Deterministic chaоs and the problem of predictability in population dynamics

The studies of the processes that can significantly influence the predictability in population dynamics are reviewed and the results of mathematical simulations of population dynamics are compared to the time series obtained in field observations. Considerable attention is given to the chaotic changes in population abundance. Some methods of numerical analysis of chaoticity and predictability of the time series are considered. The importance of comparing the results of mathematical simulation and observation data is tightly linked to problems in detecting chaos in the dynamics of natural populations and estimating the prevalence of chaotic regimes in nature. Insight into these problems can allow identification of the functional role of chaotic regimes in population dynamics.     

Observance of period-doubling bifurcation and chaos in an autonomous ODE model for malaria with vector demography

We illustrate that an autonomous ordinary differential equation model for malaria transmission can exhibit period-doubling bifurcations leading to chaos when ecological aspects of malaria transmission are incorporated into the model. In particular, when demography, feeding, and reproductive patterns of the mosquitoes that transmit the malaria-causing parasite are explicitly accounted for, the resulting model exhibits subcritical bifurcations, period-doubling bifurcations, and chaos. Vectorial and disease reproduction numbers that regulate the size of the vector population at equilibrium and the endemicity of the malaria disease, respectively, are identified and used to simulate the model to show the different bifurcations and chaotic dynamics. A subcritical bifurcation is observed when the disease reproduction number is less than unity. This highlights the fact that malaria control efforts need to be long lasting and sustained to drive the infectious populations to levels below the associated saddle-node bifurcation point at which control is feasible. As the disease reproduction number increases beyond unity, period-doubling cascades that develop into chaos closely followed by period-halving sequences are observed. The appearance of chaos suggests that characterization of the physiological status of disease vectors can provide a pathway toward understanding the complex phenomena that are known to characterize the dynamics of malaria and other indirectly transmitted infections of humans. To the best of our knowledge, there is no known unforced continuous time deterministic host-vector transmission malaria model that has been shown to exhibit chaotic dynamics. Our results suggest that malaria data may need to be critically examined for complex dynamics.     

Food chains in the chemostat: Relationships between mean yield and complex dynamics

Atritrophic food-chain chemostat model composed of a prey with Monod-type nutrient uptake, a Holling Type II predator and a Holling Type II exploited superpredator is considered in this paper. The bifurcations of the model show that dynamic complexity first increases and then decreases with the nutrient supplied to the bottom of the food chain. Extensive simulations prove that the same holds for food yield, i.e., there exists an optimum nutrient supply which maximizes mean food yield. Finally, a comparative analysis of the results points out that the optimum nutrient supply practically coincides with the nutrient supply separating chaotic dynamics from high-frequency cyclic dynamics. This reinforces the idea, already known for simpler models, that food yield maximization requires that the system behaves on the edge of chaos.     

Chaos in a periodically forced chemostat with algal mortality

We study the possibility of chaotic dynamics in the externally driven Droop model. This model describes a phytoplankton population in a chemostat under periodic nutrient supply. Previously, it has been proven under very general assumptions, that such systems are not able to exhibit chaotic dynamics. We show that the simple introduction of algal mortality may lead to chaotic oscillations of algal density in the forced chemostat. Our numerical simulations show that the existence of chaos is intimately related to plankton overshooting in the unforced model. We provide a simple measure, based on stability analysis, for estimating the amount of overshooting. These findings are not restricted to the Droop model but also hold for other chemostat models with mortality. Our results suggest periodically driven chemostats as a simple model system for the experimental verification of chaos in ecology.     

Structural perturbations to population skeletons: transient dynamics, coexistence of attractors and the rarity of chaos

BACKGROUND: Simple models of insect populations with non-overlapping generations have been instrumental in understanding the mechanisms behind population cycles, including wild (chaotic) fluctuations. The presence of deterministic chaos in natural populations, however, has never been unequivocally accepted. Recently, it has been proposed that the application of chaos control theory can be useful in unravelling the complexity observed in real population data. This approach is based on structural perturbations to simple population models (population skeletons). The mechanism behind such perturbations to control chaotic dynamics thus far is model dependent and constant (in size and direction) through time. In addition, the outcome of such structurally perturbed models is [almost] always equilibrium type, which fails to commensurate with the patterns observed in population data. METHODOLOGY/PRINCIPAL FINDINGS: We present a proportional feedback mechanism that is independent of model formulation and capable of perturbing population skeletons in an evolutionary way, as opposed to requiring constant feedbacks. We observe the same repertoire of patterns, from equilibrium states to non-chaotic aperiodic oscillations to chaotic behaviour, across different population models, in agreement with observations in real population data. Model outputs also indicate the existence of multiple attractors in some parameter regimes and this coexistence is found to depend on initial population densities or the duration of transient dynamics. Our results suggest that such a feedback mechanism may enable a better understanding of the regulatory processes in natural populations.     

