Fast Time Scale Research Articles

Effect of polymer host matrix on multi-stage isomerization kinetics of DASA photochromes

Molecular photoswitches provide a means for imparting synthetic structures with intrinsically logical and highly tunable photoresponsive properties. One variety of organic photoswitches known as Donor-Acceptor Stenhouse Adducts, or DASAs, are promising candidates for next generation light responsive materials because of their unique ability to stabilize three photochemically distinct isomeric states in solution, while their counterparts are strictly limited to binary state behavior. In this work, we show how polymethacrylate host matrices shift the energetic landscape of DASA relative to solution, prohibiting accumulation of an intermediate third isomeric state by decelerating critical steps in the photoswitching mechanism. Specifically, we employ a dual-wavelength, phase locked detection scheme to probe thermal isomerizations in the switching process that occur at fast (∼ms) time scales that are inaccessible by standard UV–Vis spectroscopic techniques. The results of this study provide valuable insight into the mechanism of multistate DASA reactivity and establish the foundation necessary to guide future efforts in offsetting kinetic matrix effects to enable dynamic, three state photoswitching in polymeric hosts.

Functional principles of genetically encoded fluorescent biosensors for metabolism and their quantitative use.

Genetically encoded fluorescent biosensors provide an attractive means of measuring chemical changes in single cells on fast timescales (milliseconds to seconds). While their most prominent application has been in tracking neural activity and neurotransmitter release, there has been growing interest in developing and deploying new versions of these tools to study brain metabolism. However, the careful use of these tools and the interpretation of the data they provide remain challenging. Many biosensors are subject to interferences that can alter sensor responses within a single cell or between cells, producing ambiguous results. This presents a challenge for quantitation and for our ability to accurately interpret sensor responses. This review describes current methods of sensor quantitation, with a focus on cellular interferences that commonly affect sensor performance, ways to avoid false inferences, and recent advances in sensor optimization to make them more robust.

Studying micro to millisecond protein dynamics using simple amide 15N CEST experiments supplemented with major-state R2 and visible peak-position constraints.

Over the last decade amide 15N CEST experiments have emerged as a popular tool to study protein dynamics that involves exchange between a 'visible' major state and sparsely populated 'invisible' minor states. Although initially introduced to study exchange between states that are in slow exchange with each other (typical exchange rates of, 10 to 400s-1), they are now used to study interconversion between states on the intermediate to fast exchange timescale while still using low to moderate (5 to 350Hz) 'saturating' B1 fields. The 15N CEST experiment is very sensitive to exchange as the exchange delay TEX can be quite long (~0.5s) allowing for a large number of exchange events to occur making it a very powerful tool to detect minor sates populated ([Formula: see text]) to as low as 1%. When systems are in fast exchange and the 15N CEST data has to be described using a model that contains exchange, the exchange parameters are often poorly defined because the [Formula: see text] versus [Formula: see text] and [Formula: see text] versus exchange rate ([Formula: see text]) plots can be quite flat with shallow or no minima and the analysis of such 15N CEST data can lead to wrong estimates of the exchange parameters due to the presence of 'spurious' minima. Here we show that the inclusion of experimentally derived constraints on the intrinsic transverse relaxation rates and the inclusion of visible state peak-positions during the analysis of amide 15N CEST data acquired with moderate B1 values (~50 to ~350Hz) results in convincing minima in the [Formula: see text] versus [Formula: see text] and the [Formula: see text] versus [Formula: see text] plots even when exchange occurs on the 100μs timescale. The utility of this strategy is demonstrated on the fast-folding Bacillus stearothermophilus peripheral subunit binding domain that folds with a rate constant ~104s-1. Here the analysis of 15N CEST data alone results in [Formula: see text] versus [Formula: see text] and [Formula: see text] versus [Formula: see text] plots that contain shallow minima, but the inclusion of visible-state peak positions and restraints on the intrinsic transverse relaxation rates of both states during the analysis of the 15N CEST data results in pronounced minima in the [Formula: see text] versus [Formula: see text] and [Formula: see text] versus [Formula: see text] plots and precise exchange parameters even in the fast exchange regime ([Formula: see text]~5). Using this strategy we find that the folding rate constant of PSBD is invariant (~10,500s-1) from 33.2 to 42.9°C while the unfolding rates (~70 to ~500s-1) and unfolded state populations (~0.7 to ~4.3%) increase with temperature. The results presented here show that protein dynamics occurring on the 10 to 104s-1 timescale can be studied using amide 15N CEST experiments.

