State-dependent Feedback Research Articles

Dynamics analysis of a nonlinear controlled predator–prey model with complex Poincaré map

In this paper, we propose a class of predator–prey models with nonlinear state-dependent feedback control in the saturated state. The nonlinear state impulse control leads to a diversity of pulse and phase sets such that the Poincaré map built on the corresponding phase sets behaves like the single-peak function and multi-peak function with multiple discontinuities. We start our study by analyzing the exact pulse and phase sets of models under various cases generated by the dependent parameter space of nonlinear state feedback control, then construct the Poincaré map that is followed by investigating their monotonicity, continuity, concavity, and immobility properties. We also explore the existence, uniqueness, and sufficient conditions for the global stability of the order-1 periodic solutions of the systems. Numerical simulations are carried out to illustrate and reveal the biological significance of our theoretical findings.

The impact of resource limitation on the pest-natural enemy ecosystem with anti-predator behavior and fear effect

Research on agricultural pest control data indicates that the actual effectiveness of insecticides may exhibit slight variations, influenced by factors such as the type and quantity of pests. Additionally, the development of resistance by pests, as a result of adaptation to the environment, can impact the precision of pest control strategies. Hence we suggest implementing a nonlinear pulse state-dependent feedback control system, considering resource limitations, to investigate the influence of varying pesticide fatality rates on pest outbreaks. This paper takes into account the fear effect and anti-predator behavior to accurately portray the farmland ecosystem. We assessed the phase set and Poincaré map under the conditions for the existence of order-k (where k=1,2,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k=1,2,3$\\end{document}) periodic solutions and examined their stability behavior. Remarkably, the system exhibited diverse bifurcation phenomena associated with pesticide-related parameters. Furthermore, the system demonstrated a multistability phenomenon involving the coexistence of the order-1 periodic solution and the limit cycle. This suggests that altering control strategies can disturb the initial coexistence status of pests and natural enemies. It also underscores that resource limitations can indeed impact the outbreak patterns and frequencies of pests. In addition to the variability in pesticide fatality rates, the inclusion of natural enemy releases in the nonlinear pulse strategy contributes to the model exhibiting complex dynamic characteristics. All these findings are substantiated by numerical simulations.

Impulsive Effects and Complexity Dynamics in the Anti-Predator Model with IPM Strategies

When confronted with the imminent threat of predation, the prey instinctively employ strategies to avoid being consumed. These anti-predator tactics involve individuals acting collectively to intimidate predators and reduce potential harm during an attack. In the present work, we propose a state-dependent feedback control predator-prey model that incorporates a nonmonotonic functional response, taking into account the anti-predator behavior observed in pest-natural enemy ecosystems within the agricultural context. The qualitative analysis of this model is presented utilizing the principles of impulsive semi-dynamical systems. Firstly, the stability conditions of the equilibria are derived by employing pertinent properties of planar systems. The precise domain of the impulsive set and phase set is determined by considering the phase portrait of the system. Secondly, a Poincaré map is constructed by utilizing the sequence of impulsive points within the phase set. The stability of the order-1 periodic solution at the boundary is subsequently analyzed by an analog of the Poincaré criterion. Additionally, this article presents various threshold conditions that determine both the existence and stability of an order-1 periodic solution. Furthermore, it investigates the existence of order-k (k≥2) periodic solutions. Finally, the article explores the complex dynamics of the model, encompassing multiple bifurcation phenomena and chaos, through computational simulations.

Dynamics analysis of a nonlinear controlled predator–prey model with complex Poincaré map

In this paper, we propose a class of predator–prey models with nonlinear state-dependent feedback control in the saturated state. The nonlinear state impulse control leads to a diversity of pulse and phase sets such that the Poincaré map built on the corresponding phase sets behaves like the single-peak function and multi-peak function with multiple discontinuities. We start our study by analyzing the exact pulse and phase sets of models under various cases generated by the dependent parameter space of nonlinear state feedback control, then construct the Poincaré map that is followed by investigating their monotonicity, continuity, concavity, and immobility properties. We also explore the existence, uniqueness, and sufficient conditions for the global stability of the order-1 periodic solutions of the systems. Numerical simulations are carried out to illustrate and reveal the biological significance of our theoretical findings.

