Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain

 The asymptotic behavior of a tri-trophic food chain model is studied. The analysis is carried out numerically, by finding both local and global bifurcations of equilibria and limit cycles. The existence of transversal homoclinic orbits to a limit cycle is shown. The appearance of homoclinic orbits, by moving through a homoclinic bifurcation point, is associated with the sudden disappearance of a chaotic attractor. A homoclinic bifurcation curve, which bounds a region of extinction, is continued through a two-dimensional parameter space. Heteroclinic orbits from an equilibrium to a limit cycle are computed. The existence of these heteroclinic orbits has important consequences on the domains of attraction. Continuation of non-transversal heteroclinic orbits through parameter space shows the existence of two codimension-two bifurcations points, where the saddle cycle is non-hyperbolic. The results are summarized by dividing the parameter space in subregions with different asymptotic behavior. Received: 25 February 1998 / Revised version: 19 August 1998     

Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect

Species establishment in a model system in a homogeneous environment can be dependent not only on the parameter setting, but also on the initial conditions of the system. For instance, predator invasion into an established prey population can fail and lead to system collapse, an event referred to as overexploitation. This phenomenon occurs in models with bistability properties, such as strong Allee effects. The Allee effect then prevents easy re-establishment of the prey species. In this paper, we deal with the bifurcation analyses of two previously published predator-prey models with strong Allee effects. We expand the analyses to include not only local, but also global bifurcations. We show the existence of a point-to-point heteroclinic cycle in these models, and discuss numerical techniques for continuation in parameter space. The continuation of such a cycle in two-parameter space forms the boundary of a region in parameter space where the system collapses after predator invasion, i.e. where overexploitation occurs. We argue that the detection and continuation of global bifurcations in these models are of vital importance for the understanding of the model dynamics.     

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.     

土壤—小麦食物链硒转移模式及其应用

利用~(75)Se为示踪剂,研究了土壤-小麦系统中Se的迁移和分布规律。土壤Se植物可利用系数SA=0.3—1.9,土壤不同,SA值也不同。土壤固液两相中Se的分配系数Kd=141—395,说明土壤对Se有强烈的吸附作用,Kd大小顺序为:暗棕色森林土>黑土>草甸棕壤。小麦对土壤Se的浓集系数CF=0.09—0.14,小麦不是浓集Se的作物;Se在小麦各器官间CF值>1,表明其在小麦体内运转比较容易。作者提出了一个食物链Se转移模式,评价了土壤有效Se水平,指出草甸棕壤有效Se充足,黑土和暗棕色森林土则不足。估算了病区每人每天摄Se量为26.6μg,远低于我国成人安全摄Se量建议值40—240μg/人·日的下限值。     

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。     

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．     

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.     

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.     

Biogeochemistry of selenium and its impact on food chain quality and human health

In areas where soils are low in bioavailable selenium (Se), potential Se deficiencies cause health risks for humans. Though higher plants have been considered not to require this element, the experience with low-Se soils in Finland has provided evidence that the supplementation of commercial fertilizers with sodium selenate affects positively not only the nutritive value of the whole food chain from soil to plants, animals and humans but also the quantity of plant yields. The level of Se addition has been optimal, and no abnormally high concentrations in plants or in foods of animal origin have been observed. Se levels in serum and human milk indicate that the average daily intake has been within limits considered to be safe and adequate. In fact, plants act as effective buffers, because their growth is reduced at high Se levels. They also tend to synthesize volatile compounds in order to reduce excess Se. On the other hand, when added at low concentrations, Se exerts a beneficial effect on plant growth via several mechanisms. As in humans and animals, Se strengthens the capacity of plants to counteract oxidative stress caused by oxygen radicals produced by internal metabolic or external factors. At proper levels it also delays some of the effects of senescence and may improve the utilization of short-wavelength light by plants. High additions are toxic and may trigger pro-oxidative reactions. Thus, the present supplementation of fertilizers with Se can be considered a very effective and readily controlled way to increase the average daily Se intake nationwide.     

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.     

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.     

Development of a combined selective enrichment method and polymerase chain reaction (PCR) assay for sensitive detection of Salmonella in food samples

PCR assays were formatted using primer pairs homologous to phoE and invA genes. The amplification conditions were optimized with pure cultures and reactions were carried out to define selectivity, specificity and sensitivity of both primer pairs. The performance of the invA primer pair was better than that of the phoE pair, making the specific detection of Salmonella serovars and strains isolated from different food samples possible. Using the invA primer pair, the combined selective enrichment method with the polymerase chain reaction assay was established and used to detect Salmonella from artificially multi-contaminated food samples. The complete procedure detected as few as three cells of Salmonella (3 c.f.u.) from milk and meat samples.     