Selection-Mutation dynamics with spatial dependence

We study the limit of many small mutations for a model of population dynamics. The population is structured by phenological traits and is spatially inhomogeneous. The various sub-populations compete for the same nutrient which diffuses through the spatial environment, with the possibility of creating fast time oscillations in the dynamics. We control those oscillations by the stability of an auxiliary system (without mutations) which guides the dynamics over fast time scales.

Multifaceted luminance gain control beyond photoreceptors in Drosophila

Animals navigating in natural environments must handle vast changes in their sensory input. Visual systems, for example, handle changes in luminance at many timescales, from slow changes across the day to rapid changes during active behavior. To maintain luminance-invariant perception, visual systems must adapt their sensitivity to changing luminance at different timescales. We demonstrate that luminance gain control in photoreceptors alone is insufficient to explain luminance invariance at both fast and slow timescales and reveal the algorithms that adjust gain past photoreceptors in the fly eye. We combined imaging and behavioral experiments with computational modeling to show that downstream of photoreceptors, circuitry taking input from the single luminance-sensitive neuron type L3 implements gain control at fast and slow timescales. This computation is bidirectional in that it prevents the underestimation of contrasts in low luminance and overestimation in high luminance. An algorithmic model disentangles these multifaceted contributions and shows that the bidirectional gain control occurs at both timescales. The model implements a nonlinear interaction of luminance and contrast to achieve gain correction at fast timescales and a dark-sensitive channel to improve the detection of dim stimuli at slow timescales. Together, our work demonstrates how a single neuronal channel performs diverse computations to implement gain control at multiple timescales that are together important for navigation in natural environments.

0060 A Classification Based Generative Approach to Selective Targeting of Global Slow Oscillations during Sleep

Abstract Introduction Interacting with sleep slow oscillations (0.5,15 Hz) is essential to understanding the mechanisms underlying their role in sleep functions, including homeostasis, synaptic reorganization, and memory consolidation. Current approaches to SO-based non-invasive closed loop alternate current stimulation (cl-tACs) focus on timing, and do not differentiate among oscillations based on their space-time profiles. In this study, we leverage machine learning and global optimization to introduce a generative model of brain stimulation protocols targeting Global SOs. Methods We identify a stimulation protocol that matches a Global SO in two stages: first focusing on the electrode montage, and then on the stimulation waveform. Using the sleep night of 22 volunteers acquired in the lab with high density EEG (175 electrodes), we labeled the SOs as Global/non-Global SOs based on their co-detection on the scalp at a small delay (0.4s). We used the current density (CSD) profile of each SO in cortical and subcortical regions to develop a well-trained classifier (based on weighted k-nearest neighbors). We identify as optimal the electrode montage that resulted in CSDs labeled as Global with highest likelihood by the classifier. We then parameterized a sinusoidal waveform and identified optimum parameters with a genetic algorithm that used feature ranking to establish a weighted distance metric to target Global SO profile built from sleep data. We repeat this approach focusing on Global SO dynamics at a slow time scale and a faster time scale (200ms vs 20ms resolution). Results We identified optimal stimulation profiles matching Global SOs at both time scales. At slow time scale, montage selection relied only on CSD near the SO trough and in the reward network and the brainstem. Linear waveforms over-performed sinusoidal waveforms for fit at low time scales. At fast time scale, CSD at both trough and 500ms before the trough were strongly selective for Global vs. non-Global SO. Conclusion Global SOs are a gating mechanism for large-scale neural communication, a necessary substrate for systems consolidation and long-term memory formation. This data-driven method builds on natural sleep data to shape a stimulation protocol that imitates Global SO dynamics and is generalizable to other SO types. Support (if any)