Antipredator behavior of a nonsmooth ecological model with a state threshold control strategy

<abstract><p>A nonsmooth ecological model was proposed and analyzed, focusing on IPM, state-dependent feedback control strategies, and anti-predator behavior. The main objective was to investigate the impact of anti-predator behavior on successful pest control, pest outbreaks, and the dynamical properties of the proposed model. First, the qualitative behaviors of the corresponding ODE model were presented, along with an accurate definition of the Poincaré map in the absence of internal equilibrium. Second, we investigated the existence and stability of order-k (where k = 1, 2, 3) periodic solutions through the monotonicity and continuity properties of the Poincaré map. Third, we conducted numerical simulations to investigate the complexity of the dynamical behaviors. Finally, we provided a precise definition of the Poincaré map in situations where an internal equilibrium existed within the model. The results indicated that when the mortality rate of the insecticide was low or high, the boundary order-1 periodic solution of the model was stable. However, when the mortality rate of the insecticide was maintained at a moderate level, the boundary order-1 periodic solution of the model became unstable; in this case, pests and natural enemies could coexist.</p></abstract>

Dynamics of a nonlinear state-dependent feedback control ecological model with fear effect

<p>Integrated pest management is a pest control strategy that combines biological and chemical methods to reduce environmental pollution and protect biodiversity. Recent research indicated that the fear caused by predators had a significant effect on the growth, development, and reproductive processes of prey. Therefore, we have proposed a pest-natrual enemy system, which is a nonlinear state-dependent feedback control model that incorporated the fear effect in the predator-prey relationship. We discussed impulsive sets and phase sets of the model and derived an expression for the Poincaré map. Furthermore, we analyzed the existence and stability of order-$ 1 $ periodic solutions and explored the existence of order-$ k $ $ (k\ge2) $ periodic solutions. Finally, numerical simulations were conducted to validate our theoretical results and reveal their biological implications.</p>

Winner-loser effects improve social network efficiency between competitors with equal resource holding power

Animal societies are structured of dominance hierarchy (DH). DH can be viewed as networks and analyzed by graph theory. We study the impact of state-dependent feedback (winner-loser effect) on the emergence of local dominance structures after pairwise contests between initially equal-ranking members (equal resource-holding-power, RHP) of small and large social groups. We simulated pairwise agonistic contests between individuals with and without a priori higher RHP by Monte-Carlo-method. Random pairwise contests between equal-ranking competitors result in random dominance structures (‘Null variant’) that are low in transitive triads and high in pass along triads; whereas state-dependent feedback (‘Winner-loser variant’) yields centralized ‘star’ structured DH that evolve from competitors with initially equal RHP and correspond to hierarchies that evolve from keystone individuals. Monte-Carlo simulated DH following state-dependent feedback show motif patterns very similar to those of a variety of natural DH, suggesting that state-dependent feedback plays a pivotal role in robust self-organizing phenomena that transcend the specifics of the individual. Self-organization based on state-dependent feedback leads to social structures that correspond to those resulting from pre-existing keystone individuals. As the efficiency of centralized social networks benefits both, the individual and the group, centralization of social networks appears to be an important evolutionary goal.

Global dynamic analysis of a nonlinear state-dependent feedback control SIR model with saturation incidence

Global dynamic analysis of a nonlinear state-dependent feedback control SIR model with saturation incidence

Attentional State-Dependent Peripheral Electrical Stimulation During Action Observation Enhances Cortical Activations in Stroke Patients.

Brain-computer interface (BCI) is a promising technique that enables patients' interaction with computers or machines by analyzing specific brain signal patterns and provides patients with brain state-dependent feedback to assist in their rehabilitation. Action observation (AO) and peripheral electrical stimulation (PES) are conventional methods used to enhance rehabilitation outcomes by promoting neural plasticity. In this study, we assessed the effects of attentional state-dependent feedback in the combined application of BCI-AO with PES on sensorimotor cortical activation in patients after stroke. Our approach involved showing the participants a video with repetitive grasping actions under four different tasks. A mu band suppression (8-13 Hz) corresponding to each task was computed. A topographical representation showed that mu suppression of the dominant (healthy) and affected hemispheres (stroke) gradually became prominent during the tasks. There were significant differences in mu suppression in the affected motor and frontal cortices of the stroke patients. The involvement of both frontal and motor cortices became prominent in the BCI-AO+triggered PES task, in which feedback was given to the patients according to their attentive watching. Our findings suggest that synchronous stimulation according to patient attention is important for neurorehabilitation of stroke patients, which can be achieved with the combination of BCI-AO feedback with PES. BCI-AO feedback combined with PES could be effective in facilitating sensorimotor cortical activation in the affected hemispheres of stroke patients.

