Complex dynamic behaviour of autonomous microbial food chains

 The dynamic behaviour of food chains under chemostat conditions is studied. The microbial food chain consists of substrate (non-growing resources), bacteria (prey), ciliates (predator) and carnivore (top predator). The governing equations are formulated at the population level. Yet these equations are derived from a dynamic energy budget model formulated at the individual level. The resulting model is an autonomous system of four first-order ordinary differential equations. These food chains resemble those occuring in ecosystems. Then the prey is generally assumed to grow logistically. Therefore the model of these systems is formed by three first-order ordinary differential equations. As with these ecosystems, there is chaotic behaviour of the autonomous microbial food chain under chemostat conditions with biologically relevant parameter values. It appears that the trajectories on the attractors consists of two superimposed oscillatory behaviours, a slow one for predator–top predator and a fast one for the prey–predator on one branch at which the top predator increases slowly. In some regions of the parameter space there are multiple attractors. Received 8 November 1995; received in revised form 7 January 1997     

Peak-to-peak dynamics in food chain models

We show in this paper that the chaotic regimes of many food chain models often enjoy a very peculiar property, known as peak-to-peak dynamics. This means that the maximum (peak) density of the populations of any trophic level can be easily forecasted provided the last two peaks of the same population are known. Moreover, extensive simulation shows that only the last peak is needed if the forecast concerns the population at the top of the food chain and that peaks variability often increases from bottom to top. All these findings bring naturally to the conclusion that top populations should be sampled in order to have higher chances to detect peak-to-peak dynamics. The analysis is carried out by studying ditrophic food chain models with seasonally varying parameters, tritrophic food chain models with constant parameters, and more complex food chain and food web models.     

Allometry and catastrophic regime shifts in food chains

Population dynamics can reflect the body mass distribution of species because there is an allometric relationship between the average body mass of species and its metabolic timescale. Since predators are generally larger than their prey, a hierarchical structure from fast timescales to slow timescales can be a general structure in food webs. In this paper, we show that changes of the metabolic timescale ratio can cause catastrophic shifts. Then, we investigate a two-dimensional parameter space with the timescale ratio and the carrying capacity of basal species, and reveal that the timescale ratio characterizes the response of the system to environmental variation. Finally, in a bistable regime, we try to clarify the relationship between the trophic position of a species and the extent to which the species induces attractor switching. We saw that, in a 4-species food chain, top predators and second consumers induce attractor switching easily compared to first consumers and basal species.     

Joint effects of nutrients and contaminants on the dynamics of a food chain in marine ecosystems

We analyze the joint effect of contaminants and nutrient loading on population dynamics of marine food chains by means of bifurcation analysis. Contaminant toxicity is assumed to alter mortality of some species with a sigmoidal dose-response relationship. A generic effect of pollutants is to delay transitions to complex dynamical states towards higher nutrient load values, but more counterintuitive consequences arising from indirect effects are described. In particular, the top predator seems to be the species more affected by pollutants, even when contaminant is toxic only to lower trophic levels.     

Top predator persistence in differential equation models of food chains: The effects of omnivory and external forcing of lower trophic levels

Persistence criteria are given for the highest trophic level predator in ordinary differential equation models of food chains exhibiting arbitrary omnivory and external supplementation of food source or an intermediate predator. The results are expressed in terms of inequalities involving the bounds on the intrinsic growth and interaction rates. Whether omnivory or external forcing enhances persistence is discussed, particularly for the examples of three-, four-, and five-link Lotka-Volterra food chains.     

Trophic exploitation in grassland food chains: simple models and a field experiment

This study provides insight into the importance of top carnivores (top-down control) and nutrient inputs (bottom-up control) in structuring food chains in a terrestrial grassland system. Qualitative predictions about food chain structure are generated using 4 simple models, each differing in assumptions about some key component in the population dynamics of the herbivore trophic level. The four model systems can be classified broadly into two groups (1) those that assume plant resource intake by herbivores is limited by search rate and handling time as described by classic Lotka-Volterra models; and (2) those that assume plant resource intake by herbivores is limited externally by the supply rate of resources as described by alternatives to Lotka-Volterra formulations. The first class of models tends to ascribe greater importance to top-down control of food chain structure whereas the second class places greater weight on bottom-up control. I evaluated the model predictions using experimentally assembled grassland food chains in which I manipulated nutrient inputs and carnivore (wolf spider) abundance to determine the degree of top-down and bottom-up control of grassland plants and herbivores (grasshoppers). The experimental results were most consistent with predictions of the second class of models implying a predominance of bottom-up control of food chain structure.     

