Qualitative Changes In Dynamics Research Articles

Thermal dependence of Daphnia life history reveals asymmetries between key vital rates

Temperature variation affects virtually every aspect of ectotherms' ecological performance, such as their foraging rate, reproduction, and survival. Although these changes influence what happens at higher levels of organizations, such as populations and communities, qualitative changes in dynamics usually require some degree of asymmetry between key vital rates, i.e. that different vital rates, such as growth, development, fecundity and mortality rates, respond differently to temperature. In order to identify possible asymmetries among vital rates and/or life stages, we characterized the thermal response of individuals a clone of Daphnia sinensis, drawn from a high-mountain environment in Taiwan, and examined the temperature dependence of growth, maturation, reproduction, and mortality rates, as well as fitness measures (r and R0) at eight temperatures. Daphnia sinensis was able to maintain reproductive success over a broad range of temperatures, much wider than the one experienced in its environment. However, negative effects of temperature were perceptible at temperatures much lower than the highest one at which they can achieve reproductive success. Adult mortality greatly increased for temperatures above 23 °C, and other vital rates started to decelerate, resulting in a large drop in lifetime reproductive success. This finding implies that D. sinensis may be able to persist over a wide range of temperatures, but also that it may become more sensitive to the detrimental effect of species interactions at increased temperatures. Different vital rates responded relatively similarly at low temperatures, but the degree of asymmetry among these rates was much more pronounced at higher temperatures. In particular, rates associated with adult performance decelerated more strongly than juveniles’ rates. These findings indicate that elevated temperatures affect the balance between juvenile and adult performance, which is known to have a crucial role in Daphnia population dynamics. We discuss the implications of these results for the dynamics of structured populations.

Nonlinear Hamiltonian analysis of new quadratic torsion theories: Cases with curvature-free constraints

It was recently found that, when linearised in the absence of matter, 58 cases of the general gravitational theory with quadratic curvature and torsion are (i) free from ghosts and tachyons and (ii) power-counting renormalisable. We inspect the nonlinear Hamiltonian structure of the eight cases whose primary constraints do not depend on the curvature tensor. We confirm the particle spectra and unitarity of all these theories in the linear regime. We uncover qualitative dynamical changes in the nonlinear regimes of all eight cases, suggesting at least a broken gauge symmetry, and possibly the activation of negative kinetic energy spin-parity sectors and acausal behaviour. Two of the cases propagate a pair of massless modes at the linear level, and were interesting as candidate theories of gravity. However, we identify these modes with vector excitations, rather than the tensor polarisations of the graviton. Moreover, we show that these theories do not support a viable cosmological background.

Probing the phase space of coupled oscillators with Koopman analysis.

With the development of probing and computing technology, the study of complex systems has become a necessity in various science and engineering problems, which may be treated efficiently with Koopman operator theory based on observed time series. In the current paper, combined with a singular value decomposition (SVD) of the constructed Hankel matrix, Koopman analysis is applied to a system of coupled oscillators. The spectral properties of the operator and the Koopman modes of a typical orbit reveal interesting invariant structures with periodic, quasiperiodic, or chaotic motion. By checking the amplitude of the principal modes along a straight line in the phase space, cusps of different sizes on the magnitude profiles are identified whenever a qualitative change of dynamics takes place. There seems to be no obstacle to extend the current analysis to high-dimensional nonlinear systems with intricate orbit structures.

On-line detection of qualitative dynamical changes in nonlinear systems: The resting-oscillation case

Motivated by neuroscience applications, we introduce the concept of qualitative detection, that is, the problem of determining on-line the current qualitative dynamical behavior (e.g., resting, oscillating, bursting, spiking etc.) of a nonlinear system. The approach is thought for systems characterized by i) large parameter variability and redundancy, ii) a small number of possible robust, qualitatively different dynamical behaviors and, iii) the presence of sharply different characteristic timescales. These properties are omnipresent in neurosciences and hamper quantitative modeling and fitting of experimental data. As a result, novel control theoretical strategies are needed to face neuroscience challenges like on-line epileptic seizure detection. The proposed approach aims at detecting the current dynamical behavior of the system and whether a qualitative change is likely to occur without quantitatively fitting any model nor asymptotically estimating any parameter. We talk of qualitative detection. We rely on the qualitative properties of the system dynamics, extracted via singularity and singular perturbation theories, to design low dimensional qualitative detectors. We introduce this concept on a general class of singularly perturbed systems and then solve the problem for an analytically tractable class of two-dimensional systems with a single unknown sigmoidal nonlinearity and two sharply separated timescales. Numerical results are provided to show the performance of the designed qualitative detector.

Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation

In a standard bifurcation of a dynamical system, the stationary points (or more generally attractors) change qualitatively when varying a control parameter. Here we describe a novel unusual effect, when the change of a parameter, e.g. a growth rate, does not influence the stationary states, but nevertheless leads to a qualitative change of dynamics. For instance, such a dynamic transition can be between the convergence to a stationary state and a strong increase without stationary states, or between the convergence to one stationary state and that to a different state. This effect is illustrated for a dynamical system describing two symbiotic populations, one of which exhibits a growth rate larger than the other one. We show that, although the stationary states of the dynamical system do not depend on the growth rates, the latter influence the boundary of the basins of attraction. This change of the basins of attraction explains this unusual effect of the qualitative change of dynamics by growth rate variation.

Predicting the onset of period-doubling bifurcations in noisy cardiac systems

Biological, physical, and social systems often display qualitative changes in dynamics. Developing early warning signals to predict the onset of these transitions is an important goal. The current work is motivated by transitions of cardiac rhythms, where the appearance of alternating features in the timing of cardiac events is often a precursor to the initiation of serious cardiac arrhythmias. We treat embryonic chick cardiac cells with a potassium channel blocker, which leads to the initiation of alternating rhythms. We associate this transition with a mathematical instability, called a period-doubling bifurcation, in a model of the cardiac cells. Period-doubling bifurcations have been linked to the onset of abnormal alternating cardiac rhythms, which have been implicated in cardiac arrhythmias such as T-wave alternans and various tachycardias. Theory predicts that in the neighborhood of the transition, the system's dynamics slow down, leading to noise amplification and the manifestation of oscillations in the autocorrelation function. Examining the aggregates' interbeat intervals, we observe the oscillations in the autocorrelation function and noise amplification preceding the bifurcation. We analyze plots--termed return maps--that relate the current interbeat interval with the following interbeat interval. Based on these plots, we develop a quantitative measure using the slope of the return map to assess how close the system is to the bifurcation. Furthermore, the slope of the return map and the lag-1 autocorrelation coefficient are equal. Our results suggest that the slope and the lag-1 autocorrelation coefficient represent quantitative measures to predict the onset of abnormal alternating cardiac rhythms.

CONTROLLING MULTISTABILITY BY SMALL PERIODIC PERTURBATION

A small perturbation of any system parameters may not in general create any significant qualitative change in dynamics of a multistable system. However, a slow-periodic modulation with properly adjusted amplitude and frequency can do so. In particular, it can control the number of coexisting attractors. The basic idea in this controlling mechanism is to introduce a collision between an attractor with its basin boundary. As a consequence, the attractor is destroyed via boundary crisis, and the chaotic transients settle down to an adjacent attractor. These features have been observed first theoretically with the Hénon map and laser rate equations, and then confirmed experimentally with a cavity-loss modulated CO 2 laser and a pump-modulated fiber laser. The number of coexisting attractors increases as the dissipativity of the system reduces. In the low-dissipative limit, the creation of attractors obeys the predictions of Gavrilov, Shilnikov and Newhouse, when the attractors, referred to as Gavrilov–Shilnikov–Newhouse (GSN) sinks, are created in various period n-tupling processes and remain organized in phase and parameter spaces in a self-similar order. We demonstrate that slow small-amplitude periodic modulation of a system parameter can even destroy these GSN sinks and the system is suitably converted again to a controllable monostable system. Such a control is robust against small noise as well. We also show the applicability of the method to control multistability in coupled oscillators and multistability induced by delayed feedback. In the latter case, it is possible to annihilate coexisting states by modulating either the feedback variable or a system parameter or the feedback strength.

Propagation of extrinsic perturbation in a negatively auto-regulated pathway

The apparent precision of the output in multi-step biochemical pathways in the face of external and intrinsic perturbations is non-obvious and conceptually difficult. Using a simple three-step negatively auto-regulated model pathway, we show that the effect of perturbation at different steps of the pathway and its transmission through the network is dependent on the context (i.e., the position) of the particular reaction step in relation to the topology of the regulatory network, stoichiometry of reactions, type of nonlinearity involved in the reactions and also on the intrinsic dynamical state of the pathway variables. We delineate the qualitative and quantitative changes in the pathway dynamics for constant (‘bias’) and random external perturbations acting on the pathway steps locally or globally to all steps. We show that constant perturbation induces qualitative change in dynamics, whereas random fluctuations cause significant quantitative variations in the concentrations of the different variables. Thus, the dynamic response of multi-step biochemical pathways to external perturbation depends on their biochemical, topological and dynamical features.