A Pulsed-Precipitation Model of Dryland Vegetation Pattern Formation

We develop a model for investigating the impact of rainstorm variability on the formation of banded vegetation patterns in dryland ecosystems. Water input, during rare rainstorms, is modeled as an instantaneous kick to the soil water. The redistribution, from surface water to soil moisture, accounts for the impact of vegetation on infiltration rate and downslope overland flow speed. These two positive feedbacks between water and biomass distributions act on the fast timescales of rain storms. During dry periods, a classic reaction-diffusion framework is used for the slow processes associated with soil water and biomass. This pulsed precipitation model predicts that the preferred spacing of the vegetation bands is determined by the characteristic distance that a storm pulse of water travels overland before infiltrating into the soil. In this way, the vegetation pattern is determined by the fast ecohydrological processes and may be attuned with its dryland precipitation pattern. We demonstrate how this modeling framework, suited for stochastic rain inputs, can be used to investigate possible collapse of a dryland pattern-forming ecosystem under different precipitation patterns with identical low annual mean. Model simulations suggest, for instance, that shorter rainy seasons and greater variability in storm depth may both hasten ecosystem collapse.

Flexible needle-type Microbiosensor for real-time monitoring traditional acupuncture-mediated adenosine release In vivo

Rapid adenosine (ADO) signaling, on the time frame of seconds, regulates physiological and pathological processes, including the therapeutic efficacy of acupuncture. Nevertheless, standard monitoring strategies are limited by poor temporal resolution. Herein, an implantable needle-type microsensor capable of monitoring ADO release in vivo in response to acupuncture in real time has been developed. Electrocatalytic Prussian Blue nanoparticles, an immobilized multienzyme system, and a permselective poly-o-phenylenediamine-based membrane were used for the sequential modification of the sensing region of the electrode. The resultant sensor can perform amperometric measurements of ADO levels in response to a very low level of applied potential (−0.05 V vs Ag/AgCl). This microsensor also functioned across a broad linear range (0–50 μM) and exhibited good sensitivity (1.1 nA/μM) with a rapid response time of under 5 s. Importantly, the sensor also exhibited good reproducibility and high selectivity. For in vivo animal studies, the microsensor was employed for the continuous assessment of instantaneous ADO release at the ST36 (Zusanli) acupoint when this acupoint was subjected to twirling-rotating acupuncture manipulation. Benefiting from superior sensor in vivo performance and stability, the positive correlation between the variability in acupuncture-induced ADO release and the stimulus intensity levels that affect the clinical benefit can be demonstrated for the first time. Overall, these results highlight a powerful approach to analyzing the in vivo physiological effects of acupuncture, expanding application realm of micro-nano sensor technology on a fast time scale.

The time scales of irreversibility in spontaneous brain activity are altered in obsessive compulsive disorder.

We study how obsessive-compulsive disorder (OCD) affects the complexity and time-reversal symmetry-breaking (irreversibility) of the brain resting-state activity as measured by magnetoencephalography (MEG). Comparing MEG recordings from OCD patients and age/sex matched control subjects, we find that irreversibility is more concentrated at faster time scales and more uniformly distributed across different channels of the same hemisphere in OCD patients than in control subjects. Furthermore, the interhemispheric asymmetry between homologous areas of OCD patients and controls is also markedly different. Some of these differences were reduced by 1-year of Kundalini Yoga meditation treatment. Taken together, these results suggest that OCD alters the dynamic attractor of the brain's resting state and hint at a possible novel neurophysiological characterization of this psychiatric disorder and how this therapy can possibly modulate brain function.

Timescale separation in the coordinated switching of bacterial flagellar motors

The output of the bacterial chemotaxis signaling pathway, the level of the intracellular regulator CheY-P, modulates the rotation direction of the flagellar motor, thereby regulating bacterial run-and-tumble behavior. The multiple flagellar motors on an E. coli cell are controlled by a common cytoplasmic pool of CheY-P. Fluctuation of the CheY-P level was thought to be able to coordinate the switching of multiple motors. Here, we measured the correlation of rotation directions between two motors on a cell, finding that it surprisingly exhibits two well separated timescales. We found that the slow timescale (∼6 s) can be explained by the slow fluctuation of the CheY-P level due to stochastic activity of the chemotactic adaptation enzymes, whereas the fast timescale (∼0.3 s) can be explained by the random pulse-like fluctuation of the CheY-P level, due probably to the activity of the chemoreceptor clusters. We extracted information on the properties of the fast CheY-P pulses based on the correlation measurements. The two well-separated timescales in the fluctuation of CheY-P level help to coordinate multiple motors on a cell and to enhance bacterial chemotactic performance.