Top-predator abundance and chaos in tritrophic food chains

Nonlinear models and laboratory experiments suggest that populations can be chaotic, whereas field data show that a fair proportion of observed populations are not too far from being chaotic. Thus, a natural question arises: do ecosystems enjoy special properties at the edge of chaos? By limiting the analysis to three classes of tritrophic food chains and to the role played by the nutrient available to the bottom of the ecosystem, we show that top-predator mean abundance is maximum at the edge of chaos. The geographical variability of species abundance along nutrient gradients and the dynamic complexity of observed natural populations are consistent with our findings. Effective guidelines for the sustainability of exploited ecosystems are also derived.     

Periodic coexistence of four species competing for three essential resources

We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.     

Effects of dispersal in a tri-trophic metapopulation model

The aim of this paper is to understand how dispersal in a patchy environment influences the stability properties of tri-trophic metapopulations. Differential equation models for tri-trophic metapopulations are formulated and analysed. The patchy nature of the metapopulations is incorporated through dispersal phases. Two variants are studied: one with a dispersal phase for the top and one with a dispersal phase for the middle level. A complete characterisation of stable and unstable equilibria is given and the possibility of invasion in these food chains is studied. A dispersal phase for the middle level can destabilize the bottom level-middle level interaction, because of the delay that dispersal causes in the reaction to the resource. When the middle level is not efficiently controlled by the top level, the unstable bottom level-middle level pair can destabilize the entire food chain. Dispersal for the top level can destabilize in the same way. A characterisation of the long term behaviour of the models is given. Bistability with a stable three species equilibrium and a stable limit cycle is one of the possibilities.     

Effects of jasmonic acid and its methylester on in vitro microtuberisation of three food yam (Dioscorea) species

The effects of jasmonic acid (JA) applied in the medium and its methylester (MeJA) applied either in the medium or as a vapour, on shoot growth and microtuber formation were evaluated in three important food yam species (Dioscorea alata, D. cayenensis, and D. rotundata). Single nodes with leaves, derived from in vitro-multiplied material, were used as explants. When delivered at higher concentrations (10 or 50 μm) both JA and MeJA suppressed node formation. Microtuberisation was supported in all three species by adding either JA or MeJA to the medium. Significant promotory effects were observed only when photoperiod, salt compositions and sucrose concentrations known to favour microtuberisation processes in yams were used. MeJA applied as a vapour strongly inhibited microtuber differentiation in D. alata on all media tested but in D. rotundata and D. cayenensis yams MeJA, also applied in the vapour phase, exhibited slight promotory effects on microtuberisation. Received: 28 January 1998 / Revision received: 7 October 1999 / Accepted: 7 January 2000     

Measuring fruit patch size for three sympatric indonesian primate species

Food availability is one of the basic factors affecting primate density and socioecology, but food availability is difficult to assess. Two different ways to obtain accurate estimates of food availability have been proposed: using phenology data or using the behaviour of animals. Phenology data can be refined by only including trees that are large enough to be used; including (potential) tree species in which by the concerned primate species forage; or including (fruiting) trees of these species that actually produce fruit. Alternatively, the sizes of the actually visited trees (foraging trees) give an estimate of fruit availability. These measures are compared for three sympatric primate species at the Ketambe Research Station, Sumatra, Indonesia: the Thomas langur, the long-tailed macaque and the orangutan. The sizes of fruiting trees and the foraging trees are larger than the potential trees. The sizes of the potential trees and of the fruiting trees are similar for the three primate species. This, however, is not reflected in the use of trees: the langurs forage on average in trees of similar size to those producing fruit, whereas the macaques and orangutans forage in trees larger than those producing fruit. The use of trees does not necessitate a different cut off point of included dbhs for the three compared primate species. The use of trees of different sizes, however, may be regulated by food competition. This indicates that sympatric primates make different foraging decisions and that behavioural measures of food availability will be less reliable.     

食物链中抗生素耐药性基因的转移

抗生素的使用,一方面起到预防和治疗疾病的作用,另一方面,饲料、食品和环境中抗生素残留增加,使抗生素耐药性菌产生并进化,从而导致将来治疗某些疾病时无有效抗生素可用,比如,结核病已经卷土重来,而且现在就有许多患者就不能用抗生素治愈。随着人们生活水平的提高,安全、健康问题受到人们广泛的关注和重视。本文初步对耐药性基因在食物链中怎样产生和转移进行综述。     

Quantification of invertebrate predation and herbivory in food chains of low complexity