马尾松毛虫种群动态的时间序列分析及复杂性动态研究

自从May(1974)指出即使是简单的种群模型也能揭示混沌动态以来,自然种群是否存在混沌一直具有争论,如何检测自然种群的混沌行为也成为种群动态研究的一个难点,通过时间序列分析和反应面模型建模的8方法分析了马尾松毛虫的复杂性动态,用自相关函数对马尾松毛虫发生的时间动态分析的结果认为动态是平衡的,其周期性不显著,而具有一定的复杂性,这种类型可以是减幅波动,有限周期或弱混沌,波动主要由系统内因引起,进一步采用反应面模型估计全局李雅普若夫指数和局域李雅普若夫指数结果均为负,显示马尾松毛虫种群动态不存在混沌现象,但是在增加一个小的噪音以后,局域李雅普若夫指数变为在0以上的波动,说明系统对噪音非常敏感,噪音对松毛虫种群动态具有很大的影响,可以将其从非混沌状态变为混沌,研究结果认为全局郴雅普若夫指数λ是一定时间内两个变动轨迹的总平均偏差,而随着种群动态的波动,指数也是波动的,所以对于检测自然种群的混沌来说不是一个好的指标,局域李雅普若夫指数λM能更好地表示自然种群混沌的存在和产生混沌的条件,对害虫管理来说对种群暴发初期的预测是尤其重要的,而此时又最难于预测,所以对种群动态的监测就尤为重要,由于马尾松毛虫的代间种群动态为第一级密度相关,前一代的虫口密度与下一代的虫口密度相关性最强,所以前一代预测下一代是最可靠的。     

Yield and dynamics of tritrophic food chains

Strong relationships between yield and dynamic behavior of tritrophic food chains are pointed out by analyzing the classical Rosenzweig-MacArthur model. On the one hand, food chains are subdivided into undersupplied and oversupplied categories, the first being those in which a marginal increase of nutrient supply to the bottom produces a marginal increase of mean yield at the top. On the other hand, a detailed bifurcation analysis proves that dynamic complexity first increases with nutrient supply (from stationary to a low-frequency cyclic regime and, finally, to chaos) and then decreases (from chaos to a high-frequency cyclic regime). A careful comparison of the two analyses supports the conclusion that food chains cycling at high frequency are oversupplied, while all others are undersupplied. A straightforward consequence of this result is that maximization of food yield requires a chaotic regime. This regime turns out to be very often on the edge of a potential catastrophic collapse of the top component of the food chain. In other words, optimality implies very complex and dangerous dynamics, as intuitively understood long ago for ditrophic food chains by Rosenzweig in his famous article on the paradox of enrichment.     

Viruses and the microbial loop

The abundance of viral-like particles in marine ecosystems ranges from <104 ml^–1 to >10⁸ ml^–1. Their distribution in time and space parallels that of other biological parameters such as bacterial abundance and chlorophyll a. There is a lack of consensus between methods used to assess viral activity, i.e., rate of change in viral abundance (increase or decrease). The highest rates, 10–100 days^–1, are observed in experiments with short sampling intervals (0.2–2 h), while lower rates, on the order of 1 day^–1, are observed in experiments with longer sampling intervals (days). Few studies have been carried out, but viruses appear, at least in some cases, to have a significant impact on carbon and nutrient flow in microbial food webs. Viruses have also been demonstrated to exert a species specific control of both bacteria and phytoplankton populations in natural waters.     

