Distributed delays stabilize ecological feedback systems

Oscillation suppression and chimera states in time-varying networks

Complex network theory has offered a powerful platform for the study of several natural dynamic scenarios, based on the synergy between the interaction topology and the dynamics of its constituents. With research in network theory being developed so fast, it has become extremely necessary to move from simple network topologies to more sophisticated and realistic descriptions of the connectivity patterns. In this context, there is a significant amount of recent works that have emerged with enormous evidence establishing the time-varying nature of the connections among the constituents in a large number of physical, biological, and social systems. The recent review article by Ghosh et al. [Phys. Rep. 949, 1-63 (2022)] demonstrates the significance of the analysis of collective dynamics arising in temporal networks. Specifically, the authors put forward a detailed excerpt of results on the origin and stability of synchronization in time-varying networked systems. However, among the complex collective dynamical behaviors, the study of the phenomenon of oscillation suppression and that of other diverse aspects of synchronization are also considered to be central to our perception of the dynamical processes over networks. Through this review, we discuss the principal findings from the research studies dedicated to the exploration of the two collective states, namely, oscillation suppression and chimera on top of time-varying networks of both static and mobile nodes. We delineate how temporality in interactions can suppress oscillation and induce chimeric patterns in networked dynamical systems, from effective analytical approaches to computational aspects, which is described while addressing these two phenomena. We further sketch promising directions for future research on these emerging collective behaviors in time-varying networks.

Complexity Collapse, Fluctuating Synchrony, and Transient Chaos in Neural Networks With Delay Clusters

Neural circuits operate with delays over a range of time scales, from a few milliseconds in recurrent local circuitry to tens of milliseconds or more for communication between populations. Modeling usually incorporates single fixed delays, meant to represent the mean conduction delay between neurons making up the circuit. We explore conditions under which the inclusion of more delays in a high-dimensional chaotic neural network leads to a reduction in dynamical complexity, a phenomenon recently described as multi-delay complexity collapse (CC) in delay-differential equations with one to three variables. We consider a recurrent local network of 80% excitatory and 20% inhibitory rate model neurons with 10% connection probability. An increase in the width of the distribution of local delays, even to unrealistically large values, does not cause CC, nor does adding more local delays. Interestingly, multiple small local delays can cause CC provided there is a moderate global delayed inhibitory feedback and random initial conditions. CC then occurs through the settling of transient chaos onto a limit cycle. In this regime, there is a form of noise-induced order in which the mean activity variance decreases as the noise increases and disrupts the synchrony. Another novel form of CC is seen where global delayed feedback causes “dropouts,” i.e., epochs of low firing rate network synchrony. Their alternation with epochs of higher firing rate asynchrony closely follows Poisson statistics. Such dropouts are promoted by larger global feedback strength and delay. Finally, periodic driving of the chaotic regime with global feedback can cause CC; the extinction of chaos can outlast the forcing, sometimes permanently. Our results suggest a wealth of phenomena that remain to be discovered in networks with clusters of delays.

Stability and Hopf bifurcation analysis of lac Operon model with distributed delay and nonlinear degradation rate

We propose a simple model of lac operon that describes the expression of B-galactosidase from lac Z gene in Escherichia coli, through the interaction among several identical mRNA. Our goal is to explore the complex dynamics (i.e. the oscillation phenomenon) of this architecture mediated by this interaction. This model was theoretically and numerically investigated using distributed time delay. We considered the average delay as a bifurcation parameter and the nonlinear degradation rate as a control parameter. Sufficient conditions for local stability were gained by using the Routh-Hurwitz criterion in the case of a weak delay kernel. Then we proved that Hopf bifurcation happened and the direction of the periodic solution was determined using multiple time scale technique. Our results suggest that the interaction among several identical mRNA plays the main role in gene regulation.