Two-timescale coordinated operation of wind-advanced adiabatic compressed air energy storage system: A bilevel stochastic dynamic programming method

Renewable energy is a promising solution to address the energy crisis and environmental issues, but it comes with challenges due to its inherent volatility and limited dispatchability. Advanced adiabatic compressed air energy storage (AA-CAES) is a favorable partner for centralized renewable integration, due to its numerous benefits, such as large capacity, long lifetime, fast response capability, and zero carbon emissions. For the economic management of a wind-AACAES system, a battery-like AA-CAES dispatch model is proposed, where the charging/discharging efficiencies and capacities are dependent on the operating power and state-of-charge (SoC) to capture the part-load characteristics. A hybrid control strategy is proposed to enhance the flexibility and efficiency of the compression side under off-design conditions caused by simultaneous changes in back pressure and load. To provide dispatch and control strategies of AA-CAES, a multi-timescale optimization problem is established. The slow timescale determines the baseline output scheduling on an hourly basis to maximize total revenue, while the fast timescale updates the real-time adjustment every five minutes to minimize the penalty of tracking error. A bi-level stochastic dynamic programming (SDP) framework is formulated to incorporate the wind uncertainties and multi timescale coordination, where terminal SoCs and value functions are essential to integrate decisions across slow and fast timescales. A non-iterative parametric programming-based method is proposed to solve the slow-timescale SDP, and a simulation-based rollout algorithm is applied to extract the fast-timescale actions, which yields evident improvement over any heuristic base policy that is sequentially consistent. The numerical results demonstrate that the proposed method has a performance gap of approximately 7.1%, which outperforms the rule-based method by 32.6%. The installation of AA-CAES improves the total revenue by 11.9%. Additionally, the hybrid control strategy enhances the charging flexibility and efficiency by 30.7% and 4.3%, respectively, resulting in a 3.9% reduction in tracking deviation penalty.

Thermal-Comfort Aware Online Co-Scheduling Framework for HVAC, Battery Systems, and Appliances in Smart Buildings

Energy management in buildings is vital for reducing electricity costs and maximizing the comfort of occupants. Excess solar generation can be used by combining a battery storage system and a heating, ventilation, and air-conditioning (HVAC) system so that occupants feel comfortable. Despite several studies on the scheduling of appliances, batteries, and HVAC, comprehensive and time scalable approaches are required that integrate such predictive information as renewable generation and thermal comfort. In this paper, we propose an thermal-comfort aware online coscheduling framework that incorporates optimal energy scheduling and a prediction model of PV generation and thermal comfort with the model predictive control (MPC) approach. We introduce a photovoltaic (PV) energy nowcasting and thermal-comfort-estimation model that provides useful information for optimization. The energy management problem is formulated as three coordinated optimization problems that cover fast and slow time-scales by considering predicted information. This approach reduces the time complexity without a significant negative impact on the result's global nature and its quality. Experimental results show that our proposed framework achieves optimal energy management that takes into account the trade-off between electricity expenses and thermal comfort. Our sensitivity analysis indicates that introducing a battery significantly improves the trade-off relationship.

Modular reconfiguration of DNA origami assemblies using tile displacement.

The power of natural evolution lies in the adaptability of biological organisms but is constrained by the time scale of genetics and reproduction. Engineeringartificial molecular machines should not only include adaptability as a core feature but also apply it within a larger design space and at a faster time scale. A lesson from engineering electromechanical robots is that modular robots can perform diverse functions through self-reconfiguration, a large-scale form of adaptation. Molecular machines made of modular, reconfigurable components may form the basis for dynamic self-reprogramming in future synthetic cells. To achieve modular reconfiguration in DNA origami assemblies, we previously developed a tile displacement mechanism in which an invader tile replaces another tile in an array with controlled kinetics. Here, we establish design principles for simultaneous reconfigurations in tile assemblies using complex invaders with distinct shapes. We present toehold and branch migration domain configurations that expand the design space of tile displacement reactions by two orders of magnitude. We demonstrate the construction of multitile invaders with fixed and variable sizes and controlled size distributions. We investigate the growth of three-dimensional (3D) barrel structures with variable cross sections and introduce a mechanism for reconfiguring them into 2D structures. Last, we show an example of a sword-shaped assembly transforming into a snake-shaped assembly, illustrating two independent tile displacement reactions occurring concurrently with minimum cross-talk. This work serves as a proof of concept that tile displacement could be a fundamental mechanism for modular reconfiguration robust to temperature and tile concentration.

