Population fluctuations, power laws and mixtures of lognormal distributions

Allen, Andrew P.; Li, B.-L.; Charnov, Eric L.

doi:10.1046/j.1461-0248.2001.00194.x

Cited by 75 publications

(66 citation statements)

References 7 publications

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“…It has been argued the tent-shaped distributions of growth rates in complex systems may emerge if the units composing the system evolve according to a random multiplicative growth process (e.g., a mixture of lognormal distributions with different variances) (37,38). However, for this explanation to hold in our system, the amount of oxygen consumed by the units composing the system (i.e., cells, tissues or organs) would need to be independent, with similar mean and different variances.…”

Section: Discussionmentioning

confidence: 99%

Scaling metabolic rate fluctuations

Labra¹,

Marquet²,

Bozinovic³

2007

Proc. Natl. Acad. Sci. U.S.A.

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Complex ecological and economic systems show fluctuations in macroscopic quantities such as exchange rates, size of companies or populations that follow non-Gaussian tent-shaped probability distributions of growth rates with power-law decay, which suggests that fluctuations in complex systems may be governed by universal mechanisms, independent of particular details and idiosyncrasies. We propose here that metabolic rate within individual organisms may be considered as an example of an emergent property of a complex system and test the hypothesis that the probability distribution of fluctuations in the metabolic rate of individuals has a ''universal'' form regardless of body size or taxonomic affiliation. We examined data from 71 individuals belonging to 25 vertebrate species (birds, mammals, and lizards). We report three main results. First, for all these individuals and species, the distribution of metabolic rate fluctuations follows a tent-shaped distribution with power-law decay. Second, the standard deviation of metabolic rate fluctuations decays as a powerlaw function of both average metabolic rate and body mass, with exponents ؊0.352 and ؊1/4 respectively. Finally, we find that the distributions of metabolic rate fluctuations for different organisms can all be rescaled to a single parent distribution, supporting the existence of general principles underlying the structure and functioning of individual organisms.allometry ͉ body mass ͉ Laplace distribution L iving organisms have been described as the most complex system in the universe, emerging from the activity of an adaptive network of interacting components that allows energy, materials, and information to be acquired, stored, distributed, and transformed (1, 2), and whose end result is the maintenance and reproduction of the network itself (3). A striking feature of complex systems is that they show regularities in the behavior of macroscopic variables, which emerge as the result of nonlinear interactions among multiple components and because of the competition of opposing control forces (4-6). These regularities commonly take the form of simple scaling relationships or power-laws (7,8). A macroscopic variable that shows scaling relationships is metabolic rate (VO 2 ) (the rate at which an animal consumes oxygen), which scales with body mass (M) such that VO 2 ϰ M a with ␣Ͻ1.For over a century, biologists have documented and tried to explain both the value of ␣, and the effects ecological factors have on it (1, 9-17). Most of these studies focus on average values of VO 2 and M, and do not consider the temporal variability in individual energy use. However, physiological variables, such as cardiac and breathing dynamics, display complex rhythms, which often show changes both with disease and aging (17)(18)(19)(20). In this context, the study of fluctuations in VO 2 can shed light on the determinants of metabolic scaling and provide a way to test competing models and explanations. Indeed, work on complex systems has shown that study of the scaling prope...

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Section: Discussionmentioning

confidence: 99%

Scaling metabolic rate fluctuations

Labra¹,

Marquet²,

Bozinovic³

2007

Proc. Natl. Acad. Sci. U.S.A.

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“…Thus the analysis of power-law and scaling relationships can help us to identify general principles that apply across a wide range of scales and levels of organizations, revealing the existence of universal principles within the seemingly idiosyncratic nature of ecological systems. However, it should be borne in mind that power-laws might emerge as a consequence of several processes not necessarily related to critical points and phase transitions (Brock, 1999;Sornette, 2000;Mitzenmacher, 2001;Allen et al, 2001) such that the claim that ecological systems are maintained near a critical state is still an open question.…”

Section: Why Bother With Scaling and Power-law Relationships?mentioning

confidence: 99%

“…It has been argued that the tent-shaped distribution of population growth rates may be the end product of a mixture of lognormal distributions in population size (Allen et al, 2001). This phenomenological explanation, however, does not account for the symmetrical nature of the distribution, nor does it provide a mechanism that accounts for its form and location.…”

Section: Power-laws In Population Growth Ratesmentioning

confidence: 99%

Scaling and power-laws in ecological systems

Marquet

Quiñones

Abades

et al. 2005

Journal of Experimental Biology

331

308

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Scaling relationships (where body size features as the independent variable) and power-law distributions are commonly reported in ecological systems. In this review we analyze scaling relationships related to energy acquisition and transformation and power-laws related to fluctuations in numbers. Our aim is to show how individual level attributes can help to explain and predict patterns at the level of populations that can propagate at upper levels of organization. We review similar relationships also appearing in the analysis of aquatic ecosystems (i.e. the biomass spectra) in the context of ecological invariant relationships (i.e. independent of size) such as the 'energetic equivalence rule' and the 'linear biomass hypothesis'. We also discuss some power-law distributions emerging in the analysis of numbers and fluctuations in ecological attributes as they point to regularities that are yet to be integrated with traditional scaling relationships and which we foresee as an exciting area of future research.

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“…Denote the total size of the object in the new system at time n by Z n , and observe that Z n forms a renewal process. Then, taking out all the empty (idle) periods of the new system and concatenating the remaining periods sequentially yields a process equal in distribution to a reflected modulated branching process {Λ n }, as defined in (2). Therefore, when the new system is in stationarity, we obtain, by the independence of {A n } and Z P , …”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Modulated Branching Processes, Origins of Power Laws, and Queueing Duality

Jelenković

Tan

2010

Mathematics of OR

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Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas. In many of the observations, e.g., city populations and sizes of living organisms, the objects of interest evolve due to the replication of their many independent components, e.g., births-deaths of individuals and replications of cells. Furthermore, the rates of the replication are often controlled by exogenous parameters causing periods of expansion and contraction, e.g., baby booms and busts, economic booms and recessions, etc. In addition, the sizes of these objects often have reflective lower boundaries, e.g., cities do not fall bellow a certain size, low income individuals are subsidized by the government, companies are protected by bankruptcy laws, etc.Hence, it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations. Indeed, our main results show that the proposed mathematical models result in power law distributions under quite general polynomial Gärtner-Ellis conditions, the generality of which could explain the ubiquitous nature of power law distributions. In addition, on a logarithmic scale, we establish an asymptotic equivalence between the reflected branching processes and the corresponding multiplicative ones. The latter, as recognized by Goldie (1991) [32], is known to be dual to queueing/additive processes. We emphasize this duality further in the generality of stationary and ergodic processes.

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Population fluctuations, power laws and mixtures of lognormal distributions

Cited by 75 publications

References 7 publications

Scaling metabolic rate fluctuations

Scaling metabolic rate fluctuations

Scaling and power-laws in ecological systems

Modulated Branching Processes, Origins of Power Laws, and Queueing Duality

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