SIS epidemic attractors in periodic environments

The demographic dynamics are known to drive the disease dynamics in constant environments. In periodic environments, we prove that the demographic dynamics do not always drive the disease dynamics. We exhibit a chaotic attractor in an SIS epidemic model, where the demograhic dynamics are asymptotically cyclic. Periodically forced SIS epidemic models are known to exhibit multiple attractors. We prove that the basins of attraction of these coexisting attractors have infinitely many components.     

Multiple attractors, saddles, and population dynamics in periodic habitats

Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable ’saddle cycle’ separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.     

Multiple attractors in a discrete competition model

The existence of multiple attractors in a competition model implies that the question of coexistence vs. extinction can depend on initial conditions. A discrete stage-structured model of two competing species is derived from a well-tested single-species model of insect populations, and is shown to exhibit multiple attractors for parameter values similar to those used in laboratory experiments which demonstrated chaos in population dynamics. The corresponding basins of attraction are investigated and shown to have very complex structures, and the initial stage structure of the populations is shown to have a significant impact on final outcomes.     

Phase switching in population cycles

Oscillatory populations may exhibit a phase change in which, for example, a high–low periodic pattern switches to a low–high pattern. We propose that phase shifts correspond to stochastic jumps between basins of attraction in an appropriate phase space which associates the different phases of a periodic cycle with distinct attractors. This mechanism accounts for two-cycle phase shifts and the occurrence of asynchronous replicates in experimental cultures of Tribolium.     

Precritical State Transition Dynamics in the Attractor Landscape of a Molecular Interaction Network Underlying Colorectal Tumorigenesis

From the perspective of systems science, tumorigenesis can be hypothesized as a critical transition (an abrupt shift from one state to another) between proliferative and apoptotic attractors on the state space of a molecular interaction network, for which an attractor is defined as a stable state to which all initial states ultimately converge, and the region of convergence is called the basin of attraction. Before the critical transition, a cellular state might transit between the basin of attraction for an apoptotic attractor and that for a proliferative attractor due to the noise induced by the inherent stochasticity in molecular interactions. Such a flickering state transition (state transition between the basins of attraction for alternative attractors from the impact of noise) would become more frequent as the cellular state approaches near the boundary of the basin of attraction, which can increase the variation in the estimate of the respective basin size. To investigate this for colorectal tumorigenesis, we have constructed a stochastic Boolean network model of the molecular interaction network that contains an important set of proteins known to be involved in cancer. In particular, we considered 100 representative sequences of 20 gene mutations that drive colorectal tumorigenesis. We investigated the appearance of cancerous cells by examining the basin size of apoptotic, quiescent, and proliferative attractors along with the sequential accumulation of gene mutations during colorectal tumorigenesis. We introduced a measure to detect the flickering state transition as the variation in the estimate of the basin sizes for three-phenotype attractors from the impact of noise. Interestingly, we found that this measure abruptly increases before a cell becomes cancerous during colorectal tumorigenesis in most of the gene mutation sequences under a certain level of stochastic noise. This suggests that a frequent flickering state transition can be a precritical phenomenon of colorectal tumorigenesis.     

Interplay between local dynamics and dispersal in discrete-time metapopulation models

The effects of synchronous dispersal on discrete-time metapopulation dynamics with local (patch) dynamics of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are explored. Single-species metapopulation models behave as single-species single-patch models, whenever all local patches are governed by compensatory dynamics. Dispersal gives rise to multiple attractors with complex basin structures, whenever some local patches are under overcompensatory dynamics. In mixed systems, dispersal is capable of altering the local dynamics from compensatory to overcompensatory dynamics and vice versa. Examples are provided of metapopulation models supporting multiple attractors with intermingled basins of attraction.     

Combined effects of LTP/LTD and synaptic scaling in formation of discrete and line attractors with persistent activity from non-trivial baseline

In this article, we analyze combined effects of LTP/LTD and synaptic scaling and study the creation of persistent activity from a periodic or chaotic baseline attractor. The bifurcations leading to the creation of new attractors have been detailed; this was achieved using a mean field approximation. Attractors encoding persistent activity can notably appear via generalized period-doubling bifurcations, tangent bifurcations of the second iterates or boundary crises, after which the basins of attraction become irregular. Synaptic scaling is shown to maintain the coexistence of a state of persistent activity and the baseline. According to the rate of change of the external inputs, different types of attractors can be formed: line attractors for rapidly changing external inputs and discrete attractors for constant external inputs.     

Boolean dynamics of biological networks with multiple coupled feedback loops

Boolean networks have been frequently used to study the dynamics of biological networks. In particular, there have been various studies showing that the network connectivity and the update rule of logical functions affect the dynamics of Boolean networks. There has been, however, relatively little attention paid to the dynamical role of a feedback loop, which is a circular chain of interactions between Boolean variables. We note that such feedback loops are ubiquitously found in various biological systems as multiple coupled structures and they are often the primary cause of complex dynamics. In this article, we investigate the relationship between the multiple coupled feedback loops and the dynamics of Boolean networks. We show that networks have a larger proportion of basins corresponding to fixed-point attractors as they have more coupled positive feedback loops, and a larger proportion of basins for limit-cycle attractors as they have more coupled negative feedback loops.     