Effects of survival thresholds upon one-dimensional dynamics of single-species populations

A simple one-dimensional model of single-species populations is studied by means of computer simulations. Although the model has a rich spectrum of dynamics including chaotic behavior, the introduction of survival thresholds makes the chaotic region so small that it can be hardly observed. Stochastic fluctuations further reduce the chaotic region because they accidentally lead populations to extinction. The model thus naturally explains the observation that the majority of natural populations do not show chaotic behavior but a monotonic return to a stable equilibrium point following a disturbance.     

银鱼的产量能预报吗

将离散Ｌogistic模型应用于银鱼种群数量变动研究,通过对滇池等４个典型湖泊或水库的银鱼年产量变动的初步分析和模拟,发现现的有的湖泊或水库银鱼产量的参数值都落入了混沌区间,在自然生态系统中找到了混沌行为的证据。同时指出：（１）混沌行为使银鱼产量长期预报不可能实现,只有短期预报才能保证必要的精度。（２）严格控制捕劳对尚未繁殖的亲鱼的影响,保留足够的繁殖亲鱼,才能保证资源的持续利用。另一方面,如谷获得相对稳定的产量,可能控制捕捞死亡率Ｆ来改变增增长率参数μ,防止银鱼产量剧烈波动。（３）水域污染和其他破坏水域饵料生物种群结构的因素能导致银鱼的内禀自然增长率γ值和最大种群数量Ｎmax发生变化,从而引起种群的数量变动。     

Time-dependent regimes of a tourism-based social–ecological system: Period-doubling route to chaos

The period-doubling route to chaos has occupied a prominent position and it is still object of great interest among the different complex phenomena observed in nonlinear dynamical systems. The reason of such interest is that such route to chaos has been observed in many physical, chemical and ecological models when they change over from simple periodic to complex aperiodic motion. In interlinked social–ecological systems (SESs) there might be an apparent great ability to cope with change and adapt if analysed only in their social dimension. However, such an adaptation may be at the expense of changes in the capacity of ecosystems to sustain the adaptation and it could affect the quality of ecosystem goods and services since it could degrade natural renewable and non-renewable resources and generate traps and breakpoints in the whole SES eventually leading to chaotic behaviour. This paper is rooted in previous results on modelling tourism-based SESs, only recently object of theoretical investigations, focusing on the dynamics of the coexistence between mass-tourists and eco-tourists. Here we describe a finer scale analysis of time-dependent regimes in the ranges of the degradation coefficient (bifurcation parameter), for which the system can exhibit coexistence. This bifurcation parameter is determined by objective changes in the real world in the quality of ecosystem goods and services together with whether and how such changes are perceived by different tourist typologies. Varying the bifurcation parameter, the dynamical system may in fact evolve toward an aperiodical dynamical state in many ways, showing that there could be different scenarios for the transition to chaos. This paper provides a further evidence for the period-doubling route to chaos with reference to tourism-based socio-ecological models, and for a period locking behaviour, where a small variation in the bifurcation parameter can lead to alternating regular and chaotic dynamics. Moreover, for many models undergoing chaos via period-doubling, it has been showed that structural perturbations with real ecological justification, may break and reverse the expected period-doublings, hence inhibiting chaos. This feature may be of a certain relevance also in the context of adaptive management of tourism-based SESs: these period-doubling reversals might in fact be used to control chaos, since they potentially act in way to suppress possibly dangerous fluctuations.     

Self-organized instability in complex ecosystems

Why are some ecosystems so rich, yet contain so many rare species? High species diversity, together with rarity, is a general trend in neotropical forests and coral reefs. However, the origin of such diversity and the consequences of food web complexity in both species abundances and temporal fluctuations are not well understood. Several regularities are observed in complex, multispecies ecosystems that suggest that these ecologies might be organized close to points of instability. We explore, in greater depth, a recent stochastic model of population dynamics that is shown to reproduce: (i) the scaling law linking species number and connectivity; (ii) the observed distributions of species abundance reported from field studies (showing long tails and thus a predominance of rare species); (iii) the complex fluctuations displayed by natural communities (including chaotic dynamics); and (iv) the species-area relations displayed by rainforest plots. It is conjectured that the conflict between the natural tendency towards higher diversity due to immigration, and the ecosystem level constraints derived from an increasing number of links, leaves the system poised at a critical boundary separating stable from unstable communities, where large fluctuations are expected to occur. We suggest that the patterns displayed by species-rich communities, including rarity, would result from such a spontaneous tendency towards instability.     