Positive complexity-stability relations in food web models without foraging adaptation

May's [1972. Will a large complex system be stable? Nature 238, 413-414] local stability analysis of random food web models showed that increasing network complexity leads to decreasing stability, a result that is contradictory to earlier empirical findings. Since this seminal work, research of complexity-stability relations became one of the most challenging issues in theoretical ecology. We investigate conditions for positive complexity-stability relations in the niche, cascade, nested hierarchy, and random models by evaluating the network robustness, i.e., the fraction of surviving species after population dynamics. We find that positive relations between robustness and complexity can be obtained when resources are large, Holling II functional response is used and interaction strengths are weighted with the number of prey species, in order to take foraging efforts into account. In order to obtain these results, no foraging dynamics needs to be included. However, the niche model does not show positive complexity-stability relations under these conditions. By comparing to empirical food web data, we show that the niche model has unrealistic distributions of predator numbers. When this distribution is randomized, positive complexity-stability relations can be found also in the niche model.     

Quantitative comparison of food niches in some freshwater zooplankton

Summary The abilities of some zooplankton (rotifers, cladocerans, copepods) to ingest different sizes and kinds of food cells were quantified by determining the relative efficiencies with which they ingested nine tracer-cell types, ranging from a coccoid bacterium (0.45 m³) to the alga Cryptomonas erosa (800–920 m³). These efficiencies were obtained by dividing the clearance rate of each zooplankton group (species population, developmental stage or size class of a species population) on each ³²P-labeled cell type by that of a simultaneously-offered, ³³P-labeled, standard cell type — Chlamydomonas reinhardtii. Similarities of efficiency patterns on these cell types (food niches) between all possible pairs of the 17 zooplankton groups from 4 ecosystems were determined by calculating correlation coefficients. Although the utilization of the tested cell types may vary greatly within a species, three feeding guilds could be distinguished — based primarily on the efficiencies with which the smallest cell types were ingested. Guild I (Poyarthra vulgaris, Keratella crassa, Diaptomus minutus nauplii) ate the smallest cells (<4 m diameter) (bacterium, Synechococcus, Nannochloris) and Ankistrodesmus very ineffifently but the three Cryptomonas species very efficiently. Guild II (Bosmina longirostris, D. minutus copepodites and Adults) had higher efficiencies on Synechococcus, Nannochloris, Ankistrodesmus, Stichococcus, and Stephanodiscus than guild I but similarly low ones on the bacterium and high ones on the Cryptomonas species. Guild III (Conochilus inicornis, Keratella cochlearis, Ceriodaphnia quadrangula, Diaphanosoma leuchtembergianum) differed from guilds I and II in having uniformly high efficiencies on all the small cells as well ad the larger ones. Principal component analysis of the matrix of correlation coefficients provided objective confirmation of the three guilds and provided a visual representation of the food niches of the 17 zooplankton groups in 3-dimensional space.     

Ecological consequences of evolution in plant defenses in a metacommunity

Dispersal can affect the assembly of local communities in a metacommunity as well as evolution of local populations in a metapopulation. These two processes may also affect each other in ways that have not yet been well studied and that may have novel effects on community structure. Here, we illustrate the interaction of these two processes on community structure with a model of adaptive evolutionary dynamics of plant defenses in a metacommunity food web involving multiple patches along a productivity gradient. We find an enhanced suite of adaptive plant types in our metacommunity model than is predicted in the absence of dispersal. We also find that this, and the movement of nutrients among patches via dispersal, alters patterns of food web architecture, trophic structure and diversity along the productivity gradient. Overall, our model illustrates that evolutionary and metacommunity dynamics may influence communities in complex interactive ways that may not be predicted by models that ignore either of these types of processes.     

Assessment of microbiological quality of food preparation process in some restaurants of Makkah city

Microbiological contamination of food processing surfaces and utensils increases considerably the risk of food-borne illnesses via cross-contamination. Hence, the safety of served meals and beverages can be evaluated through the assessment of the microbiological quality of food contact surfaces in food-serving establishments. This study carried out in Makkah city aimed to assess the microbiological contamination levels on food processing surfaces and utensils in 43 restaurants from the 9 main districts in the city. A total of 294 swab preparations were taken from 16 types of food contact surfaces including cutting boards, food containers, knives, serving dishes and other utensils were examined. Ninety samples (31%) showed more than 10 CFU/cm² which were considered positive for microbiological contamination. Meat chopping devices and cutting boards were found as the most contaminated food contact surfaces (60% and 50%), while washed serving dishes and fridge handles were the least contaminated (21% and 18%). Microorganisms detected in the study were Klebsiella spp. (18.7%), Escherichia coli (17,7%), Staphylococcus aureus (4,4%), Pseudomonas spp. (1.7%), Proteus spp. (0.7%), Bacillus cereus (0.7%), and Candida sp. (0.3%). Klebsiella spp. and E. coli were observed in at least one sample from each of the sixteen different food contact surfaces. The incidence of restaurants with contaminated food contact surfaces was significantly variable among the different districts, with a value as high as 57% in the most affected district and 20% in the less affected. No contamination with Salmonella spp. or Listeria spp. was detected, however, the detection of Bacillus cereus, a toxin-forming microorganism, in two different restaurants underlines the need for continuous microbiological assessment to ensure standard sanitation levels in restaurants and catering establishments of Makkah city.     

