Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.

One day is fine, the next is black.—The Clash     

The potential impacts of climate change on inshore squid: biology, ecology and fisheries

Squid are important components of many marine ecosystems from the poles to the equator, serving as both important predators and prey. Novel aspects of their growth and reproduction mean that they are likely to play an important role in the changing oceans due to climate change. Virtually every facet of squid life-history examined thus far has revealed an incredible capacity in this group for life-history plasticity. The extremely fast growth rates of individuals and rapid rates of turnover at the population level mean that squid can respond quickly to environmental or ecosystem change. Their ‘life-in-the-fast-lane’ life-style allows them to rapidly exploit ‘vacuums’ created in the ecosystem when predators or competitors are removed. In this way, they function as ‘weeds of the sea’. Elevated temperatures accelerate the life-histories of squid, increasing their growth rates and shortening their life-spans. At first glance, it would be logical to suggest that rising water temperatures associated with climate change (if food supply remains adequate) would be beneficial to inshore squid populations and fisheries—growth rates would increase, life spans would shorten and population turnover would accelerate. However, the response of inshore squid populations to climate change is likely to be extremely complex. The size of hatchlings emerging from the eggs becomes smaller as temperatures increase and hatchling size may have a critical influence on the size-at-age that may be achieved as adults and subsequently, population structure. The influence of higher temperatures on the egg and adult stages may thus be opposing forces on the life-history. The process of climate change will likely result in squids that hatch out smaller and earlier, undergo faster growth over shorter life-spans and mature younger and at a smaller size. Individual squid will require more food per unit body size, require more oxygen for faster metabolisms and have a reduced capacity to cope without food. It is therefore likely that biological, physiological and behavioural changes in squid due to climate change will have far reaching effects.     

Contrasting growth strategies of pond versus marine populations of nine-spined stickleback (Pungitius pungitius): a combined effect of predation and competition?

Gigantism in isolated ponds in the absence of sympatric fish species has previously been observed in nine-spined sticklebacks (Pungitius pungitius). Patterns in sexual size dimorphism suggested that fecundity selection acting on females might be responsible for the phenomenon. However, the growth strategy behind gigantism in pond sticklebacks has not been studied yet. Here, we compared von Bertalanffy growth parameters of four independent nine-spined stickleback populations reared in a common laboratory environment: two coastal marine (typical size) and two pond (giant size) populations. We found that both pond populations had larger estimated final size than marine populations, which in turn exhibited higher intrinsic growth rates than the pond populations. Female growth strategies were more divergent among marine and pond populations than those of males. Asymptotic body size and intrinsic growth rate were strongly negatively correlated. Hence, pond versus marine populations exhibited different growth strategies along a continuum. Our data suggest that quick maturation—even with the cost of being small (low fecundity)—is favoured in marine environments. On the contrary, growth to a giant final size (high fecundity)—even if it entails extended growth period—is favoured in ponds. We suggest that the absence (ponds) versus presence (marine environment) of sympatric predatory fish species, and the consequent change in the importance of intraspecific competition are responsible for the divergence in growth strategies. The sex-dependence of the patterns further emphasizes the role of females in the body size divergence in the species. Possible alternative hypotheses are also discussed.     

The Impact of Accelerating Faster than Exponential Population Growth on Genetic Variation

Current human sequencing projects observe an abundance of extremely rare genetic variation, suggesting recent acceleration of population growth. To better understand the impact of such accelerating growth on the quantity and nature of genetic variation, we present a new class of models capable of incorporating faster than exponential growth in a coalescent framework. Our work shows that such accelerated growth affects only the population size in the recent past and thus large samples are required to detect the models’ effects on patterns of variation. When we compare models with fixed initial growth rate, models with accelerating growth achieve very large current population sizes and large samples from these populations contain more variation than samples from populations with constant growth. This increase is driven almost entirely by an increase in singleton variation. Moreover, linkage disequilibrium decays faster in populations with accelerating growth. When we instead condition on current population size, models with accelerating growth result in less overall variation and slower linkage disequilibrium decay compared to models with exponential growth. We also find that pairwise linkage disequilibrium of very rare variants contains information about growth rates in the recent past. Finally, we demonstrate that models of accelerating growth may substantially change estimates of present-day effective population sizes and growth times.     