Topological Bifurcations of Attracting 2-Tori of Quasiperiodically Driven Oscillators

We examine the solutions to a damped, quasiperiodic (QP) Mathieu equation with cubic nonlinearities. The system is suspended in a four-dimensional phase space ℝ2 × T2 in which there exist attracting, knotted 2-tori called torus braids. We develop a topological classification scheme in which a torus braid is characterized by closed braids that exist in two Poincare sections, ℝ2 \times S1 × {·} and ℝ2 × {·} \times S1. Based on the classification scheme, we develop numerical invariants that describe the linkedness of attractors and provide information about the global dynamics. Numerical simulations show that changes of a single parameter lead to a global bifurcation through which the attracting torus loses stability and locally "doubles," shedding a torus of different equivalence class. We call this a topological torus bifurcation of the doubling variety (TTBD). We provide a topological analysis of the doubling produced by TTBDs and we examine the qualitative dynamical changes that result. We also examine the effect of TTBDs on the spectrum of Lyapunov exponents and the time series power spectrum.

STOCHASTIC DYNAMICS OF ELECTRORECEPTORS IN PADDLEFISH

Electroreceptors in paddlefish serve as accessible and well-defined biological models for studying the functional roles in sensory nervous systems of noisy oscillations and the nonlinear phenomena associated with them, including synchronization, noise-induced transitions, and noise-induced bursting. The spontaneous dynamics of paddlefish electroreceptors show two oscillatory modes: one associated with 26 Hz oscillations in the sensory epithelia, and another with 30-65 Hz periodicities of afferent terminals. This novel type of organization of peripheral sensory receptors, with two distinct types of embedded oscillators, results in stochastic biperiodic firing patterns of primary afferents. The biperiodicity can be explained qualitatively in terms of a simple model based on a stochastic circle map. Stimulation with broadband Gaussian noise changes the tonic firing pattern of electroreceptors to a bursting mode, indicating a noise-induced transition. This qualitative change in dynamics leads to burst synchronization among different electroreceptors.

Hybrid matrix models and their population dynamic consequences

In this paper, the main purpose is to reveal what kind of qualitative dynamical changes a continuous age-structured model may undergo as continuous reproduction is replaced with an annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as well as extensive analysis of dynamics in unstable parameter regions. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behavior of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Finally, we discuss the effects of generation delay on stability of positive equilibrium (or positive periodic solution), and show that generation delay is found to act both as a destabilizing and a stabilizing effect.

Consequences of population structure on genes under balancing selection.

This paper describes a new approach to modeling population structure for genes under strong balancing selection of the type seen in plant self-incompatibility systems and the major histocompatibility complex (MHC) system of vertebrates. Simple analytic solutions for the number of alleles maintained at equilibrium and the expected proportion of alleles shared between demes at various levels are derived and checked against simulation results. The theory accurately captures the dynamics of allele number in a subdivided population and identifies important values of m (migration rate) at which allele number and distribution change qualitatively. Starting from a panmictic population, as migration among demes decreases a qualitative change in dynamics is seen at approximately m(crit) approximately equal to the square root of(s/4piNT) where NT is the total population size and s is a measure of the strength of selection. At this point, demes can no longer maintain their panmictic allele number, due to increasing isolation from the total population. Another qualitative change occurs at a migration rate on the same order of magnitude as the mutation rate, mu. At this point, the demes are highly differentiated for allele complement, and the total number of alleles in the population is increased. Because in general u << m<(crit) at intermediate migration rates slightly fewer alleles may be maintained in the total population than are maintained at panmixia. Within this range, total allele number may not be the best indicator of whether a population is effectively panmictic, and some caution should be used when interpreting samples from such populations. The theory presented here can help to analyze data from genes under balancing selection in subdivided populations.

Analytical and numerical results for a long Josephson junction with surface losses

In this paper we show several analytical approaches to study the effect of dissipation on fluxon motion; our attention is mainly devoted to the surface impedance term which is the main quantity responsible for qualitative dynamical changes in the junction.