Allometric scaling of population variance with mean body size is predicted from Taylor’s law and density-mass allometry

Abstract

Two widely tested empirical patterns in ecology are combined here to predict how the variation of population density relates to the average body size of organisms. Taylor's law (TL) asserts that the variance of the population density of a set of populations is a power-law function of the mean population density. Density-mass allometry (DMA) asserts that the mean population density of a set of populations is a power-law function of the mean individual body mass. Combined, DMA and TL predict that the variance of the population density is a power-law function of mean individual body mass. We call this relationship "variance-mass allometry" (VMA). We confirmed the theoretically predicted power-law form and the theoretically predicted parameters of VMA, using detailed data on individual oak trees (Quercus spp.) of Black Rock Forest, Cornwall, New York. These results connect the variability of population density to the mean body mass of individuals.self-thinning | fluctuation scaling | Damuth's law | nonlinear least squares | spatial Taylor's law U nderstanding how and why living populations, natural and engineered, vary in space and time is a core problem of ecology. When a population fluctuates to a low density, its risk of extinction may rise and its genetic diversity may pass through a bottleneck with enduring evolutionary consequences. In fisheries, forests, and agriculture, population fluctuations may directly affect human supplies of food, fiber, and timber and have economic consequences. Outbreaks of arthropod and molluscan vectors of diseases of humans and animals may raise risks of disease. These scientific and practical reasons make it important to understand how and why population densities fluctuate.An important generalization about population variability is Taylor's law (1-3). Taylor's law (TL) is the subject of an estimated 1,000 papers (4) and has been confirmed for hundreds of species or groups of species in field observations and laboratory experiments (5, 6). The censused populations may include a single species, a single genus, or more loosely related organisms. TL asserts that variance of population density = aðmean population densityÞ b ; a > 0:[1]On logarithmic scales, TL becomes a linear relationship: logðvariance of population densityÞ = logðaÞ + b × logðmean population densityÞ: Typically, b > 0: The mean and variance of population density increase together. In many empirical examples, 1 ≤ b ≤ 2. Density-mass allometry (DMA) (7-13) asserts that mean population density = uðmean body mass per individualÞ v ; u > 0:Typically, bigger organisms are rarer and v < 0. DMA is sometimes called the self-thinning law (14). DMA applies to single species, collections of species (15-18), and foodwebs (19-23).Substituting DMA, Eq. 2, into TL, Eq. 1, gives variance of population density = au b ðmean body mass per individualÞ bv :We called this predicted relation variance-mass allometry (VMA). If b > 0 and v < 0, as is typical, then bv < 0: Increasing mean body mass will be associated with decreasing variance ...

Parasitism alters three power laws of scaling in a metazoan community: Taylor’s law, density-mass allometry, and variance-mass allometry

“…These relationships were a spatial TL (spatial variance in population density was a power-law function of the spatial mean population density); DMA (the spatial mean population density was a power-law function of mean body mass); and VMA (the spatial variance in population density was a power-law function of mean body mass). Third, TL and DMA, both classic relationships known for decades, accurately predicted the form and parameter values of VMA, a power-law relationship predicted only within the last decade (8,9,23), and previously tested empirically only once (9). To our knowledge, we provided here the first empirical confirmation of VMA for any animal community.…”

“…10-18), including parasitic nematodes (19) and other parasites (20), VMA has previously been confirmed empirically only for congeneric trees (Quercus spp.) in a temperate forest (9). These new data permitted us to verify the predicted VMA empirically, to our knowledge for the first time for any animals and for the first time for all metazoans in a local community.…”

“…To our knowledge, these results represent the first confirmation of VMA for any animal community, the first demonstration that VMA depends on lifestyle, and the first confirmation of VMA across a broad range of biological taxa. The only previous empirical confirmation of VMA was for oak trees (Quercus) in a temperate forest (9).…”

Random sampling of skewed distributions implies Taylor’s power law of fluctuation scaling

“…TL has been verified for hundreds of biological species and nonbiological quantities in more than a thousand papers in ecology, epidemiology, biomedical sciences, and other fields (2)(3)(4). Recently, examples of TL were found in bacterial microcosms (5,6), forest trees (7,8), human populations (9), coral reef fish populations (10), and barnacles (11,12). TL has been used practically in the design of sampling plans for the control of insect pests of soybeans (13,14) and cotton (15).…”

Parasites help find universal ecological rules

“…Although this pattern is simply and algebraically derived from the other two laws (7,13), it is lacking in empirical demonstration for anything other than oak trees! Lagrue et al provide the first documentation of this pattern for animals of any sort.…”