Zooplankton grazing impact on algae, heterotrophic flagellates and bacteria, as well as invertebrate predation on herbivorous zooplankton, were investigated in two sub-Antarctic lakes with extremely simple food chains. The two species of herbivorous zooplankton present in the lakes (the copepods boeckella michaelseni and Pseudoboeckella poppei) exerted substantial grazing pressure on algae. However, the dominant algal species exhibited properties that enabled them to avoid (large size or extruding spines, e.g. Staurastrum sp., Tribonema sp.) or compensate (recruitment from the sediment, Mallomonas sp.) grazing. There are only two potential invertebrate predators on the herbivorous copepods in the two lakes: the copepod Parabroteas sarsi and the diving beetle Lancetes claussi. Vertebrate predators are entirely abscent from sub-Antarctic lakes. Based on our experiments, we estimated that the predators would remove at most about 0.4% of the herbivorous copepods per day, whereas planktivorous fish, if present in the lakes, would have removed 5–17% of the zooplankton each day. Consequently, the invertebrate predators in these high-latitude lakes had only a marginal predation impact compared to the predation pressure on zooplankton in the presence of vertebrate predators in temperate lakes. The study of these simple systems with only two quantitatively functionally important trophic links, suggests that high grazing pressure foreces the algal community towards forms with grazer resistant adaptations such as large size, recruitment from another habitat, and grazer avoidance spines. We propose that due to such adaptations, predictions from food web theory are only partly corroborated, i.e. algal biomass actually increases with increasing productivity, although the grazer community is released from predation. In more species-rich and complex systems, e.g temperate lakes with three functionally important links, such adaptations are likely to be even more important, and, consequently, the observable effects of trophic interactions from top predators on lower trophic levels even more obscured.     

Density-dependent dispersal complicates spatial synchrony in tri-trophic food chains

Spatial synchrony can increase extinction risk and undermines metapopulation persistence. Both dispersal and biotic interactions can strongly affect spatial synchrony. Here, we explore the spatial synchrony of a tri-trophic food chain in two patches connected by density-dependent dispersal, namely the strategies of prey evasion (PE) and predator pursuit (PP). The dynamics of the food chain are depicted by both the Hastings–Powell model and the chemostat model, with synchrony measured by the Pearson correlation coefficient. We use the density-independent dispersal in the system as a baseline for comparison. Results show that the density-independent dispersal of a species in the system can promote its dynamic synchrony. Dispersal of intermediate species in the tri-trophic food chain is the strongest synchronizer. In contrast, the density-dependent PP and PE of intermediate species can desynchronize the system. Highly synchronized dynamics emerged when the basal species has a strong PE strategy or when the top species has a moderate PP strategy. Our results reveal the complex relationship between density-dependent dispersal and spatial synchrony in tri-trophic systems.     

Consequences of symbiosis for food web dynamics

Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions. We exclude such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two prey and one predator species in the chemostat. We compose a theoretical food web where a predator feeds on two prey species that have a symbiotic relationships. In addition to a species-specific resource, the two prey populations consume the products of the partner population as well. In turn, a common predator forages on these prey populations. The temporal change in the biomass and the nutrient densities in the reactor is described by ordinary differential equations (ODE). Since products are recycled, the dynamics of these abiotic materials must be taken into account as well, and they are described by odes in a similar way as the abiotic nutrients. We use numerical bifurcation analysis to assess the long-term dynamic behaviour for varying degrees of symbiosis. Attractors can be equilibria, limit cycles and chaotic attractors depending on the control parameters of the chemostat reactor. These control parameters that can be experimentally manipulated are the nutrient density of the inflow medium and the dilution rate. Bifurcation diagrams for the three species web with a facultative symbiotic association between the two prey populations, are similar to that of a bi-trophic food chain; nutrient enrichment leads to oscillatory behaviour. Predation combined with obligatory symbiotic prey-interactions has a stabilizing effect, that is, there is stable coexistence in a larger part of the parameter space than for a bi-trophic food chain. However, combined with a large growth rate of the predator, the food web can persist only in a relatively small region of the parameter space. Then, two zero-pair bifurcation points are the organizing centers. In each of these points, in addition to a tangent, transcritical and Hopf bifurcation a global heteroclinic bifurcation is emanating. This heteroclinic cycle connects two saddle equilibria where the predator is absent. Under parameter variation the period of the stable limit cycle goes to infinity and the cycle tends to the heteroclinic cycle. At this global bifurcation point this cycle breaks and the boundary of the basin of attraction disappears abruptly because the separatrix disappears together with the cycle. As a result, it becomes possible that a stable two-nutrient–two-prey population system becomes unstable by invasion of a predator and eventually the predator goes extinct together with the two prey populations, that is, the complete food web is destroyed. This is a form of over-exploitation by the predator population of the two symbiotic prey populations. When obligatory symbiotic prey-interactions are modelled with Liebigs minimum law, where growth is limited by the most limiting resource, more complicated types of bifurcations are found. This results from the fact that the Jacobian matrix changes discontinuously with respect to a varying parameter when another resource becomes most limiting.Revised version: 21 July 2003     