Multiple attractors in stage-structured population models with birth pulses

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. A single species stage-structured model with density-dependent maturation rate and birth pulse is formulated. Using the discrete dynamical system determined by its Poincaré map, we report a detailed study of the various dynamics, including (a) existence and stability of nonnegative equilibria, (b) nonunique dynamics, meaning that several attractors coexist, (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of attractor), (d) supertransients, and (e) chaotic attractors. The occurrence of these complex dynamic behaviour is related to the fact that minor changes in parameter or initial values can strikingly change the dynamic behaviours of system. Further, it is shown that periodic birth pulse, in effect, provides a natural period or cyclicity that allows multiple oscillatory solutions in the continuous dynamical systems.     

Flickering as an early warning signal

Most work on generic early warning signals for critical transitions focuses on indicators of the phenomenon of critical slowing down that precedes a range of catastrophic bifurcation points. However, in highly stochastic environments, systems will tend to shift to alternative basins of attraction already far from such bifurcation points. In fact, strong perturbations (noise) may cause the system to “flicker” between the basins of attraction of the system’s alternative states. As a result, under such noisy conditions, critical slowing down is not relevant, and one would expect its related generic leading indicators to fail, signaling an impending transition. Here, we systematically explore how flickering may be detected and interpreted as a signal of an emerging alternative attractor. We show that—although the two mechanisms differ—flickering may often be reflected in rising variance, lag-1 autocorrelation and skewness in ways that resemble the effects of critical slowing down. In particular, we demonstrate how the probability distribution of a flickering system can be used to map potential alternative attractors and their resilience. Thus, while flickering systems differ in many ways from the classical image of critical transitions, changes in their dynamics may carry valuable information about upcoming major changes.     

A dynamical systems hypothesis of schizophrenia

We propose a top-down approach to the symptoms of schizophrenia based on a statistical dynamical framework. We show that a reduced depth in the basins of attraction of cortical attractor states destabilizes the activity at the network level due to the constant statistical fluctuations caused by the stochastic spiking of neurons. In integrate-and-fire network simulations, a decrease in the NMDA receptor conductances, which reduces the depth of the attractor basins, decreases the stability of short-term memory states and increases distractibility. The cognitive symptoms of schizophrenia such as distractibility, working memory deficits, or poor attention could be caused by this instability of attractor states in prefrontal cortical networks. Lower firing rates are also produced, and in the orbitofrontal and anterior cingulate cortex could account for the negative symptoms, including a reduction of emotions. Decreasing the GABA as well as the NMDA conductances produces not only switches between the attractor states, but also jumps from spontaneous activity into one of the attractors. We relate this to the positive symptoms of schizophrenia, including delusions, paranoia, and hallucinations, which may arise because the basins of attraction are shallow and there is instability in temporal lobe semantic memory networks, leading thoughts to move too freely round the attractor energy landscape.     

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.     

Consequential classes of resources: Subtle global bifurcation with dramatic ecological consequences in a simple population model

Numerous situations exist in which a consumer uses two different kinds of resources, one fixed, the other renewable, e.g., nesting resources and food resources. With an elementary modification of the basic Lotka-Volterra consumer resource equations, we investigate the population dynamics of a consumer dependent on two resources, one fixed, the other renewable. Emerging from this structure is a situation of alternative attractors that remain qualitatively robust over a significant range of parameter values. However, a dramatic change in basins of attraction is induced by very small changes in parameters due to a global bifurcation. Noteworthy is the fact that the qualitative nature of the alternative equilibria remains constant but the dramatic change in the basins does not arise from subtle differences in initial conditions. Rather, there is a major restructuring of the vector field such that a permanent change involving large sets of initial conditions results from very small changes in parameters.     

The effect of time delays on Caribbean coral-algal interactions

The analysis of ecological models often focuses on their asymptotic behavior, but there is increasing recognition that it is important to understand the role of transient behavior. By introducing a time delay into a model of coral-algal interactions in Caribbean coral reefs that exhibits alternative stable states (a favorable coral rich state and a degraded coral-depleted state), we demonstrate the criticality of understanding the basins of attraction for stable equilibria in addition to the systems' asymptotic behavior. Specifically, we show that although the introduction of a time delay into the model does not change the asymptotic stability of the stable equilibria, there are significant changes to their basins of attraction. An understanding of these effects is necessary when determining appropriate reef management options. We then demonstrate that this is a general phenomenon by considering similar behavior underlying the changes in the basins of attraction in a simple Lotka-Volterra model of competition.     