Alternate attractors in the population dynamics of a tree-killing bark beetle

Among the most striking changes in ecosystems are those that happen abruptly and resist return to the original condition (i.e., regime shifts). This frequently involves conspicuous changes in the abundance of one species (e.g., an oubreaking pest or keystone species). Alternate attractors in population dynamics could explain switches between low and high levels of abundance, and could underlie some cases of regime shifts in ecosystems; this longstanding theoretical possibility has been difficult to test in nature. We compared the ability of an alternate attractors model versus two competing models to explain population fluctuations in the tree-killing bark beetle, Dendroctonus frontalis. Frequency distributions of abundance were distinctly bimodal, a prediction of the alternate attractors model, strongly indicating the lack of a single, noisy equilibrium. Time series abundance data refuted the existence of strong delayed density-dependence or nonlinearities, as required by the endogenous cycles model. The model of alternate attractors was further supported by the existence of positive density-dependence at intermediate beetle abundances. Experimental manipulations show that interactions with competitors and shared enemies could create a locally stable equilibrium in small populations of D. frontalis. High variation among regions and years in the abundance of predators and competitors could permit switches between alternate states. Dendroctonus frontalis now provides the strongest case that we know of for alternate attractors in natural population dynamics. The accompanying demographic instability appears to underlie spatially extensive outbreaks that have lasting impacts on forest ecosystems. Understanding feedbacks in populations with alternate attractors can help to identify thresholds underlying regime shifts, and potentially manage them to avoid undesirable impacts.     

Models of coupled population oscillators using 1-D maps

 Coupled population oscillators are investigated with the use of coupled logistic maps. Two forms of coupling are employed, reproductive and density. Three biologically distinct situations are investigated: populations independently oscillating in a two point cycle, populations independently chaotic, and populations independently approach a stable point. Both entrained and phase reversed patterns are observed along with complicated forms of chaos as the coupling parameters are varied.     

Incorporating spatial variation in density enhances the stability of simple population dynamics models

Simple discrete time models of population growth admit a wide variety of dynamic behaviors, including population cycles and chaos. Yet studies of natural and laboratory populations typically reveal their dynamics to be relatively stable. Many explanations for the apparent rarity of unstable or chaotic behavior in real populations have been developed, including the possible stabilizing roles of migration, refugia, abrupt density-dependence, and genetic variation in sensitivity to density. We develop a theoretical framework for incorporating random spatial variation in density into simple models of population growth, and apply this approach to two commonly used models in ecology: the Ricker and Hassell maps. We show that the incorporation of spatial density variation into both these models has a strong stabilizing influence on their dynamic behavior, and leads to their exhibiting stable point equilibria or stable limit cycles over a relatively much larger range of parameter values. We suggest that one reason why chaotic population dynamics are less common than the simple models indicate is, these models typically neglect the potentially stabilizing role of spatial variation in density.     

When can noise induce chaos and why does it matter: a critique

Noise‐induced chaos illustrates how small amounts of exogenous noise can have disproportionate qualitative impacts on the long term dynamics of a nonlinear system. This property is particularly clear in chaotic systems but is also important for the majority of ecological systems which are nonchaotic, and has direct implications for analyzing ecological time series and testing models against field data. Dennis et al. point out that a definition of chaos which we advocated allows a noise‐dominated system to be classified as chaotic when its Lyapunov exponent λ is positive, which misses what is really going on. As a solution, they propose to eliminate the concept of noise‐induced chaos: chaos “should retain its strictly deterministic definition”, hence “ecological populations cannot be strictly chaotic”. Instead, they suggest that ecologists ask whether ecological systems are strongly influenced by “underlying skeletons with chaotic dynamics or whatever other dynamics”– the skeleton being the hypothetical system that would result if all external and internal noise sources were eliminated. We agree with Dennis et al. about the problem – noise‐dominated systems should not be called chaotic – but not the solution. Even when an estimated skeleton predicts a system's short term dynamics with extremely high accuracy, the skeleton's long term dynamics and attractor may be very different from those of the actual noisy system. Using theoretical models and empirical data on microtine rodent cycles and laboratory populations of Tribolium, we illustrate how data analyses focusing on attributes of the skeleton and its attractor – such as the “deterministic Lyapunov exponent”λ₀ that Dennis et al. have used as their primary indicator of chaos – will frequently give misleading results. In contrast, quantitative measures of the actual noisy system, such as λ, provide useful information for characterizing observed dynamics and for testing proposed mechanistic explanations.     

