Population dynamics and the colour of environmental noise.

The effect of red, white and blue environmental noise on discrete-time population dynamics is analyzed. The coloured noise is superimposed on Moran-Ricker and Maynard Smith dynamics, the resulting power spectra are less than examined. Time series dominated by short- and long-term fluctuations are said to be blue and red, respectively. In the stable range of the Moran-Ricker dynamics, environmental noise of any colour will make population dynamics red or blue depending the intrinsic growth rate. Thus, telling apart the colour of the noise from the colour of the population dynamics may not be possible. Population dynamics subjected to red and blue environmental noises show, respectively, more red or blue power spectra than those subjected to white noise. The sensitivity to differences in the noise colours decreases with increasing complexity and ultimately disappears in the chaotic range of the population dynamics. These findings are duplicated with the Maynard Smith model for high growth rates when the strength of density dependence changes. However, for low growth rates the power spectra of the population dynamics with noise are red in stable, periodic and aperiodic ranges irrespective of the noise colour. Since chaotic population fluctuations may show blue spectra in the deterministic case, this implies that blue deterministic chaos may become red under any colour of the noise.     

Can inverse density dependence at small spatial scales produce dynamic instability in animal populations?

All else being equal, inversely density-dependent (IDD) mortality destabilizes population dynamics. However, stability has not been investigated for cases in which multiple types of density dependence act simultaneously. To determine whether IDD mortality can destabilize populations that are otherwise regulated by directly density-dependent (DDD) mortality, I used scale transition approximations to model populations with IDD mortality at smaller “aggregation” scales and DDD mortality at larger “landscape” scales, a pattern observed in some reef fish and insect populations. I evaluated dynamic stability for a range of demographic parameter values, including the degree of compensation in DDD mortality and the degree of spatial aggregation, which together determine the relative importance of DDD and IDD processes. When aggregation-scale survival was a monotonically increasing function of density (a “dilution” effect), dynamics were stable except for extremely high levels of aggregation combined with either undercompensatory landscape-scale density dependence or certain values of adult fecundity. When aggregation-scale survival was a unimodal function of density (representing both “dilution” and predator “detection” effects), instability occurred with lower levels of aggregation and also depended on the values of fecundity, survivorship, detection effect, and DDD compensation parameters. These results suggest that only in extreme circumstances will IDD mortality destabilize dynamics when DDD mortality is also present, so IDD processes may not affect the stability of many populations in which they are observed. Model results were evaluated in the context of reef fish, but a similar framework may be appropriate for a diverse range of species that experience opposing patterns of density dependence across spatial scales.     

Environmentally‐induced noise dampens and reddens with increasing trophic level in a complex food web

Stochastic variability of key abiotic factors including temperature, precipitation and the availability of light and nutrients greatly influences species’ ecological function and evolutionary fate. Despite such influence, ecologists have typically ignored the effect of abiotic stochasticity on the structure and dynamics of ecological networks. Here we help to fill that gap by advancing the theory of how abiotic stochasticity, in the form of environmental noise, affects the population dynamics of species within food webs. We do this by analysing an allometric trophic network model of Lake Constance subjected to positive (red), negative (blue), and non‐autocorrelated (white) abiotic temporal variability (noise) introduced into the carrying capacity of basal species. We found that, irrespective of the colour of the introduced noise, the temporal variability of the species biomass within the network both reddens (i.e. its positive autocorrelation increases) and dampens (i.e. the magnitude of variation decreases) as the environmental noise is propagated through the food web by its feeding interactions from the bottom to the top. The reddening reflects a buffering of the noise‐induced population variability by complex food web dynamics such that non‐autocorrelated oscillations of noise‐free deterministic dynamics become positively autocorrelated. Our research helps explain frequently observed red variability of natural populations by suggesting that ecological processing of environmental noise through food webs with a range of species’ body sizes reddens population variability in nature.     