Population Genetic Consequences of the Allee Effect and the Role of Offspring-Number Variation

A strong demographic Allee effect in which the expected population growth rate is negative below a certain critical population size can cause high extinction probabilities in small introduced populations. But many species are repeatedly introduced to the same location and eventually one population may overcome the Allee effect by chance. With the help of stochastic models, we investigate how much genetic diversity such successful populations harbor on average and how this depends on offspring-number variation, an important source of stochastic variability in population size. We find that with increasing variability, the Allee effect increasingly promotes genetic diversity in successful populations. Successful Allee-effect populations with highly variable population dynamics escape rapidly from the region of small population sizes and do not linger around the critical population size. Therefore, they are exposed to relatively little genetic drift. It is also conceivable, however, that an Allee effect itself leads to an increase in offspring-number variation. In this case, successful populations with an Allee effect can exhibit less genetic diversity despite growing faster at small population sizes. Unlike in many classical population genetics models, the role of offspring-number variation for the population genetic consequences of the Allee effect cannot be accounted for by an effective-population-size correction. Thus, our results highlight the importance of detailed biological knowledge, in this case on the probability distribution of family sizes, when predicting the evolutionary potential of newly founded populations or when using genetic data to reconstruct their demographic history.     

Spatiotemporal variation in reproductive parameters of yellow-bellied marmots

Spatiotemporal variation in reproductive rates is a common phenomenon in many wildlife populations, but the population dynamic consequences of spatial and temporal variability in different components of reproduction remain poorly understood. We used 43 years (1962–2004) of data from 17 locations and a capture–mark–recapture (CMR) modeling framework to investigate the spatiotemporal variation in reproductive parameters of yellow-bellied marmots (Marmota flaviventris), and its influence on the realized population growth rate. Specifically, we estimated and modeled breeding probabilities of two-year-old females (earliest age of first reproduction), >2-year-old females that have not reproduced before (subadults), and >2-year-old females that have reproduced before (adults), as well as the litter sizes of two-year old and >2-year-old females. Most reproductive parameters exhibited spatial and/or temporal variation. However, reproductive parameters differed with respect to their relative influence on the realized population growth rate (λ). Litter size had a stronger influence than did breeding probabilities on both spatial and temporal variations in λ. Our analysis indicated that λ was proportionately more sensitive to survival than recruitment. However, the annual fluctuation in litter size, abetted by the breeding probabilities, accounted for most of the temporal variation in λ. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Seasonal variation in growth of greater snow goose goslings: the role of food supply

Even though growth rate is an important fitness component, it is still controversial to what extent parent birds adjust the timing of offspring hatch to natural variations in food supply to maximize offspring growth. We studied the role of food availability in explaining inter- and intra-seasonal variation of growth rate in goslings of greater snow geese over 5 years. The peak of hatching generally coincided with the peak of food availability. However, early-hatched goslings usually grew faster than birds hatched at the peak, which in␣turn grew faster than late-hatched goslings, although this phenomenon was not observed in all years. There was considerable variation in growth rate among the five years, the smallest goslings produced in the best year (1991) being larger than the largest goslings of the poorest year (1994). We developed three indices of food availability, based on the cumulative availability of plant biomass and nitrogen content during the growth period, and showed that the cumulative exposure to nitrogen biomass explained up to 43% of variation (intra- and inter-annual) in body size just before fledging. In years with good feeding conditions, early-hatched goslings had access to more nitrogen during their growing period than those hatching on or after the peak and they grew faster. In years of lower food availability, early-hatched goslings had no detectable advantage over peak- or late-hatched birds for access to protein-rich food and no seasonal decline in growth rate was observed. These results confirm the critical role of food supply in the seasonal variation of growth rate in Arctic-nesting geese. Received: 27 June 1997 / Accepted: 9 October 1997     