A methyl-TROSY based 13C relaxation dispersion NMR experiment for studies of chemical exchange in proteins.

A methyl Transverse Relaxation Optimized Spectroscopy (methyl-TROSY) based, multiple quantum (MQ) 13C Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion NMR experiment is described. The experiment is derived from the previously developed MQ 13C-1H CPMG scheme (Korzhnev in J Am Chem Soc 126: 3964-73, 2004) supplemented with a CPMG train of refocusing 1H pulses applied with constant frequency and synchronized with the 13C CPMG pulse train. The optimal 1H 'decoupling' scheme that minimizes the amount of fast-relaxing methyl MQ magnetization present during CPMG intervals, makes use of an XY-4 phase cycling of the refocusing composite 1H pulses. For small-to-medium sized proteins, the MQ 13C CPMG experiment has the advantage over its single quantum (SQ) 13C counterpart of significantly reducing intrinsic, exchange-free relaxation rates of methyl coherences. For high molecular weight proteins, the MQ 13C CPMG experiment eliminates complications in the interpretation of MQ 13C-1H CPMG relaxation dispersion profiles arising from contributions to exchange from differences in methyl 1H chemical shifts between ground and excited states. The MQ 13C CPMG experiment is tested on two protein systems: (1) a triple mutant of the Fyn SH3 domain that interconverts slowly on the chemical shift time scale between the major folded state and an excited state folding intermediate; and (2) the 82-kDa enzyme Malate Synthase G (MSG), where chemical exchange at individual Ile δ1 methyl positions occurs on a much faster time-scale.

Mathematical modeling of pest resistance to insecticides in a heterogeneous environment

Insecticides are an effective tool for controlling grapevine moths such as Lobesia botrana. However, when massive use of insecticides, pests develop resistance because of selection of resistant individuals of the population and/or because of natural mutation. The main question is then to define a strategic pest management with respect to time and space. In the first part of this work, we present a model without spatial structure describing the dynamics of insects including its resistance to insecticides. In the second part, we include in the model dispersal between different space patches. Individuals can move from one patch to another at a fast time scale with respect to the demographical time scale. Using this model, we aim to understand the main characteristics leading to persistence or extinction of the insect pest population. We establish global stability results. Numerical simulations provide some interesting insights on the dynamics of the pest population.

Synchronization of inspiratory burst onset along the ventral respiratory column in the neonate mouse is mediated by electrotonic coupling

Breathing is a singularly robust behavior, yet this motor pattern is continuously modulated at slow and fast timescales to maintain blood-gas homeostasis, while intercalating orofacial behaviors. This functional multiplexing goes beyond the rhythmogenic function that is typically ascribed to medullary respiration-modulated networks and may explain lack of progress in identifying the mechanism and constituents of the respiratory rhythm generator. By recording optically along the ventral respiratory column in medulla, we found convergent evidence that rhythmogenic function is distributed over a dispersed and heterogeneous network that is synchronized by electrotonic coupling across a neuronal syncytium. First, high-speed recordings revealed that inspiratory onset occurred synchronously along the column and did not emanate from a rhythmogenic core. Second, following synaptic isolation, synchronized stationary rhythmic activity was detected along the column. This activity was attenuated following gap junction blockade and was silenced by tetrodotoxin. The layering of syncytial and synaptic coupling complicates identification of rhythmogenic mechanism, while enabling functional multiplexing.

Intrinsic neural timescales in the temporal lobe support an auditory processing hierarchy.