Density-dependent vital rates and their population dynamic consequences

We explore a set of simple, nonlinear, two-stage models that allow us to compare the effects of density dependence on population dynamics among different kinds of life cycles. We characterize the behavior of these models in terms of their equilibria, bifurcations, and nonlinear dynamics, for a wide range of parameters. Our analyses lead to several generalizations about the effects of life history and density dependence on population dynamics. Among these are: (1) iteroparous life histories are more likely to be stable than semelparous life histories; (2) an increase in juvenile survivorship tends to be stabilizing; (3) density-dependent adult survival cannot control population growth when reproductive output is high; (4) density-dependent reproduction is more likely to cause chaotic dynamics than density dependence in other vital rates; and (5) changes in development rate have only small effects on bifurcation patterns. Received: 12 April 1999 / Published online: 3 August 2000     

Comparative population dynamics of large and small mammals in the Northern Hemisphere: deterministic and stochastic forces

Deterministic feedbacks within populations interact with extrinsic, stochastic processes to generate complex patterns of animal abundance over time and space. Animals inherently differ in their responses to fluctuating environments due to differences in body sizes and life history traits. However, controversy remains about the relative importance of deterministic and stochastic forces in shaping population dynamics of large and small mammals. We hypothesized that effects of environmental stochasticity and density dependence are stronger in small mammal populations relative to their effects in large mammal populations and thus differentiate the patterns of population dynamics between them. We conducted an extensive, comparative analysis of population dynamics in large and small mammals to test our hypothesis, using seven population parameters to describe general dynamic patterns for 23 (14 species) time series of observations of abundance of large mammals and 38 (21 species) time series for small mammals. We used state‐space models to estimate the strength of direct and delayed density dependence as well as the strength of environmental stochasticity. We further used phylogenetic comparative analysis to detect differences in population dynamic patterns and individual population parameters, respectively, between large and small mammals. General population dynamic patterns differed between large and small mammals. However, the strength of direct and delayed density dependence was comparable between large and small mammals. Moreover, the variances of population growth rates and environmental stochasticity were greater in small mammals than in large mammals. Therefore, differences in population response to stochastic forces and strength of environmental stochasticity are the primary factor that differentiates population dynamic patterns between large and small mammal species.     

The impact of stochasticity on the behaviour of nonlinear population models: synchrony and the Moran effect

Environmental variation is ubiquitous, but its effects on nonlinear population dynamics are poorly understood. Using simple (unstructured) nonlinear models we investigate the effects of correlated noise on the dynamics of two otherwise independent populations (the Moran effect), i.e. we focus on noise rather than dispersion or trophic interaction as the cause of population synchrony. We find that below the bifurcation threshold for periodic behaviour (1) synchrony between populations is strongly dependent on the shape of the noise distribution but largely insensitive to which model is studied, (2) there is, in general, a loss of synchrony as the noise is filtered by the model, (3) for specially structured noise distributions this loss can be effectively eliminated over a restricted range of distribution parameter values even though the model might be nonlinear, (4) for unstructured models there is no evidence of correlation enhancement, a mechanism suggested by Moran, but above the bifurcation threshold enhancement is possible for weak noise through phase-locking, (5) rapid desynchronisation occurs as the chaotic regime is approached. To carry out the investigation the stochastic models are (a) reformulated in terms of their joint asymptotic probability distributions and (b) simulated to analyse temporal patterns.     

Dispersal evolution diminishes the negative density dependence in dispersal

In many organisms, dispersal varies with the local population density. Such patterns of density-dependent dispersal (DDD) are expected to shape the dynamics, spatial spread, and invasiveness of populations. Despite their ecological importance, empirical evidence for the evolution of DDD patterns remains extremely scarce. This is especially relevant because rapid evolution of dispersal traits has now been empirically confirmed in several taxa. Changes in DDD of dispersing populations could help clarify not only the role of DDD in dispersal evolution, but also the possible pattern of subsequent range expansion. Here, we investigate the relationship between dispersal evolution and DDD using a long-term experimental evolution study on Drosophila melanogaster. We compared the DDD patterns of four dispersal-selected populations and their non-selected controls. The control populations showed negative DDD, which was stronger in females than in males. In contrast, the dispersal-selected populations showed DDD, where neither males nor females exhibited DDD. We compare our results with previous evolutionary predictions that focused largely on positive DDD, and highlight how the direction of evolutionary change depends on the initial DDD pattern of a population. Finally, we discuss the implications of DDD evolution for spatial ecology and evolution.     

Persistence of structured populations in random environments

Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and density-dependent interactions influence population dynamics, we provide a general theory for persistence for density-dependent matrix models in random environments. For populations with compensating density dependence, exhibiting “bounded” dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating density-dependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stage-structured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space.     