Dynamic analysis of two fishery capture models with a variable search rate and fuzzy biological parameters.

The fishery resource is a kind of important renewable resource and it is closely connected with people's production and life. However, fishery resources are not inexhaustible, so it has become an important research topic to develop fishery resources reasonably and ensure their sustainability. In the current study, considering the environment changes in the system, a fishery model with a variable predator search rate and fuzzy biological parameters was established first and then two modes of capture strategies were introduced to achieve fishery resource exploitation. For the fishery model in a continuous capture mode, the dynamic properties were analyzed and the results show that predator search rate, imprecision indexes and capture efforts have a certain impact on the existence and stability of the coexistence equilibrium. The bionomic equilibrium and optimal capture strategy were also discussed. For the fishery model in a state-dependent feedback capture mode, the complex dynamics including the existence and stability of the periodic solutions were investigated. Besides the theoretical results, numerical simulations were implemented step by step and the effects of predator search rate, fuzzy biological parameters and capture efforts on the system were demonstrated. This study not only enriched the related content of fishery dynamics, but also provided certain reference for the development and utilization of fishery resources under the environment with uncertain parameters.

Learning quantum-state feedback control with backpropagation-free stochastic optimization

High-fidelity state preparation represents a fundamental challenge in the application of quantum technology. While the majority of optimal control approaches use feedback to improve the controller, the controller itself often does not incorporate explicit state dependence. Here, we present a general framework for training deep feedback networks for open quantum systems with continuous weak measurement that allows a variety of system and control structures that are prohibitive by many other techniques and can in effect react to unmodeled effects through nonlinear filtering. Our approach benefits from characteristics of both stochastic sampling and gradient-based optimization methods yet does not require differentiability as in backpropagation approaches. We demonstrate that this method is efficient due to inherent parallelizability, robust to open system interactions, and outperforms landmark state-dependent feedback control results in simulation.

Motion state-dependent motor learning based on explicit visual feedback is quickly recalled, but is less stable than adaptation to physical perturbations.

Recent studies have shown that adaptation to visual feedback perturbations during arm reaching movements involves implicit and explicit learning components. Evidence also suggests that explicit, intentional learning mechanisms are largely responsible for savings-a faster recalibration compared with initial training. However, the extent explicit learning mechanisms facilitate learning and early savings (i.e., the rapid recall of previous performance) for motion state-dependent learning is generally unknown. To address this question, we compared the early savings/recall achieved by two groups of human subjects. One experienced physical perturbations (a velocity-dependent force-field, vFF) to promote adaptation that is thought to be a largely implicit process. The second was only given visual feedback of the required force-velocity relationship; subjects moved in force channels and we provided visual feedback of the lateral force exerted during the movement, as well as the required force pattern based on the movement velocity. Thus, subjects were shown explicit information on the extent the applied temporal pattern of force matched the required velocity-dependent force profile if the force-field perturbation had been applied. After training, both groups experienced a decay and washout period, which was followed by a reexposure block to assess early savings/recall. Although decay was faster for the explicit visual feedback group, the single-trial recall was similar to the physical perturbation group. Thus, compared with visual feedback perturbations, conscious modification of motor output based on motion state-dependent feedback demonstrates rapid recall, but this adjustment is less stable than adaptation based on experiencing the multisensory errors that accompany physical perturbations.NEW & NOTEWORTHY The extent explicit feedback facilitates motion state-dependent changes to motor output is largely unknown. Here, we examined motor adaptation for subjects that experienced physical perturbations and another that made adjustments based on explicit visual feedback information of the required force-velocity relationship. Our results suggest that adjustment of motor output can be based on explicit motion state-dependent information and demonstrates rapid recall, but this learning is less stable than adaptation based on physical perturbations to movement.

Bifurcation Analysis of a Tumour-Immune Model with Nonlinear Killing Rate as State-Dependent Feedback Control

Impulsive control strategies have been widely used in cancer treatment and linear impulsive control has always been considered in previous studies. We propose a novel tumour-immune model with nonlinear killing rate as state-dependent feedback control, which can better reflect the saturation effects of the tumour and immune cell mortalities due to chemotherapy, and its dynamic behaviors are investigated. The paper aims to discuss the transcritical and subcritical bifurcations of the model. To begin with, the threshold conditions for tumour eradication and tumour persistence in the model without pulse interventions are provided. We define the Poincaré map of the proposed model and then address the existence and orbital asymptotically stability of the model’s tumour-free periodic solution. Furthermore, by using the bifurcation theory of the discrete one-parameter family of maps, which is determined by the Poincaré mapping, we investigate the model’s transcritical and subcritical pitchfork bifurcations with respect to the key parameter.

