Diversity of temporal self-organized behaviors in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay-differential equations revealed a notable richness of temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states. The Feigenbaum, intermitency and quasiperiodicity routes to chaos can emerge in the biochemical oscillator. Moreover, different types of birhythmicity, trirhythmicity and hard excitation emerge in the phase space. For a single range of the control parameter it can be observed the coexistence of two quasiperiodicity routes to chaos, the coexistence of a stable steady state with a stable torus, and the coexistence of a strange attractor with different stable regimes such as chaos with different periodic regimes, chaos with bursting behavior, and chaos with torus. In most of the numerical studies, the biochemical oscillator has been considered under periodic input flux being the mean input flux rate 6 mM/h. On the other hand, several investigators have observed quasiperiodic time patterns and chaotic oscillations by monitoring the fluorescence of NADH in glycolyzing yeast under sinusoidal glucose input flux. Our numerical results match well with these experimental studies.     

Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

Temporal variability in a system of coupled mitotic timers

 Cell proliferation is considered a periodic process governed by a relaxation timer. The collective behavior of a system composed of three identical relaxation oscillators in numerically studied under the condition that diffusion of the slow mode dominates. We demonstrate: (1) the existence of three periodic regimes with different periods and phase relations and an unsymmetrical, stable steady-state (USSS); (2) the coexistence of in-phase oscillations and USSS; (3) the coexistence of periodic attractors; and (4) the emergence of a two-loop limit cycle coexisting with both in-phase oscillations and a stable steady-state. The qualitative reasons for such a diversitiy and its possible role in the generation of cell cycle variability are discussed. Received: 18 March 1992/Accepted in revised form: 16 April 1994     

Coexistence of multiple propagating wave-fronts in a regulated enzyme reaction model: link with birhythmicity and multi-threshold excitability

We analyze the spatial propagation of wave-fronts in a biochemical model for a product-activated enzyme reaction with non-linear recycling of product into substrate. This model was previously studied as a prototype for the coexistence of two distinct types of periodic oscillations (birhythmicity). The system is initially in a stable steady state characterized by the property of multi-threshold excitability, by which it is capable of amplifying in a pulsatory manner perturbations exceeding two distinct thresholds. In such conditions, when the effect of diffusion is taken into account, two distinct wave-fronts are shown to propagate in space, with distinct amplitudes and velocities, for the same set of parameter values, depending on the magnitude of the initial perturbation. Such a multiplicity of propagating wave-fronts represents a new type of coexistence of multiple modes of dynamic behavior, besides the coexistence involving, under spatially homogeneous conditions, multiple steady states, multiple periodic regimes, or a combination of steady and periodic regimes.     

State selection in coupled identical biochemical systems with coexisting stable states

This work examines state selection for coupled biochemical systems with coexisting stable states. For biochemically identical biochemical systems, different coupled systems are examined for the coexistence of (a) one steady state and one oscillatory state or (b) two oscillatory states. For case (a), it is revealed that state selection is always governed by two key factors: the values of kinetic parameters and the coupling strength. When the coupling strength is small, the coupled systems remain in the basin of attraction of their original states. When it is sufficiently large, all coupled systems are always entrained, independently of their original states. Furthermore, for the entrainment, which of the two coexisting states is selected depends sensitively on the activity of recycling enzyme (one of kinetic parameters). It is shown that this is because changing the activity of recycling enzyme alters the size of basin of attraction of each state. When both systems in the same oscillatory state are coupled, an additional factor, namely phase shift between two oscillations, may also affect state selection, and coupling may cause the systems to select either the original oscillatory state or the coexisting steady state. In addition to the features of case (a), case (b) also supports quasiperiodic oscillations and synchronisation of two periodic oscillations. Implications of the results for understanding state selection during the evolution of coupled biochemical systems with coexisting stable states are discussed.     

The dynamics of production and destruction: Analytic insight into complex behavior

This paper analytically explores the properties of simple differential-difference equations that represent dynamic processes with feedback dependent on prior states of the system. Systems with pure negative and positive feedback are examined, as well as those with mixed (positive/negative) feedback characteristics. Very complex time dependent behaviors may arise from these processes. Indeed, the same mechanism may, depending on system parameters and initial conditions, produce simple, regular, repetitive patterns and completely irregular random-like fluctuations.For the differential-delay equations considered here we prove the existence of: (i) stable and unstable limit cycles, where the stable cycles may have an arbitrary number of extrema per period; and (ii) chaos, meaning the presence of infinitely many periodic solutions of different period and of infinitely many irregular and mixing solutions.     

Complex Dynamics in an Eco-epidemiological Model

The presence of infectious diseases can dramatically change the dynamics of ecological systems. By studying an SI-type disease in the predator population of a Rosenzweig–MacArthur model, we find a wealth of complex dynamics that do not exist in the absence of the disease. Numerical solutions indicate the existence of saddle–node and subcritical Hopf bifurcations, turning points and branching in periodic solutions, and a period-doubling cascade into chaos. This means that there are regions of bistability, in which the disease can have both a stabilising and destabilising effect. We also find tristability, which involves an endemic torus (or limit cycle), an endemic equilibrium and a disease-free limit cycle. The endemic torus seems to disappear via a homoclinic orbit. Notably, some of these dynamics occur when the basic reproduction number is less than one, and endemic situations would not be expected at all. The multistable regimes render the eco-epidemic system very sensitive to perturbations and facilitate a number of regime shifts, some of which we find to be irreversible.     

