasticity.
2.  We analyse a stochastic environment model of the red kangaroo ( Macropus rufus ), a species inhabiting an environment characterized by unpredictable and highly variable rainfall, and calculate the elasticity of the stochastic growth rate with respect to the mean and variability in vital rates.
3.  Juvenile survival is the most variable vital rate but a proportional change in the mean adult survival rate has a much stronger effect on the stochastic growth rate.
4.  Even if changes in average rainfall have a larger impact on population growth rate, increased variability in rainfall may still be important also in long-lived species. The elasticity with respect to the standard deviation of rainfall is comparable to the mean elasticities of all vital rates but the survival in age class 3 because increased variation in rainfall affects both the mean and variability of vital rates.
5.  Red kangaroos are harvested and, under the current rainfall pattern, an annual harvest fraction of c . 20% would yield a stochastic growth rate about unity. However, if average rainfall drops by more than c . 10%, any level of harvesting may be unsustainable, emphasizing the need for integrating climate change predictions in population management and increase our understanding of how environmental stochasticity translates into population growth rate.     

Partitioning the effects of deterministic and stochastic processes on species extinction risk

Species populations are subjected to deterministic and stochastic processes, both of which contribute to their risk of extinction. However, current understanding of the relative contributions of these processes to species extinction risk is far from complete. Here, we address this knowledge gap by analyzing a suite of models representing species populations with negative intrinsic growth rates, to partition extinction risk according to deterministic processes and two broad classes of stochastic processes – demographic and environmental variance. Demographic variance refers to random variations in population abundance arising from random sampling of events given a particular set of intrinsic demographic rates, whereas environmental variance refers to random abundance variations arising from random changes in intrinsic demographic rates over time. When the intrinsic growth rate was not close to zero, we found that deterministic growth was the main driver of mean time to extinction, even when population size was small. This contradicts the intuition that demographic variance is always an important determinant of extinction risk for small populations. In contrast, when the intrinsic growth rate was close to zero, stochastic processes exerted substantial negative effects on the mean time to extinction. Demographic variance had a greater effect than environmental variance at low abundances, with the reverse occurring at higher abundances. In addition, we found that the combined effects of demographic and environmental variance were often substantially lower than the sum of their effects in isolation from each other. This sub-additivity indicates redundancy in the way the two stochastic processes increase extinction risk, and probably arises because both processes ultimately increase extinction risk by boosting variation in abundance over time.     

Extinction conditions for isolated populations affected by environmental stochasticity

We determine the critical patch size below which extinction occurs for populations living in one-dimensional habitats surrounded by completely hostile environments in the presence of environmental fluctuations. The population dynamics is reformulated in terms of a stochastic reaction–diffusion equation and is reduced to a deterministic equation that incorporates the systematic contributions of the noise. We obtain bifurcation diagrams and relations for the mean population density at the stationary state, the critical patch size, and the mean number of individuals in the habitat. The effect of the noise differs, depending on whether it affects the net growth rate or the intraspecific competition term. Fluctuations in the net growth rate decrease the critical patch size, whereas fluctuations in the competition term do not change the critical patch size. We compare our analytical results with numerical solutions of the stochastic partial differential equations and show that our procedure proves useful in dealing with reaction–diffusion equations with multiplicative noise.     

Community assembly of a euryhaline fish microbiome during salinity acclimation

Microbiomes play a critical role in promoting a range of host functions. Microbiome function, in turn, is dependent on its community composition. Yet, how microbiome taxa are assembled from their regional species pool remains unclear. Many possible drivers have been hypothesized, including deterministic processes of competition, stochastic processes of colonization and migration, and physiological ‘host‐effect’ habitat filters. The contribution of each to assembly in nascent or perturbed microbiomes is important for understanding host–microbe interactions and host health. In this study, we characterized the bacterial communities in a euryhaline fish and the surrounding tank water during salinity acclimation. To assess the relative influence of stochastic versus deterministic processes in fish microbiome assembly, we manipulated the bacterial species pool around each fish by changing the salinity of aquarium water. Our results show a complete and repeatable turnover of dominant bacterial taxa in the microbiomes from individuals of the same species after acclimation to the same salinity. We show that changes in fish microbiomes are not correlated with corresponding changes to abundant taxa in tank water communities and that the dominant taxa in fish microbiomes are rare in the aquatic surroundings, and vice versa. Our results suggest that bacterial taxa best able to compete within the unique host environment at a given salinity appropriate the most niche space, independent of their relative abundance in tank water communities. In this experiment, deterministic processes appear to drive fish microbiome assembly, with little evidence for stochastic colonization.     