New results on stabilization for neutral type descriptor hybrid systems with time-varying delays

This paper revisits the issue of the stabilization for neutral type descriptor hybrid systems (NTDHSs) with time-varying delays. By a novel Lyapunov–Krasovskii functional (LKF) we constructed, a mode dependent and delay dependent criterion is acquired for the unforced NTDHSs to be stochastically admissible. On the basis of the criterion, a mode dependent state feedback controller (SFC) is designed such that the resulting closed-loop system is regular, impulse free and stochastically stable. By singular-value decomposition method, sufficient conditions are established via strict linear matrix inequality (LMI). And the gain of the SFC can be derived. It is worth noting that our results generalize the corresponding results published in latest literatures to some degree. The superiority and validity of our methods are displayed via two numerical examples.

A State-Dependent Impulsive Nonlinear System with Ratio-Dependent Action Threshold for Investigating the Pest-Natural Enemy Model

Based on the Lotka–Volterra system, a pest-natural enemy model with nonlinear feedback control as well as nonlinear action threshold is introduced. The model characterizes the implementation of comprehensive prevention and control measures when the pest density reaches the nonlinear action threshold level depending on the pest density and its change rate. The mortality rate of the pest is a saturation function that strictly depends on their density while the release of natural enemies is also a nonlinear pulse term depending on the density of real-time natural enemies. The exact impulsive and phase sets are given. The definition and properties of the Poincaré map corresponding to the pulse points on the phase set are provided. We investigate the existence and stability of boundary and interior order-1 periodic solution. The theoretical analysis developed in the present paper combined with nonlinear controlling measures as well as nonlinear action threshold methods and techniques laid the foundation for the establishment and analysis of other state-dependent feedback control models.

Nonlinear effects of instantaneous and delayed state dependence in a delayed feedback loop

<p style='text-indent:20px;'>We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduces nonlinearity into an otherwise linear system. We investigate the ensuing nonlinear dynamics with the case of instantaneous state dependence as our starting point. We present the bifurcation diagram in the parameter plane of the two feedback strengths showing how periodic orbits bifurcate from a curve of Hopf bifurcations and disappear along a curve where both period and amplitude grow beyond bound as the orbits become saw-tooth shaped. We then 'switch on' the delay within the state-dependent feedback term, reflected by a parameter <inline-formula><tex-math id="M1">\begin{document}$ b>0 $\end{document}</tex-math></inline-formula>. Our main conclusion is that the new parameter <inline-formula><tex-math id="M2">\begin{document}$ b $\end{document}</tex-math></inline-formula> has an immediate effect: as soon as <inline-formula><tex-math id="M3">\begin{document}$ b>0 $\end{document}</tex-math></inline-formula> the bifurcation diagram for <inline-formula><tex-math id="M4">\begin{document}$ b = 0 $\end{document}</tex-math></inline-formula> changes qualitatively and, specifically, the nature of the limiting saw-tooth shaped periodic orbits changes. Moreover, we show — numerically and through center manifold analysis — that a degeneracy at <inline-formula><tex-math id="M5">\begin{document}$ b = 1/3 $\end{document}</tex-math></inline-formula> of an equilibrium with a double real eigenvalue zero leads to a further qualitative change and acts as an organizing center for the bifurcation diagram. Our results demonstrate that state dependence in delayed feedback terms may give rise to new dynamics and, moreover, that the observed dynamics may change significantly when the state-dependent feedback depends on past states of the system. This is expected to have implications for models arising in different application contexts, such as models of human balancing and conceptual climate models of delayed action oscillator type.</p>

Necessary conditions for feedback stabilization and safety

<p style='text-indent:20px;'>Brockett's necessary condition yields a test to determine whether a system can be made to stabilize about some operating point via continuous, purely state-dependent feedback. For many real-world systems, however, one wants to stabilize sets which are more general than a single point. One also wants to control such systems to operate safely by making obstacles and other "dangerous" sets repelling.</p><p style='text-indent:20px;'>We generalize Brockett's necessary condition to the case of stabilizing general compact subsets having a nonzero Euler characteristic in general ambient state spaces (smooth manifolds). Using this generalization, we also formulate a necessary condition for the existence of "safe" control laws. We illustrate the theory in concrete examples and for some general classes of systems including a broad class of nonholonomically constrained Lagrangian systems. We also show that, for the special case of stabilizing a point, the specialization of our general stabilizability test is stronger than Brockett's.</p>