Quasiperiodicity route to chaos in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay differential equations reveals a quasiperiodicity route to chaos. When the delay changes in our biochemical system, we can observe the emergence of a strange attractor that replaces a previous torus. This behavior happens both under a constant input flux and when the frequency of the periodic substrate input flux changes. The results obtained under periodic input flux are in agreement with experimental observations.     

Phase resetting and bifurcation in the ventricular myocardium.

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.     

Analysis of stability of different regimes of heart rate dynamics by computer modeling

Computer modeling revealed the following three regimes of heart rate dynamics: linear dynamics, “1st degree chaos,” and “2nd degree chaos.” This study investigated a stability of these regimes with respect to changes in initial conditions. The results show that the greatest stability is notable for the linear regime. For this regime small errors in values of initial conditions can not sharply change the initial dynamics of RR intervals. Both nonlinear regimes of heart rate dynamics are unstable, and a degree of instability of regime “2nd degree chaos” is higher in comparison with regime “1st degree chaos.” The results of computer modeling are in agreement with experimental data pointing to the existence of a relationship between the degree of heart rate irregularity and cardiac electrical stability.     

Time-dependent regimes of a tourism-based social–ecological system: Period-doubling route to chaos

The period-doubling route to chaos has occupied a prominent position and it is still object of great interest among the different complex phenomena observed in nonlinear dynamical systems. The reason of such interest is that such route to chaos has been observed in many physical, chemical and ecological models when they change over from simple periodic to complex aperiodic motion. In interlinked social–ecological systems (SESs) there might be an apparent great ability to cope with change and adapt if analysed only in their social dimension. However, such an adaptation may be at the expense of changes in the capacity of ecosystems to sustain the adaptation and it could affect the quality of ecosystem goods and services since it could degrade natural renewable and non-renewable resources and generate traps and breakpoints in the whole SES eventually leading to chaotic behaviour. This paper is rooted in previous results on modelling tourism-based SESs, only recently object of theoretical investigations, focusing on the dynamics of the coexistence between mass-tourists and eco-tourists. Here we describe a finer scale analysis of time-dependent regimes in the ranges of the degradation coefficient (bifurcation parameter), for which the system can exhibit coexistence. This bifurcation parameter is determined by objective changes in the real world in the quality of ecosystem goods and services together with whether and how such changes are perceived by different tourist typologies. Varying the bifurcation parameter, the dynamical system may in fact evolve toward an aperiodical dynamical state in many ways, showing that there could be different scenarios for the transition to chaos. This paper provides a further evidence for the period-doubling route to chaos with reference to tourism-based socio-ecological models, and for a period locking behaviour, where a small variation in the bifurcation parameter can lead to alternating regular and chaotic dynamics. Moreover, for many models undergoing chaos via period-doubling, it has been showed that structural perturbations with real ecological justification, may break and reverse the expected period-doublings, hence inhibiting chaos. This feature may be of a certain relevance also in the context of adaptive management of tourism-based SESs: these period-doubling reversals might in fact be used to control chaos, since they potentially act in way to suppress possibly dangerous fluctuations.     

Excitability with multiple thresholds. A new mode of dynamic behavior analyzed in a regulated biochemical system

We analyze in a biochemical model the phenomenon of excitability in which suprathreshold perturbations of a stable steady state are amplified in a pulsatory manner. The two-variable model is that of an autocatalytic enzyme reaction with recycling of product into the substrate. This model was previously studied for the coexistence between two stable periodic regimes (birhythmicity). We show that the multiplicity of dynamic behavioral modes extends to the phenomenon of excitability. Whereas excitable behavior is generally characterized by a single threshold for excitation, two distinct thresholds may coexist in this model. Moreover, in these conditions, two different plateaux are obtained for the response amplitude when the stimulus is gradually increased. By means of phase plane analysis we explain the origin of multiple thresholds for excitability and predict the conditions for their occurrence. Implications of the phenomenon for excitable cells, in particular for neurons, are discussed.     

Competition in a turbidostat for an inhibitory nutrient

A model of competition in a turbidostat between two species for an inhibitory growth-limiting nutrient is considered. It is shown that the model has rich dynamics. A coexistence equilibrium and the washout equilibrium can be asymptotically stable simultaneously so that coexistence may depend on initial conditions. Under certain conditions, periodic coexistence of the two species occurs. There is a possibility that two species coexist, whereas one species dies out in the absence of its rival.     

