Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincareacute return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing delta(1). However, for higher values of delta(1) the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter zeta) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.

How Complex, Probable, and Predictable is Genetically Driven Red Queen Chaos?

Coevolution between two antagonistic species has been widely studied theoretically for both ecologically- and genetically-driven Red Queen dynamics. A typical outcome of these systems is an oscillatory behavior causing an endless series of one species adaptation and others counter-adaptation. More recently, a mathematical model combining a three-species food chain system with an adaptive dynamics approach revealed genetically driven chaotic Red Queen coevolution. In the present article, we analyze this mathematical model mainly focusing on the impact of species rates of evolution (mutation rates) in the dynamics. Firstly, we analytically proof the boundedness of the trajectories of the chaotic attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. By using symbolic dynamics theory, we quantify the complexity of genetically driven Red Queen chaos computing the topological entropy of existing one-dimensional iterated maps using Markov partitions. Co-dimensional two bifurcation diagrams are also built from the period ordering of the orbits of the maps. Then, we study the predictability of the Red Queen chaos, found in narrow regions of mutation rates. To extend the previous analyses, we also computed the likeliness of finding chaos in a given region of the parameter space varying other model parameters simultaneously. Such analyses allowed us to compute a mean predictability measure for the system in the explored region of the parameter space. We found that genetically driven Red Queen chaos, although being restricted to small regions of the analyzed parameter space, might be highly unpredictable.

Chaos and crises in a model for cooperative hunting: A symbolic dynamics approach

In this work we investigate the population dynamics of cooperative hunting extending the McCann and Yodzis model for a three-species food chain system with a predator, a prey, and a resource species. The new model considers that a given fraction sigma of predators cooperates in prey's hunting, while the rest of the population 1-sigma hunts without cooperation. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of the kneading sequences associated with one-dimensional maps that reproduce significant aspects of the dynamics of the species under several degrees of cooperative hunting. Our model also allows us to investigate the so-called deterministic extinction via chaotic crisis and transient chaos in the framework of cooperative hunting. The symbolic sequences allow us to identify a critical boundary in the parameter spaces (K, C-0) and (K, sigma) which separates two scenarios: (i) all-species coexistence and (ii) predator's extinction via chaotic crisis. We show that the crisis value of the carrying capacity K-c decreases at increasing sigma, indicating that predator's populations with high degree of cooperative hunting are more sensitive to the chaotic crises. We also show that the control method of Dhamala and Lai [Phys. Rev. E 59, 1646 (1999)] can sustain the chaotic behavior after the crisis for systems with cooperative hunting. We finally analyze and quantify the inner structure of the target regions obtained with this control method for wider parameter values beyond the crisis, showing a power law dependence of the extinction transients on such critical parameters.

OC/EC ratio observations in Europe: Re-thinking the approach for apportionment between primary and secondary organic carbon

This study explores a large set of OC and EC measurements in PM(10) and PM(2.5) aerosol samples, undertaken with a long term constant analytical methodology, to evaluate the capability of the OC/EC minimum ratio to represent the ratio between the OC and EC aerosol components resulting from fossil fuel combustion (OC(ff)/EC(ff)). The data set covers a wide geographical area in Europe, but with a particular focus upon Portugal, Spain and the United Kingdom, and includes a great variety of sites: urban (background, kerbside and tunnel), industrial, rural and remote. The highest minimum ratios were found in samples from remote and rural sites. Urban background sites have shown spatially and temporally consistent minimum ratios, of around 1.0 for PM(10) and 0.7 for PM(2.5).The consistency of results has suggested that the method could be used as a tool to derive the ratio between OC and EC from fossil fuel combustion and consequently to differentiate OC from primary and secondary sources. To explore this capability, OC and EC measurements were performed in a busy roadway tunnel in central Lisbon. The OC/EC ratio, which reflected the composition of vehicle combustion emissions, was in the range of 03-0.4. Ratios of OC/EC in roadside increment air (roadside minus urban background) in Birmingham, UK also lie within the range 03-0.4. Additional measurements were performed under heavy traffic conditions at two double kerbside sites located in the centre of Lisbon and Madrid. The OC/EC minimum ratios observed at both sites were found to be between those of the tunnel and those of urban background air, suggesting that minimum values commonly obtained for this parameter in open urban atmospheres over-predict the direct emissions of OC(ff) from road transport. Possible reasons for this discrepancy are explored. (C) 2011 Elsevier Ltd. All rights reserved.

Endothelial lipase correlated positively with EPA/AA ratio in RBCs membrane

The fatty acid profile of erythrocyte membranes has been considered a good biomarker for several pathologic situations. Dietary intake, digestion, absorption, metabolism, storage and exchange amongst compartments, greatly influence the fatty acids composition of different cells and tissues. Lipoprotein and hepatic lipases were also involved in fatty acid availability. In the present work we examined the correlations between fatty acid in Red Blood Cells (RBCs) membranes, the fatty acid desaturase and elongase activities, glycaemia, blood lipids, lipoproteins and apoproteins, and the endothelial lipase (EL) mass in plasma. Twenty one individuals were considered in the present study, with age >18 y. RBCs membranes were obtained and analysed for fatty acid composition by gas chromatography. The amount of fatty acids (as percentage) were analysed, and the ratios between fatty acid 16:1/16:0; 18:1/18:0; 18:0/16:0; 22:6 n-3/20:5 n-3 and 20:4 n-6/18:2 n-6 were calculated. Bivariate analysis (rs) and partial correlations were determined. SCD16 estimation activity correlated positively with BMI (rs=0.466, p=0.043) and triacylglycerols (TAG) (rs=0.483, p=0.026), and negatively with the ratio ApoA1/ApoB (rs=-0.566, p=0.007). Endothelial lipase (EL) correlated positively with the EPA/AA ratio in RBCs membranes (rs=0.524, p=0.045). After multi-adjustment for BMI, age, hs-CRP and dietary n3/n6 ratio, the correlations remained significant between EL and EPA/AA ratio. At the best of our knowledge this is the first report that correlated EL with the fatty acid profile of RBCs plasma membranes. The association found here can suggest that the enzyme may be involved in the bioavailability and distribution of n-3/n-6 fatty acids, suggesting a major role for EL in the pathophysiological mechanisms involving biomembranes’ fatty acids, such as in inflammatory response and eicosanoids metabolites pathways.