Enhancing noise-induced switching times in systems with distributed delays

The paper addresses the problem of calculating the noise-induced switching rates in systems with delay-distributed kernels and Gaussian noise. A general variational formulation for the switching rate is derived for any distribution kernel, and the obtained equations of motion and boundary conditions represent the most probable, or optimal, path, which maximizes the probability of escape. Explicit analytical results for the switching rates for small mean time delays are obtained for the uniform and bi-modal (or two-peak) distributions. They suggest that increasing the width of the distribution leads to an increase in the switching times even for longer values of mean time delays for both examples of the distribution kernel, and the increase is higher in the case of the two-peak distribution. Analytical predictions are compared to the direct numerical simulations and show excellent agreement between theory and numerical experiment.

Delay models for the early embryonic cell cycle oscillator

Time delays are known to play a crucial role in generating biological oscillations. The early embryonic cell cycle in the frog Xenopus laevis is one such example. Although various mathematical models of this oscillating system exist, it is not clear how to best model the required time delay. Here, we study a simple cell cycle model that produces oscillations due to the presence of an ultrasensitive, time-delayed negative feedback loop. We implement the time delay in three qualitatively different ways, using a fixed time delay, a distribution of time delays, and a delay that is state-dependent. We analyze the dynamics in all cases, and we use experimental observations to interpret our results and put constraints on unknown parameters. In doing so, we find that different implementations of the time delay can have a large impact on the resulting oscillations.

Synchronization of networks of oscillators with distributed delay coupling

This paper studies the stability of synchronized states in networks, where couplings between nodes are characterized by some distributed time delay, and develops a generalized master stability function approach. Using a generic example of Stuart-Landau oscillators, it is shown how the stability of synchronized solutions in networks with distributed delay coupling can be determined through a semi-analytic computation of Floquet exponents. The analysis of stability of fully synchronized and of cluster or splay states is illustrated for several practically important choices of delay distributions and network topologies.

Synchronization-desynchronization transitions in complex networks: an interplay of distributed time delay and inhibitory nodes

We investigate the combined effects of distributed delay and the balance between excitatory and inhibitory nodes on the stability of synchronous oscillations in a network of coupled Stuart-Landau oscillators. To this end a symmetric network model is proposed for which the stability can be investigated analytically. It is found that beyond a critical inhibition ratio, synchronization tends to be unstable. However, increasing distributional widths can counteract this trend, leading to multiple resynchronization transitions at relatively high inhibition ratios. The extended applicability of the results is confirmed by numerical studies on asymmetrically perturbed network topologies. All investigations are performed on two distribution types, a uniform distribution and a Γ distribution.

Analytical criterion for amplitude death in nonautonomous systems with piecewise nonlinear coupling

We investigate the amplitude death phenomenon in a nonautonomous chained network with complicated piecewise nonlinear coupling functions. An analytical criterion for the boundary of the amplitude death region is derived by using the average method. The mechanism of the amplitude death in the nonautonomous networks is very different from that of autonomous systems and rapid dynamic transitions could halt the amplitude death. Numerical verifications are carried out to check jump transitions among different solution branches and further confirm the correctness of the theoretical results.

Amplitude death in oscillator networks with variable-delay coupling

We study the conditions of amplitude death in a network of delay-coupled limit cycle oscillators by including time-varying delay in the coupling and self-feedback. By generalizing the master stability function formalism to include variable-delay connections with high-frequency delay modulations (i.e., the distributed-delay limit), we analyze the regimes of amplitude death in a ring network of Stuart-Landau oscillators and demonstrate the superiority of the proposed method with respect to the constant delay case. The possibility of stabilizing the steady state is restricted by the odd-number property of the local node dynamics independently of the network topology and the coupling parameters.