Trophic interactions of fish communities at midwater depths enhance long-term carbon storage and benthic production on continental slopes

Biological transfer of nutrients and materials between linked ecosystems influences global carbon budgets and ecosystem structure and function. Identifying the organisms or functional groups that are responsible for nutrient transfer, and quantifying their influence on ecosystem structure and carbon capture is an essential step for informed management of ecosystems in physically distant, but ecologically linked areas. Here, we combine natural abundance stable isotope tracers and survey data to show that mid-water and bentho-pelagic-feeding demersal fishes play an important role in the ocean carbon cycle, bypassing the detrital particle flux and transferring carbon to deep long-term storage. Global peaks in biomass and diversity of fishes at mid-slope depths are explained by competitive release of the demersal fish predators of mid-water organisms, which in turn support benthic fish production. Over 50% of the biomass of the demersal fish community at depths between 500 and 1800 m is supported by biological rather than detrital nutrient flux processes, and we estimate that bentho-pelagic fishes from the UK–Irish continental slope capture and store a volume of carbon equivalent to over 1 million tonnes of CO₂ every year.     

Natural vegetation cover in the landscape and edge effects: differential responses of insect orders in a fragmented forest

Human activities have led to global simplification of ecosystems, among which Neotropical dry forests are some of the most threatened. Habitat loss as well as edge effects may affect insect communities. Here, we analyzed insects sampled with pan traps in 9 landscapes (at 5 scales, in 100–500 m diameter circles) comprising cultivated fields and Chaco Serrano forests, at overall community and taxonomic order level. In total 7043 specimens and 456 species of hexapods were captured, with abundance and richness being directly related to forest cover at 500 m and higher at edges in comparison with forest interior. Community composition also varied with forest cover and edge/interior location. Different responses were detected among the 8 dominant orders. Collembola, Hemiptera, and Orthoptera richness and/or abundance were positively related to forest cover at the larger scale, while Thysanoptera abundance increased with forest cover only at the edge. Hymenoptera abundance and richness were negatively related to forest cover at 100 m. Coleoptera, Diptera, and Hymenoptera were more diverse and abundant at the forest edge. The generally negative influence of forest loss on insect communities could have functional consequences for both natural and cultivated systems, and highlights the relevance of forest conservation. Higher diversity at the edges could result from the simultaneous presence of forest and matrix species, although “resource mapping” might be involved for orders that were richer and more abundant at edges. Adjacent crops could benefit from forest proximity since natural enemies and pollinators are well represented in the orders showing positive edge effects.     

Chaotic Red Queen coevolution in three-species food chains

Coevolution between two antagonistic species follows the so-called ‘Red Queen dynamics’ when reciprocal selection results in an endless series of adaptation by one species and counteradaptation by the other. Red Queen dynamics are ‘genetically driven’ when selective sweeps involving new beneficial mutations result in perpetual oscillations of the coevolving traits on the slow evolutionary time scale. Mathematical models have shown that a prey and a predator can coevolve along a genetically driven Red Queen cycle. We found that embedding the prey–predator interaction into a three-species food chain that includes a coevolving superpredator often turns the genetically driven Red Queen cycle into chaos. A key condition is that the prey evolves fast enough. Red Queen chaos implies that the direction and strength of selection are intrinsically unpredictable beyond a short evolutionary time, with greatest evolutionary unpredictability in the superpredator. We hypothesize that genetically driven Red Queen chaos could explain why many natural populations are poised at the edge of ecological chaos. Over space, genetically driven chaos is expected to cause the evolutionary divergence of local populations, even under homogenizing environmental fluctuations, and thus to promote genetic diversity among ecological communities over long evolutionary time.     

Phenotypic structure and bifurcation behavior of population models

Discrete time models for density-regulated populations have been shown to exhibit periodic and chaotic motion in the absence of any external signal. We show how the genetic structure of a population can initiate bifurcations to periodic and chaotic trajectories. We investigate by simulation the dependence of this phenomenon on the strength of assortative mating, the level of heterozygosity, and the intensity of selection. The implications of internally generated chaos for population modeling are discussed.