Dynamic complexities in host-parasitoid interaction

In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and not on higher dimensional ecological systems. Here we report a detailed study of the complicated dynamics occurring in a basic discrete-time model of host-parasitoid interaction. The complexities include (a) non-unique dynamics, meaning that several attractors coexist, (b) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties (pattern of self-similarity and fractal basin boundaries), (c) intermittency, (d) supertransients, (e) chaotic attractors, and (f) "transient chaos". Because of these complexities minor changes in parameter or initial values may strikingly change the dynamic behavior of the system. All the phenomena presented in this paper should be kept in mind when examining and interpreting the dynamics of ecological systems. Copyright 1999 Academic Press.     

Evidence for and against the existence of alternate attractors on coral reefs

Synthesis Coral reefs are widely thought to exhibit multiple attractors which have profound implications for people that depend on them. If reefs become ‘stuck’ within a self‐reinforcing state dominated by seaweed, it becomes disproportionately difficult and expensive for managers to shift the system back towards its natural, productive, coral state. The existence of multiple attractors is controversial. We assess various forms of evidence and conclude that there remains no incontrovertible proof of multiple attractors on reefs. However, the most compelling evidence, which combines ecological models and field data, is far more consistent with multiple attractors than the competing hypothesis of only a single, coral attractor. Managers should exercise caution and assume that degraded reefs can become stuck there. Testing for the existence of alternate attractors in ecosystems that possess slow dynamics and frequent pulse perturbation is exceptionally challenging. Coral reefs typify such conditions and the existence of alternate attractors is controversial. We analyse different forms of evidence and assess whether they support or challenge the existence of multiple attractors on Caribbean reefs, many of which have shown profound phase shifts in community structure from coral to algal dominance. Field studies alone provide no insight into multiple attractors because the non‐equilibrial nature of reef dynamics prevents equilibria from being observed. Statistical models risk failing to sample the parameter space in which multiple attractors occur, and have failed to account for the confounding effects of heterogeneous environments, anthropogenic drivers (e.g. fishing), and major disturbances (e.g. hurricanes). Simple and complex models all find multiple attractors over some – though not all – regions of a system driver (fishing). Tests of model predictions with field data closely match theory of alternate attractors but a forward‐leaning monotonic curve with only a single attractor can also be fitted to these data. Deeper consideration of the assumptions of this monotonic relationship reveal significant ecological problems which disappear under a model of multiple attractors. To date, there is no evidence against the existence of multiple attractors on Caribbean reefs and while there remains no definitive proof, the balance of evidence and ecological reasoning favours their existence. Theory predicts that Caribbean reefs do not exhibit alternate attractors in their natural state but that disease‐induced loss of two key functional groups has generated bistability. Whether alternate attractors becomes a persistent element of reef dynamics or a brief moment in their geological history will depend, in part, on the ability of functional groups to recover and the impacts of climate change and ocean acidification on coral growth and mortality.     

Stochastic Sensitivity Analysis of Noise-Induced Extinction in the Ricker Model with Delay and Allee Effect

A susceptibility of population systems to the random noise is studied on the base of the conceptual Ricker-type model taking into account the delay and Allee effect. This two-dimensional discrete model exhibits the persistence in the form of equilibria, discrete cycles, closed invariant curves, and chaotic attractors. It is shown how the Allee effect constrains the persistence zones with borders defined by crisis bifurcations. We study the role of random noise on the contraction and destruction of these zones. This phenomenon of the noise-induced extinction is investigated with the help of direct numerical simulations and semi-analytical approach based on the stochastic sensitivity functions. Stochastic transitions from the persistence regimes to the extinction are studied by the analysis of the mutual arrangement of the basins of attraction and confidence domains.     

Boolean dynamics of genetic regulatory networks inferred from microarray time series data

MOTIVATION: Methods available for the inference of genetic regulatory networks strive to produce a single network, usually by optimizing some quantity to fit the experimental observations. In this article we investigate the possibility that multiple networks can be inferred, all resulting in similar dynamics. This idea is motivated by theoretical work which suggests that biological networks are robust and adaptable to change, and that the overall behavior of a genetic regulatory network might be captured in terms of dynamical basins of attraction. RESULTS: We have developed and implemented a method for inferring genetic regulatory networks for time series microarray data. Our method first clusters and discretizes the gene expression data using k-means and support vector regression. We then enumerate Boolean activation-inhibition networks to match the discretized data. Finally, the dynamics of the Boolean networks are examined. We have tested our method on two immunology microarray datasets: an IL-2-stimulated T cell response dataset and a LPS-stimulated macrophage response dataset. In both cases, we discovered that many networks matched the data, and that most of these networks had similar dynamics. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Multiple mixed-type attractors in a competition model

We show that a discrete-time, two-species competition model with Ricker (exponential) nonlinearities can exhibit multiple mixed-type attractors. By this is meant dynamic scenarios in which there are simultaneously present both coexistence attractors (in which both species are present) and exclusion attractors (in which one species is absent). Recent studies have investigated the inclusion of life-cycle stages in competition models as a casual mechanism for the existence of these kinds of multiple attractors. In this paper we investigate the role of nonlinearities in competition models without life-cycle stages.