Amplitude and phase dynamics in oscillators with distributed-delay coupling

This paper studies the effects of distributed-delay coupling on the dynamics in a system of non-identical coupled Stuart-Landau oscillators. For uniform and gamma delay distribution kernels, the conditions for amplitude death are obtained in terms of average frequency, frequency detuning and the parameters of the coupling, including coupling strength and phase, as well as the mean time delay and the width of the delay distribution. To gain further insights into the dynamics inside amplitude death regions, the eigenvalues of the corresponding characteristic equations are computed numerically. Oscillatory dynamics of the system is also investigated, using amplitude and phase representation. Various branches of phase-locked solutions are identified, and their stability is analysed for different types of delay distributions.

Developmental variability and stability in continuous-time host-parasitoid models

Insect host-parasitoid systems are often modeled using delay-differential equations, with a fixed development time for the juvenile host and parasitoid stages. We explore here the effects of distributed development on the stability of these systems, for a random parasitism model incorporating an invulnerable host stage, and a negative binomial model that displays generation cycles. A shifted gamma distribution was used to model the distribution of development time for both host and parasitoid stages, using the range of parameter values suggested by a literature survey. For the random parasitism model, the addition of biologically plausible levels of developmental variability could potentially double the area of stable parameter space beyond that generated by the invulnerable host stage. Only variability in host development time was stabilizing in this model. For the negative binomial model, development variability reduced the likelihood of generation cycles, and variability in host and parasitoid was equally stabilizing. One source of stability in these models may be aggregation of risk, because hosts with varying development times have different vulnerabilities. High levels of variability in development time occur in many insects and so could be a common source of stability in host-parasitoid systems.

Stochastic development in biologically structured population models

Variation in organismal development is ubiquitous in nature but omitted from most age- and stage-structured population models. I give a general approach for formulating and analyzing its role in density-independent population models using the framework of integral projection models. The approach allows flexible assumptions, including correlated development times among multiple life stages. I give a new Monte Carlo numerical integration approach to calculate long-term growth rate, its sensitivities, stable age-stage distributions and reproductive value. This method requires only simulations of individual life schedules, rather than iteration of full population dynamics, and has practical and theoretical appeal because it ties easily implemented simulations to numerical solution of demographic equations. I show that stochastic development is demographically important using two examples. For a desert cactus, many stochastic development models, with independent or correlated stage durations, can generate the same stable stage distribution (SSD) as the real data, but stable age-within-stage distributions and sensitivities of growth rate to demographic rates differ greatly among stochastic development scenarios. For Mediterranean fruit flies, empirical variation in maturation time has a large impact on population growth. The systematic model formulation and analysis approach given here should make consideration of variable development models widely accessible and readily extendible.

Distributed delays stabilize neural feedback systems

We consider the effect of distributed delays in neural feedback systems. The avian optic tectum is reciprocally connected with the isthmic nuclei. Extracellular stimulation combined with intracellular recordings reveal a range of signal delays from 3 to 9 ms between isthmotectal elements. This observation together with prior mathematical analysis concerning the influence of a delay distribution on system dynamics raises the question whether a broad delay distribution can impact the dynamics of neural feedback loops. For a system of reciprocally connected model neurons, we found that distributed delays enhance system stability in the following sense. With increased distribution of delays, the system converges faster to a fixed point and converges slower toward a limit cycle. Further, the introduction of distributed delays leads to an increased range of the average delay value for which the system's equilibrium point is stable. The system dynamics are determined almost exclusively by the mean and the variance of the delay distribution and show only little dependence on the particular shape of the distribution.

Synchronized patterns induced by distributed time delays

Considering the fact that the distances between coupled oscillators may delay the receiving of signals, we study here the influence of uniformly distributed delays in an array of coupled pendulums instead of studying the influence of the coupling strength. We find that with an increase of the range of distributed delays, the chaotic behaviors of the coupled arrays may be controlled and different synchronized patterns can be induced. An analytic solution is given to confirm the numerical results. This finding may provide further insight into information processing in neurons.

Can distributed delays perfectly stabilize dynamical networks?