Stochastic variation in food availability influences weight and age at maturity

Variation in mean food availability, and in the variance around the mean, affects the growth rate during development. Previous theoretical work on the influence of environmental quality or growth rates on the phenotypic traits age and size at maturation assumed that there is no variation in growth rate or food availability within a generation. We develop a stochastic dynamic programming (SDP) model of the foraging behaviour of aphidophagous syrphids, and use this model to predict when syrphids should pupate (mature) when average food availability changes, or varies stochastically, during development.The optimal strategy takes into account not only the availability of food, but also the timing of its availability. Food availability, when small, influences developmental time, but not weight at pupation. Food availability, when large, influences weight at pupation, but not developmental time. When the food supply is low, the optimal strategy adjusts the size at pupation downwards for stochastic as opposed to deterministic availability of food. The conclusions reinforce the need for life-history studies to consider state dependence and short-term variability in growth rates.     

Factors influencing size inequality in peatland black spruce and tamarack: evidence from post-drainage release growth

1 We used tree ring analysis to determine stem radius and thus examine size variation over time in two even-aged (approximately 40-year-old) mixed populations of black spruce and tamarack established on peatlands in a boreal forest. We also followed the response of one of these populations to improved edaphic conditions over 8 years following drainage.
2 Populations of trees in undrained areas showed a decline in size variability over time until age 20–25 years, after which size heterogeneity was relatively stable.
3 For trees in undrained areas there was a relationship between age and size for the first 20–25 years, but this relationship then broke down due to a period where relative growth rate and size were inversely related.
4 For the population of trees in the drained area, smaller trees (i.e. those that had been growing more slowly prior to drainage) showed significantly greater drainage-induced release growth, while larger trees (those growing faster prior to drainage) showed an initial reduction in growth following drainage and, overall, less release growth.
5 The response of tamarack to drainage was more dramatic than for black spruce.
6 Despite extensive variation in tree size, drainage dramatically reduced variability in growth rate among trees of each species, such that size variability in the populations declined.
7 We postulate that the heterogeneity of microsites with respect to edaphic conditions, perhaps associated with the hummock to hollow microtopographic gradient, has a major influence on growth variation, and hence size inequality, in peatland populations of black spruce and tamarack.     

Selective mortality of larval Japanese seabass in Ariake Bay, Japan

This article describes the growth, mortality, and selection patterns in early larval stage of Japanese seabass Lateolabrax japonicus in Ariake Bay, Kyushu, Japan. Japanese seabass larvae were collected from the spawning ground in December 2007, and juveniles were collected from the nursery ground in March 2008. Otoliths were analyzed to produce daily records of size-at-age and growth rate that were compared between larvae and juveniles to determine whether selective mortality occurred. A weight-specific growth coefficient (G) and instantaneous mortality coefficient (M) were computed, and the recruitment potential was evaluated from the ratio of M:G. Selection for fast-growing and bigger larvae was evident during the 5–14 days after hatching (DAH). Selective mortality acted to preferentially remove fish that were slow growing and/or relatively small members of the cohort at least during the period examined. Trends in the growth rate differences between larvae and juveniles suggested that the selection process continued beyond 14 DAH although the exact duration over which selective mortality occurred was unknown. Survivors of Japanese seabass exhibited traits consistent with all aspects of the ‘growth–mortality’ hypothesis: faster growth, larger size-at-age, and shorter larval stage duration (LSD), i.e., larvae with faster growth, bigger size-at-age and a shorter LSD selectively survived the larval period. Larvae had higher M (0.323) than G (0.159), resulting in the M:G ratio of >1.0 (M:G = 2.031), suggesting that the larval cohort was rapidly losing biomass. Future studies should look to determine the duration over which selective mortality occurs and the timing of transition between M and G (M = G) and accumulation of cohort biomass.     