Bifurcation analysis of an ecological model with nonlinear state–dependent feedback control by Poincaré map defined in phase set

Ecosystems often need to integrate instantaneous interventions such as biological resource management or integrated pest control. The implementation of interventions, in turn, depends on the state and its change rate of the system, which brings great challenges for the stability and bifurcation analyses. To develop qualitative analysis techniques, we propose a predator–prey model with state-dependent pulse interventions, including spraying insecticides and releasing natural enemies, in which the implementation of control measures depends on whether the weighted value of pest density and its change rate reaches the action threshold. We first address the threshold condition for the existence and stability of the boundary periodic solutions and then define a one-parameter family of discrete (Poincaré) maps. By analyzing the properties of those discrete maps, we conclude that if there is no interior equilibrium for the system without control measures and the releasing amount of natural enemies is greater than zero, then there is at least one positive periodic solution. While if the system without control measures has an internal equilibrium and only the chemical tactic is applied, then there could be an unstable positive periodic solution near the boundary periodic solution. Consequently, backward bifurcation and bi-stability occur. The analytical techniques developed here could be applied to analyze more generalized models and other fields, including infectious disease control.

Bifurcation Analysis of an Ecological System with State-Dependent Feedback Control and Periodic Forcing

By assuming a periodic variation in the intrinsic growth rate of the prey, a nonlinear ecological system with periodic forcing and state-dependent feedback control is proposed. The main purpose of the present paper is to study the dynamical behavior generated by periodic forcing and nonlinear impulse perturbations and their effects on pest control. To do this, we first investigate the existence and stability of the boundary periodic solution, and then we employ the numerical bifurcation techniques, mainly including one-dimensional and two-dimensional parameter bifurcation analyses, to reveal that the system exhibits rich and complex dynamic behaviors. Especially, period-adding bifurcation with chaos is found in the two-parameter bifurcation plane. Moreover, we find the periodic structure similar to Arnold tongues, and they are arranged according to the sequence of a Farey tree. In addition, one-dimensional bifurcation diagrams reveal the existence of order-[Formula: see text] periodic, and chaotic solutions, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the effects of the initial population density of pests and natural enemies on the pulse frequency and the biological significance related to the numerical results are studied and discussed.

Eco-evolutionary dynamics with payoff-dependent environmental feedback

• A general payoff-dependent feedback is proposed to depict how the environment is altered by the game dynamics. • Limiting behavior of the payoff-dependent eco-evolutionary dynamics are rigorously analyzed. The sensitivity of degradation rates on defectors’ payoffs promotes cooperation, whereas the sensitivity of enhancement rates on cooperators’ payoffs inhibits cooperation. • The time scale of environment and strategy can qualitatively alter the dynamical outcomes, which cannot for the state-dependent feedback. • The sensitivity of degradation rates on defectors’ payoffs promotes cooperation, whereas the sensitivity of enhancement rates on cooperators’ payoffs inhibits cooperation. In recent years, the eco-evolutionary game theory which captures the coupled dynamics of cooperation and environment attracts considerable attention. For strategic updating, it is assumed that individuals’ behaviors are driven by natural selection. For the evolution of environment, it is typically described by state-dependent environmental feedback. This state-dependent feedback is heuristic. In this work, we propose an alternative heuristic assumption on the feedback, i.e., payoff-dependent environmental feedback. For a general class of payoff-dependent feedback dynamics, we show the condition under which the internal equilibria exist and are stable, respectively. Furthermore, we focus on scenarios in which the environmental enhancement rates and degradation rates are linear mapping and exponential mapping of the cooperator’s (defector’s) payoffs, respectively. Our analysis and numerical results show that the sensitivity of environmental degradation rates to defector’s payoffs promotes cooperation, whereas the sensitivity of enhancement rates to cooperator’s payoffs inhibits cooperation. In addition, we find that the relative speed of environmental and strategy dynamics can qualitatively influence system dynamics. These results show that nonlinear dynamics of the co-evolution system can be dramatically altered when the underlying feedback is not state-dependent. Our work might shed light on the exploring cooperation mechanism in the nonlinear system with environmental feedback.