Signal transmission delays tend to destabilize dynamical networks leading to oscillation, but their dispersion contributes oppositely toward stabilization. We analyze an integrodifferential equation that describes the collective dynamics of a neural network with distributed signal delays. With the Gamma distributed delays less dispersed than exponential distribution, the system exhibits reentrant phenomena, in which the stability is once lost but then recovered as the mean delay is increased. With delays dispersed more highly than exponential, the system never destabilizes.

Intracellular delay limits cyclic changes in gene expression

Based on previously published experimental observations and mathematical models for Hes1, p53 and NF-kappaB gene expression, we improve these models through a distributed delay formulation of the time lag between transcription factor binding and mRNA production. This description of natural variability for delays introduces a transition from a stable steady state to limit cycle oscillations and then a second transition back to a stable steady state which has not been observed in previously published models. We demonstrate our approach for two models. The first model describes Hes1 autorepression with equations for Hes1 mRNA production and Hes1 protein translation. The second model describes Hes1 repression by the protein complex Gro/TLE1/Hes1, where Gro/TLE1 is activated by Hes1 phosphorylation. Finally, we discuss our analytical and numerical results in relation to experimental data.

Periodic and aperiodic traveling pulses in population dynamics: an example from the occurrence of epidemic infections

The dynamics of the occurrence of the dengue hemorrhagic fever in the 72 provinces of Thailand is investigated by performing a proper orthogonal decomposition (POD) on spatiotemporal data. Using this technique, we are able to identify and select the contribution of different modes, selected according to the energy content, to the evolution of the epidemic during 14 years. We found that the phenomenon is characterized by periodic cycles of yearly occurrence characterized by spatial scales of about 420 km. Superimposed on this basic mode, POD analysis is able to reveal the presence of high-energetic aperiodic traveling pulses of the epidemic, which extend spatially for about 510 km from Bangkok.

The impact of mortality on predator population size and stability in systems with stage-structured prey

The relationships between a predator population's mortality rate and its population size and stability are investigated for several simple predator-prey models with stage-structured prey populations. Several alternative models are considered; these differ in their assumptions about the nature of density dependence in the prey's population growth; the nature of stage-transitions; and the stage-selectivity of the predator. Instability occurs at high, rather than low predator mortality rates in most models with highly stage-selective predation; this is the opposite of the effect of mortality on stability in models with homogeneous prey populations. Stage-selective predation also increases the range of parameters that lead to a stable equilibrium. The results suggest that it may be common for a stable predator population to increase in abundance as its own mortality rate increases in stable systems, provided that the predator has a saturating functional response. Sufficiently strong density dependence in the prey generally reverses this outcome, and results in a decrease in predator population size with increasing predator mortality rate. Stability is decreased when the juvenile stage has a fixed duration, but population increases with increasing mortality are still observed in large areas of stable parameter space. This raises two coupled questions which are as yet unanswered; (1) do such increases in population size with higher mortality actually occur in nature; and (2) if not, what prevents them from occurring? Stage-structured prey and stage-related predation can also reverse the 'paradox of enrichment', leading to stability rather than instability when prey growth is increased.

Steady-state stabilization due to random delays in maps with self-feedback loops and in globally delayed-coupled maps

We study the stability of the fixed-point solution of an array of mutually coupled logistic maps, focusing on the influence of the delay times, , of the interaction between the and maps. Two of us recently reported [Phys. Rev. Lett. 94, 134102 (2005)] that if are random enough, the array synchronizes in a spatially homogeneous steady state. Here we study this behavior by comparing the dynamics of a map of an array of delayed-coupled maps with the dynamics of a map with self-feedback delayed loops. If is sufficiently large, the dynamics of a map of the array is similar to the dynamics of a map with self-feedback loops with the same delay times. Several delayed loops stabilize the fixed point, when the delays are not the same; however, the distribution of delays plays a key role; if the delays are all odd a periodic orbit (and not the fixed point) is stabilized. We present a linear stability analysis and apply some mathematical theorems that explain the numerical results.