How trophic interaction strength depends on traits

A key problem in community ecology is to understand how individual-level traits give rise to population-level trophic interactions. Here, we propose a synthetic framework based on ecological considerations to address this question systematically. We derive a general functional form for the dependence of trophic interaction coefficients on trophically relevant quantitative traits of consumers and resources. The derived expression encompasses—and thus allows a unified comparison of—several functional forms previously proposed in the literature. Furthermore, we show how a community’s, potentially low-dimens ional, effective trophic niche space is related to its higher-dimensional phenotypic trait space. In this manner, we give ecological meaning to the notion of the “dimensionality of trophic niche space.” Our framework implies a method for directly measuring this dimensionality. We suggest a procedure for estimating the relevant parameters from empirical data and for verifying that such data matches the assumptions underlying our derivation.     

Spatially distributed stochasticity and the constancy of ecosystems

Ifconstancy is a measure of an ecosystem's (in) variability through time andstability is a measure of the system's ability to damp and recover from environmental perturbations, then constancy depends not only on stability but also on the frequency and amplitude of perturbations—theenvironmental “noise level”. The stability of an ecosystem reflects its texture, extent, and viscosity (fine-scale structure); the noise level experienced by the system (“effective” noise level) reflects the level at any point (“ambient” noise level), the spectrum of stochastic scale (regional distribution of stochasticity), and the system's spatial extent (size, or number of patches included). The coefficient of variation of a limiting stochastic variate is a measure of the effective noise level. Ifp is the total number of patches in the system (its extent) andn is the number of contiguous patches with noise signals correlated through time (its stochastic scale), then the coefficient of variation is directly proportional to whenevern<p. Thus ecosystems of small stochastic scalen or large sizep damp out environmental noise by “spreading the risk” in space, thereby reducing their variability in time.     

Patterns of growth in wild bottlenose dolphins, Tursiops truncatus

The growth of bottlenose dolphins is described from observations made during a capture release programme that has operated in coastal waters of the eastern Gulf of Mexico from 1970 to the present. Measurements of standard length, girth and body mass were recorded from 47 female and 49 male dolphins, some captured as many as nine times. Ages were known from approximate birth dates or estimated from counts of dentinal growth layers. In all three measurements. females grew at a faster initial rate than males, but reached asymptotic size at an earlier age. This extended period of growth in males resulted in significant sexual dimorphism in length, girth and mass at physical maturity. The growth of both sexes was well described by three-parameter Gompertz models using either cross-sectional data or a mixture of longitudinal and cross-sectional data. There was considerable variation in size-at-age for both sexes in all year classes. Residuals of size measurements were used to derive measures of relative size for individual dolphins; most dolphins demonstrated little ontogenetic change in relative size. Body mass was adequately predicted by multiple regression equations that incorporated both length and girth as independent variables.     

Determining the expected variability of immune responses using the cyton model

During an adaptive immune response, lymphocytes proliferate for 5–20 cell divisions, then stop and die over a period of weeks. The cyton model for regulation of lymphocyte proliferation and survival was introduced by Hawkins et al. (Proc. Natl. Acad. Sci. USA 104, 5032–5037, 2007) to provide a framework for understanding this response and its regulation. The model assumes stochastic values for division and survival times for each cell in a responding population. Experimental evidence indicates that the choice of times is drawn from a skewed distribution such as the lognormal, with the fate of individual cells being potentially highly variable. For this reason we calculate the higher moments of the model so that the expected variability can be determined. To do this we formulate a new analytic framework for the cyton model by introducing a generalization to the Bellman–Harris branching process. We use this framework to introduce two distinct approaches to predicting variability in the immune response to a mitogenic signal. The first method enables explicit calculations for certain distributions and qualitatively exhibits the full range of observed immune responses. The second approach does not facilitate analytic solutions, but allows simple numerical schemes for distributions for which there is little prospect of analytic formulae. We compare the predictions derived from the second method to experimentally observed lymphocyte population sizes from in vivo and in vitro experiments. The model predictions for both data sets are remarkably accurate. The important biological conclusion is that there is limited variation around the expected value of the population size irrespective of whether the response is mediated by small numbers of cells undergoing many divisions or for many cells pursuing a small number of divisions. Therefore, we conclude the immune response is robust and predictable despite the potential for great variability in the experience of each individual cell.      