During rest, intrinsic neural dynamics manifest at multiple timescales, which progressively increase along visual and somatosensory hierarchies. Theoretically, intrinsic timescales are thought to facilitate processing of external stimuli at multiple stages. However, direct links between timescales at rest and sensory processing, as well as translation to the auditory system are lacking. Here, we measured intracranial electroencephalography in 11 human patients with epilepsy (4 women), while listening to pure tones. We show that in the auditory network, intrinsic neural timescales progressively increase, while the spectral exponent flattens, from temporal to entorhinal cortex, hippocampus, and amygdala. Within the neocortex, intrinsic timescales exhibit spatial gradients that follow the temporal lobe anatomy. Crucially, intrinsic timescales at baseline can explain the latency of auditory responses: as intrinsic timescales increase, so do the single-electrode response onset and peak latencies. Our results suggest that the human auditory network exhibits a repertoire of intrinsic neural dynamics, which manifest in cortical gradients with millimeter resolution and may provide a variety of temporal windows to support auditory processing.SIGNIFICANCE STATEMENT:Endogenous neural dynamics are often characterized by their intrinsic timescales. These are thought to facilitate processing of external stimuli. However, a direct link between intrinsic timing at rest and sensory processing is missing. Here, with intracranial electroencephalography (iEEG), we show that intrinsic timescales progressively increase from temporal to entorhinal cortex, hippocampus, and amygdala. Intrinsic timescales at baseline can explain the variability in the timing of iEEG responses to sounds: cortical electrodes with fast timescales also show fast and short-lasting responses to auditory stimuli, which progressively increase in the hippocampus and amygdala. Our results suggest that a hierarchy of neural dynamics in the temporal lobe manifests across cortical and limbic structures and can explain the temporal richness of auditory responses.

Dynamic synchronization between hippocampal representations and stepping

The hippocampus is a mammalian brain structure that expresses spatial representations1 and is crucial for navigation2,3. Navigation, in turn, intricately depends on locomotion; however, current accounts suggest a dissociation between hippocampal spatial representations and the details of locomotor processes. Specifically, the hippocampus is thought to represent mainly higher-order cognitive and locomotor variables such as position, speed and direction of movement4–7, whereas the limb movements that propel the animal can be computed and represented primarily in subcortical circuits, including the spinal cord, brainstem and cerebellum8–11. Whether hippocampal representations are actually decoupled from the detailed structure of locomotor processes remains unknown. To address this question, here we simultaneously monitored hippocampal spatial representations and ongoing limb movements underlying locomotion at fast timescales. We found that the forelimb stepping cycle in freely behaving rats is rhythmic and peaks at around 8 Hz during movement, matching the approximately 8 Hz modulation of hippocampal activity and spatial representations during locomotion12. We also discovered precisely timed coordination between the time at which the forelimbs touch the ground (‘plant’ times of the stepping cycle) and the hippocampal representation of space. Notably, plant times coincide with hippocampal representations that are closest to the actual position of the nose of the rat, whereas between these plant times, the hippocampal representation progresses towards possible future locations. This synchronization was specifically detectable when rats approached spatial decisions. Together, our results reveal a profound and dynamic coordination on a timescale of tens of milliseconds between central cognitive representations and peripheral motor processes. This coordination engages and disengages rapidly in association with cognitive demands and is well suited to support rapid information exchange between cognitive and sensory–motor circuits.

Morphology, Energy Level Alignment, and Charge Transfer at the Protoporphyrin IX-Semiconductor Interface.

The combination of molecular catalysts and semiconductor substrates in hybrid heterogeneous photo- or electrocatalytic devices could yield synergistic effects that result in enhanced activity and long-term stability. The extent of synergy strongly depends on the electronic interactions and energy level alignment between the molecular states and the valence and conduction band of the substrate. These properties of hybrid interfaces are investigated for a model system composed of protoporphyrin IX (PPIX) as a stand-in for molecular catalysts and a variety of semiconductor substrates. Monolayers of PPIX are deposited using Langmuir-Blodgett deposition. Their morphology is studied in dependence of the deposition surface pressure to achieve a high-quality, dense coverage. By making use of ultraviolet-visible spectroscopy and ultraviolet photoelectron spectroscopy, the band alignment is determined by the vacuum level and incorporates an interface dipole of 0.4 eV independent of the substrate. The HOMO, LUMO, and LUMO+1 levels were determined to be at 5.6, 3.7, and 2.7 eV below the vacuum level, respectively. The quenching of PPIX photoluminescence in dependence of the potential gradient between excited state and electron affinity of the semiconductor substrates is overall in good agreement with electron transfer processes occurring at very fast time scales on the order of femtoseconds. Nevertheless, deviations from this model become apparent for narrower band gap semiconductors, which points to an additional relevance of other processes, such as energy transfer. These findings highlight the importance of matching the semiconductor to the molecular catalyst to prevent undesirable deactivation pathways.

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.