Restricted dispersal reduces the strength of spatial density dependence in a tropical bird population

Spatial processes could play an important role in density-dependent population regulation because the disproportionate use of poor quality habitats as population size increases is widespread in animal populations-the so-called buffer effect. While the buffer effect patterns and their demographic consequences have been described in a number of wild populations, much less is known about how dispersal affects distribution patterns and ultimately density dependence. Here, we investigated the role of dispersal in spatial density dependence using an extraordinarily detailed dataset from a reintroduced Mauritius kestrel (Falco punctatus) population with a territorial (despotic) breeding system. We show that recruitment rates varied significantly between territories, and that territory occupancy was related to its recruitment rate, both of which are consistent with the buffer effect theory. However, we also show that restricted dispersal affects the patterns of territory occupancy with the territories close to release sites being occupied sooner and for longer as the population has grown than the territories further away. As a result of these dispersal patterns, the strength of spatial density dependence is significantly reduced. We conclude that restricted dispersal can modify spatial density dependence in the wild, which has implications for the way population dynamics are likely to be impacted by environmental change.     

Relative importance of density-dependent regulation and environmental stochasticity for butterfly population dynamics

The relative contribution of density-dependent regulation and environmental stochasticity to the temporal dynamics of animal populations is one of the central issues of ecology. In insects, the primary role of the latter factor, typically represented by weather patterns, is widely accepted. We have evaluated the impact of density dependence as well as density-independent factors, including weather and mowing regime, on annual fluctuations of butterfly populations. As model species, we used Maculinea alcon and M. teleius living in sympatry and, consequently, we also analysed the effect of their potential competition. Density dependence alone explained 62 and 42% of the variation in the year-to-year trends of M. alcon and M. teleius, respectively. The cumulative Akaike weight of models with density dependence, which can be interpreted as the probability that this factor should be contained in the most appropriate population dynamics model, exceeded 0.97 for both species. In contrast, the impacts of inter-specific competition, mowing regime and weather were much weaker, with their cumulative weights being in the range of 0.08–0.21; in addition, each of these factors explained only 2–5% of additional variation in Maculinea population trends. Our results provide strong evidence for density-dependent regulation in Maculinea, while the influence of environmental stochasticity is rather minor. In the light of several recent studies on other butterflies that detected significant density-dependent effects, it would appear that density-dependent regulation may be more widespread in this group than previously thought, while the role of environmental stochasticity has probably been overestimated. We suggest that this misconception is the result of deficiencies in the design of most butterfly population studies in the past, including (1) a strong focus on adults and a neglect of the larval stage in which density-dependent effects are most likely to occur; (2) an almost exclusive reliance on transect count results that may confound the impact of environmental stochasticity on butterfly numbers with its impact on adult longevity. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

有色环境噪音对空间异质种群动态同步性的影响

随着种群动态和空间结构研究兴趣的增加,激发了大量的有关空间同步性的理论和实验的研究工作。空间种群的同步波动现象在自然界广泛存在,它的影响和原因引起了很多生态学家的兴趣。Moran定理是一个非常重要的解释。但以往的研究大多假设环境变化为空间相关的白噪音。越来越多的研究表明很多环境变化的时间序列具有正的时间自相关性,也就是说用红噪音来描述更加合理。因此,推广经典的Moran效应来处理空间相关红噪音的情形很有必要。利用线性的二阶自回归过程的种群模型,推导了两种群空间同步性与种群动态异质性和环境变化的时间相关性(即环境噪音的颜色)之间的关系。深入分析了种群异质性和噪音颜色对空间同步性的影响。结果表明种群动态异质性不利于空间同步性,但详细的关系比较复杂。而红色噪音的同步能力体现在两方面:一方面,本身的相关性对同步性有贡献;另一方面,环境变化时间相关性可以通过改变种群密度依赖来影响同步性,但对同步性的影响并无一致性的结论,依赖于种群的平均动态等因素。这些结果对理解同步性的机理、利用同步机理来制定物种保护策略和害虫防治都有重要的意义。     

Density-dependent population regulation detected in short time series of saproxylic beetles

