Importance Of Stochastic Processes Research Articles

Patterns and assembly processes of planktonic and sedimentary bacterial community differ along a trophic gradient in freshwater lakes

Abstract The water column (fluidic) and sediments (non-fluidic) represent two distinct habitats in lakes. Planktonic and sedimentary bacterial communities are fundamental components of the biogeochemical processes within freshwater lakes. Anthropogenic activities induced eutrophication has been recognized as one of the most severe environmental problems for the lake ecosystem. However, little is known regarding the responses of bacterial communities in fluidic and non-fluidic lake habitats to eutrophication nor the underlying ecological mechanisms. Here, we investigated the planktonic and sedimentary bacterial communities from a series of freshwater lakes across a range of trophic states. We observed a high heterogeneity in bacterial community composition between habitats and along the trophic gradient. Alpha diversity of the sedimentary bacterial community decreased monotonically with increased eutrophication, whereas that of planktonic bacterial community exhibited a downward arched (parabolic) pattern along the trophic gradient. Co-occurrence network analysis revealed that the network of planktonic bacterial community is more susceptible to eutrophication than the network of sedimentary bacterial community. Null-model analysis suggested that eutrophication enhanced the contributions of stochastic processes in controlling the assembly of planktonic and sedimentary bacterial communities. However, the importance of stochastic processes in the assembly of planktonic bacterial community reduced within lakes experiencing extreme eutrophication. The differences in environmental characteristics and community assembly processes between water and sediment may account for the discrepancy in bacterial diversity patterns of the two lacustrine habitat types along the same trophic gradient. These results offer insights into the successional patterns of bacterial communities in freshwater lakes subjected to anthropogenic eutrophication. Moreover, the different distribution patterns and assembly processes of bacterial communities act as indicators of specific habitat responding to environmental gradients.

Spatial Variation in Fish Species Richness on Coral Reefs: Habitat Fragmentation and Stochastic Structuring Processes

Most studies of coral reef fish communities have been restricted to site-attached species on small, isolated patches of habitat. Few have investigated spatial variation in fish species richness in relation to predictions based on stochastic or deterministic processes of community organisation. Our aims were to: (1) compare species richness on contiguous and fragmented reef habitats, and (2) investigate the mechanisms underlying spatial variation in species richness. Quantitative comparison of species-area curves for contiguous and patchy coral reef indicated that patch reefs support more species than equivalent areas of contiguous reef. However, Monte-Carlo simulated rarefaction curves indicated little difference in the species-individuals relationship for both habitats. Rarefaction was employed to eliminate variation in species richness among sites due to differences in sample size (number of fish present). After removal of sample-size effects, multiple regression models explained 30% and 25% of total variability in species richness on contiguous and patchy coral reef based on variation in habitat structure (e.g., depth, shelter availability, substratum characteristics). To investigate the likely importance of stochastic processes in determining spatial variation in species richness, we compared the species-individuals relationship from contiguous reef sites with the relationships derived from null models involving the random reallocation of fish among sites. Comparisons of the observed data with the outcomes of the null models indicated that spatial variation in species richness was not wholly attributable to stochastic processes. We suggest that the observed patterns of species richness may reflect species interactions (e.g., competition and predation) within fish communities.

Accretion in the inner nebula: The relationship between terrestrial planetary compositions and meteorites*

Abstract— The bulk compositions of the terrestrial planets are assessed. Venus and Earth probably have similar bulk compositions, but Mars is enriched in volatile elements. The inner planets are all depleted in volatile elements, as shown by K/U ratios, relative to most meteorites and the CI primordial values. Terrestrial upper mantle Mg/Si ratios are high compared with CI data. If they are representative of the bulk Earth, then the Earth accreted from a segregated suite of planetesimals that had non‐chondritic major element abundances. The CI meteorite abundances, despite aqueous alteration, match the solar data and provide the best estimate for the composition of the solar nebula, including the iron abundance. The widespread depletion of volatile elements in the inner solar nebula is most likely caused by heating related to early violent solar activity (e.g., T Tauri and FU Orionis stages) which, for example, drove water out to a “snow line” in the vicinity of Jupiter. The variation in composition among the meteorites and the apparent lack of mixing among the groups indicates accretion from narrow feeding zones. There appears to have been little mixing between meteorite and planetary formation zones, as shown by the oxygen isotope variations, lack of mixing of meteorite groups, and differences in K/U ratios. In summary, it appears that the final accretion of planets did not result in widespread homogenization, and that mixing zones were not more than about 0.3 A.U. wide. Although the composition of the Moon is unique, and its origin due to an essentially random event, its presence reinforces the planetesimal hypothesis and the importance of stochastic processes during planetary accretion in the inner solar system.

On the Generalized "Birth-and-Death" Process

The importance of stochastic processes in relation to problems of population growth was pointed out by W. Feller [1] in 1939. He considered among other examples the "birth-and-death" process in which the expected birth and death rates (per head of population per unit of time) were constants, $\lambda_o$ and $\mu_o$, say. In this paper, I shall give the complete solution of the equations governing the generalised birth-and-death process in which the birth and death rates $\lambda(t)$ and $\mu(t)$ may be any specified functions of the time. The mathematical method employed starts from M. S. Bartlett's idea of replacing the differential-difference equations for the distribution of the population size by a partial differential equation for its generating function. For an account of this technique,$^1$ reference may be made to Bartlett's North Carolina lectures [2]. The formulae obtained lead to an expression for the probability of the ultimate extinction of the population, and to the necessary and sufficient condition for a birth-and-death process to be of "transient" type. For transient processes the distribution of the cumulative population is also considered, but here in general it is not found possible to do more than evaluate its mean and variance as functions of $t$, although a complete solution (including the determination of the asymptotic form of the distribution as $t$ tends to infinity) is obtained for the simple process in which the birth and death rates are independent of the time. It is shown that a birth-and-death process can be constructed to give an expected population size $\bar n_t$ which is any desired function of the time $t$, and among the many possible solutions the unique one is determined which makes the fluctuation, Var$(n_t)$, a minimum for all. The general theory is illustrated with reference of two examples. The first of these is the $(\lambda_0, \mu_1t)$ process introduced by N. Arley [3] in his study of the cascade showers associated with cosmic radiation; here the birth rate is constant and the death rate is a constant multiple of the "age, $t$, of the process. The $\bar n_t$-curve is then Gaussian in form, and the process is always of transient type. The second example is provided by the family of "periodic" processes, in which the birth and death rates are periodic functions of the time $t$. These appear well adapted to describe the response of population growth (or epidemic spread) to the influence of the seasons.