Selection between multiple periodic regimes in a biochemical system: complex dynamic behaviour resolved by use of one-dimensional maps

We analyse a model biochemical system in which two autocatalytic enzyme reactions are coupled in series, in conditions where multiple stable periodic regimes coexist for the same set of parameter values. We determine how the periodic regimes are reached from different initial conditions. The structure of the attraction basins is generally simple in the case of two coexisting limit cycles (birhythmicity). This structure and the associated behaviour may, however, become highly complex. In particular, the system exhibits enhanced sensitivity to initial conditions when the boundaries of the attraction basins are fractal. In the latter case, it becomes difficult to predict the evolution towards either one of two limit cycles, a phenomenon known as final state sensitivity. We show how these complex phenomena can be explained in a unified and simple manner by means of one-dimensional return maps derived from the time evolution of the model and from fifth degree polynomial equations. We suggest experimental tests of the sensitivity to initial conditions in chemical systems presenting birhythmicity. The physiological significance of the results is discussed with respect to the sensitivity of regulatory systems admitting multiple stable biological rhythms.     

Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations

We analyze the behavior of a two-variable biochemical model in conditions where it admits multiple oscillatory domains in parameter space. The model represents an autocatalytic enzyme reaction with input of substrate both from a constant source and from non-linear recycling of product into substrate. This system was previously studied for birhythmicity, i.e. the coexistence between two stable periodic regimes (Moran and Goldbeter 1984), and for multithreshold excitability (Moran and Goldbeter 1985). When two distinct oscillatory domains obtain as a function of the substrate injection rate, the system is capable of exhibiting two markedly different modes of oscillations for slightly different values of this control parameter. Phase plane analysis shows how the multiplicity of oscillatory domains depends on the parameters that govern the underlying biochemical mechanism of product recycling. We analyze the response of the model to various kinds of transient perturbations and to periodic changes in the substrate input that bring the system through the two ranges of oscillatory behavior. The results provide a qualitative explanation for experimental observations (Jahnsen and Llinas 1984b) related to the occurrence of two different modes of oscillations in thalamic neurones.     

Microbial expansion–collision dynamics promote cooperation and coexistence on surfaces

Microbes colonizing a surface often experience colony growth dynamics characterized by an initial phase of spatial clonal expansion followed by collision between neighboring colonies to form potentially genetically heterogeneous boundaries. For species with life cycles consisting of repeated surface colonization and dispersal, these spatially explicit “expansion‐collision dynamics” generate periodic transitions between two distinct selective regimes, “expansion competition” and “boundary competition,” each one favoring a different growth strategy. We hypothesized that this dynamic could promote stable coexistence of expansion‐ and boundary‐competition specialists by generating time‐varying, negative frequency‐dependent selection that insulates both types from extinction. We tested this experimentally in budding yeast by competing an exoenzyme secreting “cooperator” strain (expansion–competition specialists) against nonsecreting “defectors” (boundary–competition specialists). As predicted, we observed cooperator–defector coexistence or cooperator dominance with expansion–collision dynamics, but only defector dominance otherwise. Also as predicted, the steady‐state frequency of cooperators was determined by colonization density (the average initial cell–cell distance) and cost of cooperation. Lattice‐based spatial simulations give good qualitative agreement with experiments, supporting our hypothesis that expansion–collision dynamics with costly public goods production is sufficient to generate stable cooperator–defector coexistence. This mechanism may be important for maintaining public–goods cooperation and conflict in microbial pioneer species living on surfaces.     

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

Complex dynamics in a model microbial system

The forced double-Monod model (for a chemostat with a predator, a prey and periodically forced inflowing substrate) displays quasiperiodicity, phase locking, period doubling and chaotic dynamics. Stroboscopic sections reveal circle maps for the quasiperiodic regimes and noninvertible maps of the interval for the chaotic regimes. Criticality in the circle maps sets the stage for chaos in the model. This criticality may arise with an increase in the period or amplitude of forcing.     

The dynamics of coupled populations subject to control

The dynamics of coupled populations have mostly been studied in the context of metapopulation viability with application to, for example, species at risk. However, when considering pests and pathogens, eradication, not persistence, is often the end goal. Humans may intervene to control nuisance populations, resulting in reciprocal interactions between the human and natural systems that can lead to unexpected dynamics. The incidence of these human-natural couplings has been increasing, hastening the need to better understand the emergent properties of such systems in order to predict and manage outbreaks of pests and pathogens. For example, the success of the growing aquaculture industry depends on our ability to manage pathogens and maintain a healthy environment for farmed and wild fish. We developed a model for the dynamics of connected populations subject to control, motivated by sea louse parasites that can disperse among salmon farms. The model includes exponential population growth with a forced decline when populations reach a threshold, representing control interventions. Coupling two populations with equal growth rates resulted in phase locking or synchrony in their dynamics. Populations with different growth rates had different periods of oscillation, leading to quasiperiodic dynamics when coupled. Adding small amounts of stochasticity destabilized quasiperiodic cycles to chaos, while stochasticity was damped for periodic or stable dynamics. Our analysis suggests that strict treatment thresholds, although well intended, can complicate parasite dynamics and hinder control efforts. Synchronizing populations via coordinated management among farms leads to more effective control that is required less frequently. Our model is simple and generally applicable to other systems where dispersal affects the management of pests and pathogens.