Individualism in plant populations: using stochastic differential equations to model individual neighbourhood-dependent plant growth

We study individual plant growth and size hierarchy formation in an experimental population of Arabidopsis thaliana, within an integrated analysis that explicitly accounts for size-dependent growth, size- and space-dependent competition, and environmental stochasticity. It is shown that a Gompertz-type stochastic differential equation (SDE) model, involving asymmetric competition kernels and a stochastic term which decreases with the logarithm of plant weight, efficiently describes individual plant growth, competition, and variability in the studied population. The model is evaluated within a Bayesian framework and compared to its deterministic counterpart, and to several simplified stochastic models, using distributional validation. We show that stochasticity is an important determinant of size hierarchy and that SDE models outperform the deterministic model if and only if structural components of competition (asymmetry; size- and space-dependence) are accounted for. Implications of these results are discussed in the context of plant ecology and in more general modelling situations.     

Individual size variation and population stability in a seasonal environment: a discrete-time model and its calibration using grasshoppers

Much recent literature is concerned with how variation among individuals (e.g., variability in their traits and fates) translates into higher-level (i.e., population and community) dynamics. Although several theoretical frameworks have been devised to deal with the effects of individual variation on population dynamics, there are very few reports of empirically based estimates of the sign and magnitude of these effects. Here we describe an analytical model for size-dependent, seasonal life cycles and evaluate the effect of individual size variation on population dynamics and stability. We demonstrate that the effect of size variation on the population net reproductive rate varies in both magnitude and sign, depending on season length. We calibrate our model with field data on size- and density-dependent growth and survival of the generalist grasshopper Melanoplus femurrubrum. Under deterministic dynamics (fixed season length), size variation impairs population stability, given naturally occurring densities. However, in the stochastic case, where season length exhibits yearly fluctuations, size variation reduces the variance in population growth rates, thus enhancing stability. This occurs because the effect of size variation on net reproductive rate is dependent on season length. We discuss several limitations of the current model and outline possible routes for future model development.     

Age-specific fitness components and their temporal variation in the barn owl

Theory predicts that temporal variability plays an important role in the evolution of life histories, but empirical studies evaluating this prediction are rare. In constant environments, fitness can be measured by the population growth rate lambda, and the sensitivity of lambda to changes in fitness components estimates selection on these traits. In variable environments, fitness is measured by the stochastic growth rate lambda(S), and stochastic sensitivities estimate selection pressure. Here we examine age-specific schedules for reproduction and survival in a barn owl population (Tyto alba). We estimated how temporal variability affected fitness and selection, accounting for sampling variance. Despite large sample sizes of old individuals, we found no strong evidence for senescence. The most variable fitness components were associated with reproduction. Survival was less variable. Stochastic simulations showed that the observed variation decreased fitness by about 30%, but the sensitivities of lambda and lambda(S) to changes in all fitness components were almost equal, suggesting that temporal variation had negligible effects on selection. We obtained these results despite high observed variability in the fitness components and relatively short generation time of the study organism, a situation in which temporal variability should be particularly important for natural selection and early senescence is expected.     

A note on a paper by Erik Volz: SIR dynamics in random networks

Recent work by Volz (J Math Biol 56:293–310, 2008) has shown how to calculate the growth and eventual decay of an SIR epidemic on a static random network, assuming infection and recovery each happen at constant rates. This calculation allows us to account for effects due to heterogeneity and finiteness of degree that are neglected in the standard mass-action SIR equations. In this note we offer an alternate derivation which arrives at a simpler—though equivalent—system of governing equations to that of Volz. This new derivation is more closely connected to the underlying physical processes, and the resulting equations are of comparable complexity to the mass-action SIR equations. We further show that earlier derivations of the final size of epidemics on networks can be reproduced using the same approach, thereby providing a common framework for calculating both the dynamics and the final size of an epidemic spreading on a random network. Under appropriate assumptions these equations reduce to the standard SIR equations, and we are able to estimate the magnitude of the error introduced by assuming the SIR equations.