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.     

A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

The von Bertalanffy growth curve has been commonly used for modeling animal growth (particularly fish). Both deterministic and stochastic models exist in association with this curve, the latter allowing for the inclusion of fluctuations or disturbances that might exist in the system under consideration which are not always quantifiable or may even be unknown. This curve is mainly used for modeling the length variable whereas a generalized version, including a new parameter b≥1, allows for modeling both length and weight for some animal species in both isometric (b=3) and allometric (b≠3) situations.In this paper a stochastic model related to the generalized von Bertalanffy growth curve is proposed. This model allows to investigate the time evolution of growth variables associated both with individual behaviors and mean population behavior. Also, with the purpose of fitting the above-mentioned model to real data and so be able to forecast and analyze particular characteristics, we study the maximum likelihood estimation of the parameters of the model. In addition, and regarding the numerical problems posed by solving the likelihood equations, a strategy is developed for obtaining initial solutions for the usual numerical procedures. Such strategy is validated by means of simulated examples. Finally, an application to real data of mean weight of swordfish is presented.     

Random variation and concentration effects in PCR

Even though the efficiency of the polymerase chain reaction (PCR) reaction decreases, analyses are made in terms of Galton-Watson processes, or simple deterministic models with constant replication probability (efficiency). Recently, Schnell and Mendoza have suggested that the form of the efficiency, can be derived from enzyme kinetics. This results in the sequence of molecules numbers forming a stochastic process with the properties of a branching process with population size dependence, which is supercritical, but has a mean reproduction number that approaches one. Such processes display ultimate linear growth, after an initial exponential phase, as is the case in PCR. It is also shown that the resulting stochastic process for a large Michaelis-Menten constant behaves like the deterministic sequence x(n) arising by iterations of the function f(x)=x+x/(1+x).     

Demography of Verreaux’s sifaka in a stochastic rainfall environment

In this study, we use deterministic and stochastic models to analyze the demography of Verreaux’s sifaka (Propithecus verreauxi verreauxi) in a fluctuating rainfall environment. The model is based on 16 years of data from Beza Mahafaly Special Reserve, southwest Madagascar. The parameters in the stage-classified life cycle were estimated using mark-recapture methods. Statistical models were evaluated using information-theoretic techniques and multi-model inference. The highest ranking model is time-invariant, but the averaged model includes rainfall-dependence of survival and breeding. We used a time-series model of rainfall to construct a stochastic demographic model. The time-invariant model and the stochastic model give a population growth rate of about 0.98. Bootstrap confidence intervals on the growth rates, both deterministic and stochastic, include 1. Growth rates are most elastic to changes in adult survival. Many demographic statistics show a nonlinear response to annual rainfall but are depressed when annual rainfall is low, or the variance in annual rainfall is high. Perturbation analyses from both the time-invariant and stochastic models indicate that recruitment and survival of older females are key determinants of population growth rate.     

Environmental changes affect the assembly of soil bacterial community primarily by mediating stochastic processes

Both ‘species fitness difference’‐based deterministic processes, such as competitive exclusion and environmental filtering, and ‘species fitness difference’‐independent stochastic processes, such as birth/death and dispersal/colonization, can influence the assembly of soil microbial communities. However, how both types of processes are mediated by anthropogenic environmental changes has rarely been explored. Here we report a novel and general pattern that almost all anthropogenic environmental changes that took place in a grassland ecosystem affected soil bacterial community assembly primarily through promoting or restraining stochastic processes. We performed four experiments mimicking 16 types of environmental changes and separated the compositional variation of soil bacterial communities caused by each environmental change into deterministic and stochastic components, with a recently developed method. Briefly, because the difference between control and treatment communities is primarily caused by deterministic processes, the deterministic change was quantified as (mean compositional variation between treatment and control) – (mean compositional variation within control). The difference among replicate treatment communities is primarily caused by stochastic processes, so the stochastic change was estimated as (mean compositional variation within treatment) – (mean compositional variation within control). The absolute of the stochastic change was greater than that of the deterministic change across almost all environmental changes, which was robust for both taxonomic and functional‐based criterion. Although the deterministic change may become more important as environmental changes last longer, our findings showed that changes usually occurred through mediating stochastic processes over 5 years, challenging the traditional determinism‐dominated view.     