Understanding the regulation of natural populations has been a long-standing problem in ecology. Here we analyze the population dynamics of 17 species of saproxylic beetles in Shizuoka Prefecture, Japan collected over 11–12 years using autoregressive integrated moving average (ARIMA) models. We first examined the dynamics for indications of the order of the ARIMA models and evaluated the time series to determine that it was not simply a random, white noise sequence. All species dynamics were not mere random noise, and ARIMA models up to lag 3 were considered. The best model was selected from the possible ones using several criteria: model convergence, weak residual autocorrelation, the small sample AIC must be among the smallest that were not significantly different, and the lag indicated by the cutoff values in the detrended partial autocorrelation function. We found significant and nearly significant direct density-dependence for 14 of the 17 species, varying from −0.709 and stronger. The characteristic return rates were strong and only one species had a weak return rate (>0.9), implying that these species were strongly regulated by density-dependent factors. We found that populations with higher order ARIMA models (lag 2 and 3) had weaker return rates than populations with ARIMA models with only one lag, suggesting that species with more complex dynamics were more weakly regulated. These results contrast with previous suggestions that 20+ years are needed to detect density dependence from population time series and that most populations are weakly regulated.     

Allee effects in stochastic populations

The Allee effect, or inverse density dependence at low population sizes, could seriously impact preservation and management of biological populations. The mounting evidence for widespread Allee effects has lately inspired theoretical studies of how Allee effects alter population dynamics. However, the recent mathematical models of Allee effects have been missing another important force prevalent at low population sizes: stochasticity. In this paper, the combination of Allee effects and stochasticity is studied using diffusion processes, a type of general stochastic population model that accommodates both demographic and environmental stochastic fluctuations. Including an Allee effect in a conventional deterministic population model typically produces an unstable equilibrium at a low population size, a critical population level below which extinction is certain. In a stochastic version of such a model, the probability of reaching a lower size a before reaching an upper size b , when considered as a function of initial population size, has an inflection point at the underlying deterministic unstable equilibrium. The inflection point represents a threshold in the probabilistic prospects for the population and is independent of the type of stochastic fluctuations in the model. In particular, models containing demographic noise alone (absent Allee effects) do not display this threshold behavior, even though demographic noise is considered an "extinction vortex". The results in this paper provide a new understanding of the interplay of stochastic and deterministic forces in ecological populations.     

Fluctuations in the Size of Isolated Single-Species Populations and Natural Selection

This paper discusses the basic types of dynamical behavior of populations obtained in discrete models, such as monotonous dynamics, stable limited cycles, and chaotic variations. All these modes are shown to have possibly arisen in the evolution of limited populations under the effect of density-independent selection. This effect together with that of density-dependent non-selective factors has been termed F-selection, which is characterized by independence of relative fitnesses from population density, whereas populations may be ecologically limited; in other words, absolute fitnesses prove to be a function of population size. The characteristic of F-selection is to be not sensitive to changes in population size but to lead to fluctuations, that create conditions for achieving density-dependent selection.     

Inter- and intra-specific patterns of density dependence and population size variability in Salmoniformes

Population dynamics are typically affected by a combination of density-independent and density-dependent factors, the latter of which have been conceptually and theoretically linked with how variable population sizes are over time—which in turn has been tied to how prone populations are to extinction. To address evidence for the occurrence of density dependence and its relationship with population size variability (pv), we quantified each of these for 126 populations of 8 species of Salmoniformes. Using random-effects models, we partitioned variation in the strength of density dependence and the magnitude of pv between and within species and estimated the correlation of density dependence and population size variability at both the between- and within-species levels. We found that variation in the strength of density dependence was predominately within species (I ² = 0.47). In contrast, variation in population size variability was distributed both between and within species (I ² = 0.40). Contrary to theoretical and conceptual expectations, the strength of density dependence and the magnitude of population size variability were positively correlated at the between species level (r = 0.90), although this estimate had 95 % credibility intervals (Bayesian analogues to confidence intervals) that overlapped zero. The within-species correlation between density dependence and population size variability was not distinguishable from zero. Given that density dependence for Salmoniformes was highly variable within species, we next determined the joint effects of intrinsic (density-dependent) and extrinsic (density-independent) factors on the population dynamics of a threatened salmonid, the Lahontan cutthroat trout (Oncorhynchus clarkii henshawi). We found that density-dependent and -independent factors additively contributed to population dynamics. This finding suggests that the observed within-species variability in density dependence might be attributable to local differences in the strength of density-independent factors.     