混沌理论及其在建立神经网络模型中的应用

随着许多学科的相互紧密交叉以及混沌理论的日益深入的研究,人们从生物现象中提出了许多与混沌有关的神经网络模型,本文对混沌理论的基本原理做了简要概述,并着重介绍了四种有代表性的混沌神经网络模型及其应用．同时指出这一研究方向无论在理论还是在应用方面都具有十分诱人的前景．     

Differential effects of food availability on population growth and fitness of three species of estuarine, bacterial-feeding nematodes

The significance of bottom-up controls on biological communities has been a long-standing topic of interest in ecology. However, before environmental effects on communities can be properly assessed, a thorough knowledge of the individual species' responses is required. We studied effects of food availability on population development and on different life-history traits in three species of bacterial-feeding nematodes, Diplolaimelloides oschei, Diplolaimelloides meyli (both Monhysteridae) and Pellioditis marina (Rhabditidae), which co-occur on macrophyte detritus in the Westerschelde Estuary (SW Netherlands). The bacteria Escherichia coli was offered in five food-availability treatments corresponding to initial cell densities from 3 × 10¹⁰ cells ml^− 1 to 3 × 10⁷ cells ml^− 1. The three bacterial-feeding nematode species studied here showed differential responses to food availability, which agreed with the general idea that Rhabditidae have extreme colonization abilities under very high food availability, while Monhysteridae tend to have a somewhat slower population development and comparatively lower food requirements. Several life-history traits, including juvenile mortality and development time, did not exhibit a clear food-availability dependence, but bioenergetics-related parameters did. Results on the F1 generation may, however, be affected by strong maternal effects on life-history traits of their progeny. Patterns of food-availability dependence of population increase and size at maturity were similar in P. marina. Both Diplolaimelloides species, however, exhibited a large body size at maturity but a very low population increase at the highest food availability, suggesting a trade-off between biomass and reproduction. Comparison with published data on other nematode species reveals that nematode responses to food availability as well as to other environmental factors are highly species-specific.     

一个海洋食物链能流的初步研究

选择我国北方海岸带水域中属于第一、二、三、四营养级的４个经济种,构成一个简单的人工食物链,即金藻→卤虫→玉筋鱼→黑鱼君．测定了各营养级间物质（湿重、干重）、能量的转换效率,求出了食物链各环节的生产量比值．结果表明,在这个食物链运转过程中,生产１ｋｇ湿重的黑鱼君（第四营养级）需要消耗相当于初级生产力２３５．２ｋｇ湿重的金藻（第一营养级）;生产１ｋｇ干重的黑鱼君需要消耗相当于１４８．３ｋｇ干重的金藻;黑鱼君富集１ｋＪ能量需要消耗含有１１０．７ｋＪ的金藻．     

Contaminated food and uptake of heavy metals by fish: a review and a proposal for further research

Summary 1. The uptake of heavy metals via the alimentary tract can be an important factor for the metal budget of fish. 2. Concepts such as biomagnification, bioaccumulation, biotransference, or concentration factors, convey little information about the real threat originating from heavy metals in an aquatic food chain. 3. In polluted aquatic ecosystems the transfer of metals through food chains can be high enough to bring about harmful concentrations in the tissues of fish. This relationship is called the food chain effect. 4. Two kinds of ecological factors influence the food chain effect: firstly, high levels of contamination of the food, and, secondly, the reduction of species diversity. When susceptible species are eliminated, metal-tolerant food organisms may become dominant. Their tolerance may be based either on their ability to accumulate excessive amounts of metals or to exclude heavy metals from the tissues. These two strategies represent feedback mechanisms which may enhance or weaken the food chain effect. 5. It is concluded that future investigations on transference of heavy metals to fish must take into more careful consideration the specific ecological situation of a given environment.     

引起神经元“非周期敏感现象”的分岔机制

实验发现神经元平均发放率变化率在放电脉冲序列串(ISI序列)呈非周期节律时普遍大于ISI呈周期节律时的平均发放率变化率,称为“非周期敏感现象”。以HR神经元模型和胰腺β-细胞模型为例,在合适的参数改变量作用下观察到了“非周期敏感现象”,并进一步讨论了平均发放率变化率与ISI序列动力学性质的关系。发现当ISI序列经历混沌-周期分岔时“非周期敏感现象”表现明显,尤其在ISI序列经历从混沌到嵌入在混沌中的周期窗口的分岔时表现最为显著。进一步的分析表明周期窗口在整个混沌带中所占测度较大,故混沌．周期分岔及从混沌到嵌入在混沌中的周期窗口分岔是引起神经元“非周期敏感现象”的一种重要动力学机制。实验结果支持上述结论。