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.

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.

Topological Complexity and Predictability in the Dynamics of a Tumor Growth Model with Shilnikov's Chaos

Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability.

On chaos, transient chaos and ghosts in single populations models with allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)(infinity) occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects. (C) 2011 Elsevier Ltd. All rights reserved.

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.

Impact of a 1755-like tsunami in Huelva, Spain

Coastal areas are highly exposed to natural hazards associated with the sea. In all cases where there is historical evidence for devastating tsunamis, as is the case of the southern coasts of the Iberian Peninsula, there is a need for quantitative hazard tsunami assessment to support spatial planning. Also, local authorities must be able to act towards the population protection in a preemptive way, to inform 'what to do' and 'where to go' and in an alarm, to make people aware of the incoming danger. With this in mind, we investigated the inundation extent, run-up and water depths, of a 1755-like event on the region of Huelva, located on the Spanish southwestern coast, one of the regions that was affected in the past by several high energy events, as proved by historical documents and sedimentological data. Modelling was made with a slightly modified version of the COMCOT (Cornell Multi-grid Coupled Tsunami Model) code. Sensitivity tests were performed for a single source in order to understand the relevance and influence of the source parameters in the inundation extent and the fundamental impact parameters. We show that a 1755-like event will have a dramatic impact in a large area close to Huelva inundating an area between 82 and 92 km(2) and reaching maximum run-up around 5 m. In this sense our results show that small variations on the characteristics of the tsunami source are not too significant for the impact assessment. We show that the maximum flow depth and the maximum run-up increase with the average slip on the source, while the strike of the fault is not a critical factor as Huelva is significantly far away from the potential sources identified up to now. We also show that the maximum flow depth within the inundated area is very dependent on the tidal level, while maximum run-up is less affected, as a consequence of the complex morphology of the area.

The importance of quality of life in patients with multiple sclerosis: do they need auxiliary walking devices?

Objective: To describe the importance of training multiple sclerosis (MS) patients with auxiliary walking devices (walking stick, crutch, or wheelchair) in a way that achieves a better quality of life (QOL). Design: Exploratory and descriptive. Setting: General hospital in Portugal.

Measuring and Controlling the Chaotic Motion of Profits

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

Toplological Entropy in the Syncronization of Piecewise Linear and Monotone Maps. Coupled Duffing Oscillators

In this paper is presented a relationship between the synchronization and the topological entropy. We obtain the values for the coupling parameter, in terms of the topological entropy, to achieve synchronization of two unidirectional and bidirectional coupled piecewise linear maps. In addition, we prove a result that relates the synchronizability of two m-modal maps with the synchronizability of two conjugated piecewise linear maps. An application to the unidirectional and bidirectional coupled identical chaotic Duffing equations is given. We discuss the complete synchronization of two identical double-well Duffing oscillators, from the point of view of symbolic dynamics. Working with Poincare cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized.

Thick dyke emplacement and internal flow: A structural and magnetic fabric study of the deep-seated dolerite dyke of Foum Zguid (southern Morocco)

Knowledge on forced magma injection and magma flow in dykes is crucial for the understanding of how magmas migrate through the crust to the Earth's surface. Because many questions still persist, we used the long, thick, and deep-seated Foum Zguid dyke (Morocco) to investigate dyke emplacement and internal flow by means of magnetic methods, structural analysis, petrography, and scanning electron microscopy. We also investigated how the host rocks accommodated the intrusion. Regarding internal flow: 1. Important variations of the rock magnetic properties and magnetic fabric occur with distance from dyke wall; 2. anisotropy of anhysteretic remanent magnetization reveals that anisotropy of magnetic susceptibility (AMS) results mainly from the superposition of subfabrics with distinct coercivities and that the imbrication between magnetic foliation and dyke plane is more reliable to deduce flow than the orientation of the AMS maximum principal axis; and 3. a dominant upward flow near the margins can be inferred. The magnetic fabric closest to the dyke wall likely records magma flow best due to fast cooling, whereas in the core the magnetic properties have been affected by high-temperature exsolution and metasomatic effects due to slow cooling. Regarding dyke emplacement, this study shows that the thick forceful intrusion induced deformation by homogeneous flattening and/or folding of the host sedimentary strata. Dewatering related to heat, as recorded by thick quartz veins bordering the dyke in some localities, may have also helped accommodating dyke intrusion. The spatial arrangement of quartz veins and their geometrical relationship with the dyke indicate a preintrusive to synintrusive sinistral component of strike slip.