Visibility of the impact of environmental noise: a response to Kaitala and Ranta

Based on simulation modelling, Kaitala and Ranta (2001 Proc. R. Soc. Lond. B 268, 1769-1774) have argued that detecting the statistical relationships between environmental variability and population fluctuations will be difficult. However, their study was limited in that only one pattern of density dependence and one detection method were used. Here, we show that their conclusion is in part a consequence of their choice of population model and in part a consequence of using relatively weak or inappropriate statistical methods. Other patterns of density dependence respond differently to environmental fluctuations, and the impact of the disturbance on these is clearly visible using their methods. For some patterns of population dynamics, environmental impacts are more readily detectable by correlating running-average environmental conditions with the population time-series or by correlating the first differences of the population time-series with environmental noise. When more appropriate statistical methods are used, environmental forcing is detectable in the majority of cases used by Kaitala and Ranta. The interplay between environmental stochasticity and density-dependent population growth means that there is no single best method to detect the influence of environmental forcing, even when population dynamics are approximately linear. But environmental forcing will often be detectable, contrary to Kaitala and Ranta's assertions.     

Effects on population persistence: the interaction between environmental noise colour,intraspecific competition and space

It is accepted that accurate estimation of risk of population extinction, or persistence time, requires prediction of the effect of fluctuations in the environment on population dynamics. Generally, the greater the magnitude, or variance, of environmental stochasticity, the greater the risk of population extinction. Another characteristic of environmental stochasticity, its colour, has been found to affect population persistence. This is important because real environmental variables, such as temperature, are reddened or positively temporally autocorrelated. However, recent work has disagreed about the effect of reddening environmental stochasticity. Ripa and Lundberg (1996) found increasing temporal autocorrelation (reddening) decreased the risk of extinction, whereas a simple and powerful intuitive argument (Lawton 1988) predicts increased risk of extinction with reddening. This study resolves the apparent contradiction, in two ways, first, by altering the dynamic behaviour of the population models. Overcompensatory dynamics result in persistence times increasing with increased temporal autocorrelation; undercompensatory dynamics result in persistence times decreasing with increased temporal autocorrelation. Secondly, in a spatially subdivided population, with a reasonable degree of spatial heterogeneity in patch quality, increasing temporal autocorrelation in the environment results in decreasing persistence time for both types of competition. Thus, the inclusion of coloured noise into ecological models can have subtle interactions with population dynamics.     

Floater dynamics can explain positive patterns of density-dependent fecundity in animal populations

After some 70 years of debate on density-dependent regulation of animal populations, there is still poor understanding of where spatial and temporal density dependence occurs. Clearly defining the portion of the population that shapes density-dependent patterns may help to solve some of the ambiguities that encircle density dependence and its patterns. In fact, individuals of the same species and population can show different dynamics and behaviors depending on their locations (e.g., breeding vs. dispersal areas). Considering this form of intrapopulation heterogeneity may improve our understanding of density dependence and population dynamics in general. We present the results of individual-based simulations on a metapopulation of the Spanish imperial eagle Aquila adalberti. Our results suggest that high rates of floater mortality within settlement areas can determine a shift in the classical relationship (from negative to positive) between the fecundity (i.e., fledglings per pair) and density (i.e., number of pairs) of the breeding population. Finally, we proved that different initial conditions affecting the breeder portion of the population can lead to the same values of fecundity. Our results can represent a starting point for new and more complex approaches studying the regulation of animal populations, where the forgotten and invisible component--the floater--is taken into account.     

Population dynamics with a refuge: fractal basins and the suppression of chaos

We consider the effect of coupling an otherwise chaotic population to a refuge. A rich set of dynamical phenomena is uncovered. We consider two forms of density dependence in the active population: logistic and exponential. In the former case, the basin of attraction for stable population growth becomes fractal, and the bifurcation diagrams for the active and refuge populations are chaotic over a wide range of parameter space. In the case of exponential density dependence, the dynamics are unconditionally stable (in that the population size is always positive and finite), and chaotic behavior is completely eradicated for modest amounts of dispersal. We argue that the use of exponential density dependence is more appropriate, theoretically as well as empirically, in a model of refuge dynamics.