Allee effects and resilience in stochastic populations

Allee effects, or positive functional relationships between a population’s density (or size) and its per unit abundance growth rate, are now considered to be a widespread if not common influence on the growth of ecological populations. Here we analyze how stochasticity and Allee effects combine to impact population persistence. We compare the deterministic and stochastic properties of four models: a logistic model (without Allee effects), and three versions of the original model of Allee effects proposed by Vito Volterra representing a weak Allee effect, a strong Allee effect, and a strong Allee effect with immigration. We employ the diffusion process approach for modeling single-species populations, and we focus on the properties of stationary distributions and of the mean first passage times. We show that stochasticity amplifies the risks arising from Allee effects, mainly by prolonging the amount of time a population spends at low abundance levels. Even weak Allee effects become consequential when the ubiquitous stochastic forces affecting natural populations are accounted for in population models. Although current concepts of ecological resilience are bound up in the properties of deterministic basins of attraction, a complete understanding of alternative stable states in ecological systems must include stochasticity.     

Role of gestation delay in a plankton-fish model under stochastic fluctuations

The present paper studies a minimal prey-predator model in the context of marine plankton interaction together with predation by planktivorous fish. The time lag required for gestation of the predator is incorporated and the resulting delayed model is analyzed for stability and bifurcation phenomena. A stochastic extension of the model is considered by perturbing the growth process of phytoplankton using colored noise process known to be more appropriate for the marine environment. The stochastic models with and without gestation delay are analyzed for stability aspects and a threshold value of gestation delay is obtained; this threshold is then compared with that of the deterministic model.     

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.     

Spread rate for a nonlinear stochastic invasion

Despite the recognized importance of stochastic factors, models for ecological invasions are almost exclusively formulated using deterministic equations [29]. Stochastic factors relevant to invasions can be either extrinsic (quantities such as temperature or habitat quality which vary randomly in time and space and are external to the population itself) or intrinsic (arising from a finite population of individuals each reproducing, dying, and interacting with other individuals in a probabilistic manner). It has been long conjectured [27] that intrinsic stochastic factors associated with interacting individuals can slow the spread of a population or disease, even in a uniform environment. While this conjecture has been borne out by numerical simulations, we are not aware of a thorough analytical investigation. In this paper we analyze the effect of intrinsic stochastic factors when individuals interact locally over small neighborhoods. We formulate a set of equations describing the dynamics of spatial moments of the population. Although the full equations cannot be expressed in closed form, a mixture of a moment closure and comparison methods can be used to derive upper and lower bounds for the expected density of individuals. Analysis of the upper solution gives a bound on the rate of spread of the stochastic invasion process which lies strictly below the rate of spread for the deterministic model. The slow spread is most evident when invaders occur in widely spaced high density foci. In this case spatial correlations between individuals mean that density dependent effects are significant even when expected population densities are low. Finally, we propose a heuristic formula for estimating the true rate of spread for the full nonlinear stochastic process based on a scaling argument for moments. Received: 19 October 1998 / Revised version: 1 September 1999 / Published online: 4 October 2000     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Turbulent transport of suspended particles and dispersing benthic organisms: the hitting-time distribution for the local exchange model

Fine particles suspended in turbulent water exhibit highly irregular trajectories as they are buffeted by fluid eddies. The Local Exchange Model provides a stochastic diffusion approximation to the randomlike motion of such particles (e.g. dispersing benthic organisms in a stream). McNair et al. (1997, J. theor. Biol.188, 29) used this model to derive equations governing the mean hitting time, which is the expected time until a particle hits bottom for the first time from a given initial elevation. The present paper derives equations governing the probability distribution of the hitting time, then studies the distribution's dependence on a particle's initial elevation and two dimensionless parameters. The results show that for fine particles suspended in moderately to highly turbulent water, the hitting-time distribution is strongly skewed to the right, with mode相似文献