The gamma distribution and weighted multimodal gamma distributions as models of population abundance

Dennis, B. R.; Patil, G. P.

doi:10.1016/0025-5564(84)90031-2

Cited by 128 publications

(128 citation statements)

References 33 publications

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“…The Gamma model has a long history in statistical ecology (e.g., Fisher et al, 1943), not only for analytical convenience but also because it may possess a deeper biological interpretation. Dennis and Patil (1984) showed that the Gamma is the approximate stationary distribution for the abundance of a population uctuating around a stable equilibrium. In Section 6 we comment on the sensitivity of our results to the Gamma model assumption.…”

Section: A Sampling Model For Measured Expressionmentioning

confidence: 99%

On Differential Variability of Expression Ratios: Improving Statistical Inference about Gene Expression Changes from Microarray Data

Newton

Kendziorski²,

Richmond³

et al. 2001

Journal of Computational Biology

636

458

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We consider the problem of inferring fold changes in gene expression from cDNA microarray data. Standard procedures focus on the ratio of measured uorescent intensities at each spot on the microarray, but to do so is to ignore the fact that the variation of such ratios is not constant. Estimates of gene expression changes are derived within a simple hierarchical model that accounts for measurement error and uctuations in absolute gene expression levels. Signi cant gene expression changes are identi ed by deriving the posterior odds of change within a similar model. The methods are tested via simulation and are applied to a panel of Escherichia coli microarrays.

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Section: A Sampling Model For Measured Expressionmentioning

confidence: 99%

On Differential Variability of Expression Ratios: Improving Statistical Inference about Gene Expression Changes from Microarray Data

Newton

Kendziorski²,

Richmond³

et al. 2001

Journal of Computational Biology

636

458

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“…The long-term behavior of (1.1) is determined by the stochastic growth rate a − σ 2 2 in the following way (see Evans et al 2015;Dennis and Patil 1984):…”

Section: Du (T) = U (T)(a − Bu (T))dt + σ U (T)dw (T) T ≥ 0mentioning

confidence: 99%

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Buono

Eftimie

2014

J. Math. Biol.

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This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout n patches whose population abundances are modeled as the solutions of a system of nonlinear stochastic differential equations living on [0, ∞) n . We prove that r , the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter r can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if r > 0, the population abundances converge polynomially fast to a unique invariant probability measure on (0, ∞) n , while when r < 0, the population abundances of the patches converge almost surely to 0 exponentially fast. 2013) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. We also show that the stochastic growth rate depends continuously on the coefficients. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. We show how one can adapt the nondegenerate results to the degenerate setting.As an example we fully analyze the two-patch case, n = 2, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the nondegenerate setting treated by Evans et al. which show that sometimes coupling by dispersal can make the system persistent.

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“…In particular, the model in which a is not equal to 0 and b is Ͻ0 represents a stochastic logistic growth. Under this model, the population no longer attains a single deterministic equilibrium, as in the Ricker equation, but instead, it approaches a "cloud of points" (60), a stationary distribution which can be approximated by a gamma probability density function (20,24) whose mean is Ϫa/b. The point Ϫa/b represents a center for return tendencies: it is the population abundance at which the average change is N t , conditional on N tϪ1 being zero, thus accounting for the stationary phase.…”

Section: Methodsmentioning

confidence: 99%

“…Primary models describe the basic rules of how microbial numbers change over time (52). Next, these simple models are used to derive secondary models that account for the effect of a set of factors in microbial growth.The forces of the environment and the events of reproduction and growth are themselves stochastic in nature (3,21,24,32,36,39,44,45,51,53), yet simple ecological models, such as the Verhulst logistic equation, result in deterministic predictions. However, smooth convergence to asymptotic results is not what is usually seen, even in rigorous experimental settings (17, 31).…”

mentioning

confidence: 99%

Use of Stochastic Models To Assess the Effect of Environmental Factors on Microbial Growth

Ponciano¹,

Vandecasteele

Hess³

et al. 2005

Appl Environ Microbiol

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We present a novel application of a stochastic ecological model to the study and analysis of microbial growth dynamics as influenced by environmental conditions in an extensive experimental data set. The model proved to be useful in bridging the gap between theoretical ideas in ecology and an applied problem in microbiology. The data consisted of recorded growth curves of Escherichia coli grown in triplicate in a base medium with all 32 possible combinations of five supplements: glucose, NH 4 Cl, HCl, EDTA, and NaCl. The potential complexity of 2 5 experimental treatments and their effects was reduced to 2 2 as just the metal chelator EDTA, the presumed osmotic pressure imposed by NaCl, and the interaction between these two factors were enough to explain the variability seen in the data. The statistical analysis showed that the positive and negative effects of the five chemical supplements and their combinations were directly translated into an increase or decrease in time required to attain stationary phase and the population size at which the stationary phase started. The stochastic ecological model proved to be useful, as it effectively explained and summarized the uncertainty seen in the recorded growth curves. Our findings have broad implications for both basic and applied research and illustrate how stochastic mathematical modeling coupled with rigorous statistical methods can be of great assistance in understanding basic processes in microbial ecology.Mathematical modeling coupled with rigorous statistical methods can be of great assistance in understanding the interaction of organisms with their physical and biological environment (8,47,48,50,52,56). Studies in the field of predictive microbiology have shown that successful modeling requires both adequate models and thorough data sets (57; for extensive reviews, see references 10, 42, and 55). In predictive microbiology a two-step modeling approach is used (reference 55 and citations therein). Primary models describe the basic rules of how microbial numbers change over time (52). Next, these simple models are used to derive secondary models that account for the effect of a set of factors in microbial growth.The forces of the environment and the events of reproduction and growth are themselves stochastic in nature (3,21,24,32,36,39,44,45,51,53), yet simple ecological models, such as the Verhulst logistic equation, result in deterministic predictions. However, smooth convergence to asymptotic results is not what is usually seen, even in rigorous experimental settings (17, 31). Hence, a more realistic alternative to population growth modeling is to confront stochastic equations with the data at hand (references 5, 14, and 17 and citations therein and references 41 and 47).In this paper, we describe a novel application of a stochastic population model to analyze how environmental conditions influence microbial growth dynamics using an extensive experimental data set. The primary model used was the stochastic Ricker (SR) equation (27, 51). Our secondary model...

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The gamma distribution and weighted multimodal gamma distributions as models of population abundance

Cited by 128 publications

References 33 publications

On Differential Variability of Expression Ratios: Improving Statistical Inference about Gene Expression Changes from Microarray Data

On Differential Variability of Expression Ratios: Improving Statistical Inference about Gene Expression Changes from Microarray Data

Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Use of Stochastic Models To Assess the Effect of Environmental Factors on Microbial Growth

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