On the concept of endogenous volatility

Most financial and economic time-series display a strong volatility around their trends. The difficulty in explaining this volatility has led economists to interpret it as exogenous, i.e., as the result of forces that lie outside the scope of the assumed economic relations. Consequently, it becomes hard or impossible to formulate short-run forecasts on asset prices or on values of macroeconomic variables. However, many random looking economic and financial series may, in fact, be subject to deterministic irregular behavior, which can be measured and modelled. We address the notion of endogenous volatility and exemplify the concept with a simple business-cycles model.

Dynamical analysis in growth models: Blumberg’s equation

We present a new dynamical approach to the Blumberg's equation, a family of unimodal maps. These maps are proportional to Beta(p, q) probability densities functions. Using the symmetry of the Beta(p, q) distribution and symbolic dynamics techniques, a new concept of mirror symmetry is defined for this family of maps. The kneading theory is used to analyze the effect of such symmetry in the presented models. The main result proves that two mirror symmetric unimodal maps have the same topological entropy. Different population dynamics regimes are identified, when the intrinsic growth rate is modified: extinctions, stabilities, bifurcations, chaos and Allee effect. To illustrate our results, we present a numerical analysis, where are demonstrated: monotonicity of the topological entropy with the variation of the intrinsic growth rate, existence of isentropic sets in the parameters space and mirror symmetry.

Tsunamigenic earthquakes in the Gulf of Cadiz: fault model and recurrence

The Gulf of Cadiz, as part of the Azores-Gibraltar plate boundary, is recognized as a potential source of big earthquakes and tsunamis that may affect the bordering countries, as occurred on 1 November 1755. Preparing for the future, Portugal is establishing a national tsunami warning system in which the threat caused by any large-magnitude earthquake in the area is estimated from a comprehensive database of scenarios. In this paper we summarize the knowledge about the active tectonics in the Gulf of Cadiz and integrate the available seismological information in order to propose the generation model of destructive tsunamis to be applied in tsunami warnings. The fault model derived is then used to estimate the recurrence of large earthquakes using the fault slip rates obtained by Cunha et al. (2012) from thin-sheet neotectonic modelling. Finally we evaluate the consistency of seismicity rates derived from historical and instrumental catalogues with the convergence rates between Eurasia and Nubia given by plate kinematic models.

An Extension of Gompertzian Growth Dynamics Weibull and Frechet Models

In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta* (p, q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p = 2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta* (2, q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.

Strong and weak Allee effects and chaotic dynamics in Richards' growths

In this paper we define and investigate generalized Richards' growth models with strong and weak Allee effects and no Allee effect. We prove the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, depending on the implicit conditions, which involve the several parameters considered in the models. New classes of functions describing the existence or not of Allee effect are introduced, a new dynamical approach to Richards' populational growth equation is established. These families of generalized Richards' functions are proportional to the right hand side of the generalized Richards' growth models proposed. Subclasses of strong and weak Allee functions and functions with no Allee effect are characterized. The study of their bifurcation structure is presented in detail, this analysis is done based on the configurations of bifurcation curves and symbolic dynamics techniques. Generically, the dynamics of these functions are classified in the following types: extinction, semi-stability, stability, period doubling, chaos, chaotic semistability and essential extinction. We obtain conditions on the parameter plane for the existence of a weak Allee effect region related to the appearance of cusp points. To support our results, we present fold and flip bifurcations curves and numerical simulations of several bifurcation diagrams.