Immune impairment in HIV infection: Existence of risky and immunodeficiency thresholds

Results of several studies show that some DC populations are susceptible to HIV. Modulation of DCs by HIV infection, in particular interference of the antigen-presenting function of DCs, is a key aspect in viral pathogenesis and contributes to viral evasion from immunity because the loss of the DC function engenders some impairment effects for a proliferation of CTL responses, which play an important role in the immune response to HIV. As described herein, we use a simple mathematical model to examine virus-immune dynamics over the course of HIV infection in the context of the immune impairment effects. A decrease of the DC number and function during the course of HIV-1 infection is observed. Therefore, we simply assumed that the immune impairment rate increases over the HIV infection. Under the assumption, four processes of the disease progression dynamics of our model are classifiable according to their virological properties. It is particularly interesting a typical disease progression presents a “risky threshold” and an “immunodeficiency threshold”. Regarding the former, the immune system might collapse when the impairment rate of HIV exceeds a threshold value (which corresponds to a transcritical bifurcation point). For the latter, the immune system always collapses when the impairment rate exceeds the value (which corresponds to a saddle-node bifurcation point). To test our theoretical framework, we investigate the existence and distribution of these thresholds in 10 patients.     

Ecological consequences of scaling of chew cycle duration and daily feeding time in Primates

Feeding systems and behaviors must evolve to satisfy the metabolic needs of organisms. This includes modifications to feeding systems as body size and metabolic needs change. Using our own data and data from the literature, we examine how size-related changes in metabolic needs are met by size-related changes in daily feeding time, chew cycle duration, volume of food processed per chew, and daily food volume intake in primates. Increases in chew cycle duration with body mass in haplorhine primates are described by a simple power function (cycle time α body mass^0.181). Daily feeding time increases with body mass when analyzed using raw data from the “tips” of the primate phylogenetic tree, but not when using phylogenetically independent contrasts. Whether or not daily feeding time remains constant or increases with body mass, isometry of ingested bite size and the slow rate of increase in chew cycle time with body size combine to allow daily ingested food volume to scale faster than predicted by metabolic rate. This positive allometry of daily ingested food volume may compensate for negative allometry of nutrient concentration in primate foods. Food material properties such as toughness and hardness have little impact on scaling of chew cycle durations, sequence durations, or numbers of chews in a sequence. Size-related changes in food processing abilities appear to accommodate size-related changes in food material properties, and primates may alter ingested bite sizes in order to minimize the impacts of food material properties on temporal variables such as chew cycle duration and chew sequence duration.     

Role of incidence function in vaccine-induced backward bifurcation in some HIV models

The phenomenon of backward bifurcation in disease models, where a stable endemic equilibrium co-exists with a stable disease-free equilibrium when the associated reproduction number is less than unity, has important implications for disease control. In such a scenario, the classical requirement of the reproduction number being less than unity becomes only a necessary, but not sufficient, condition for disease elimination. This paper addresses the role of the choice of incidence function in a vaccine-induced backward bifurcation in HIV models. Several examples are given where backward bifurcations occur using standard incidence, but not with their equivalents that employ mass action incidence. Furthermore, this result is independent of the type of vaccination program adopted. These results emphasize the need for further work on the incidence functions used in HIV models.     

Convergence properties of the degree distribution of some growing network models

In this article we study a class of randomly grown graphs that includes some preferential attachment and uniform attachment models, as well as some evolving graph models that have been discussed previously in the literature. The degree distribution is assumed to form a Markov chain; this gives a particularly simple form for a stochastic recursion of the degree distribution. We show that for this class of models the empirical degree distribution tends almost surely and in norm to the expected degree distribution as the size of the graph grows to infinity and we provide a simple asymptotic expression for the expected degree distribution. Convergence of the empirical degree distribution has consequences for statistical analysis of network data in that it allows the full data to be summarized by the degree distribution of the nodes without losing the ability to obtain consistent estimates of parameters describing the network.