Stability Control Method of Hybrid Impulsive System Based on M-Matrix

Aiming at the problems of poor control effect and poor stability of the mixed pulse system with the traditional method, this paper introduces the M-matrix to establish the pulse delay differential indefinite formula and realize stability control of the mixed pulse system. The synchronization problem of mixed-pulse systems in complex networks is analyzed using M matrix. The local coupling strength of the impulsive system is controlled according to the adaptive method. A class of Multi-Lyapunov functions is constructed for stability control of hybrid impulsive systems. The proposed method is proved to have better control effect through experiments.

Baptism of fire: Modeling the effects of prescribed fire on tick-borne disease

Recently, tick-borne illnesses have been trending upward and are an increasing source of risk to people's health in the United States. This is due to range expansion in tick habitats as a result of climate change. Thus, it is imperative to find a practical and cost-efficient way of managing tick populations. Prescribed burns are a common form of land management that can be cost efficient if properly managed and can be applied across large amounts of land. In this study, we present a compartmental model for ticks carrying Lyme disease and uniquely incorporate the effects of prescribed fire using an impulsive system to investigate the effects of prescribed fire intensity (high and low) and the duration between burns. Our study found that fire intensity has a larger impact in reducing tick population than the frequency between burns. Furthermore, burning at high intensity is preferable to burning at low intensity whenever possible, although high intensity burns may be unrealistic due to environmental factors. Annual burns resulted in the most significant reduction of infectious nymphs, which are the primary carriers of Lyme disease.

Dynamics of an Impulsive Stochastic Predator–Prey System with the Beddington–DeAngelis Functional Response

Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of quadratic functions, sufficient conditions on the existence of periodic Markovian processes are provided. The uniform continuity and global attractivity of solutions are also investigated. Additionally, we investigate the extinction and permanence in the mean of all species with the help of comparison methods and inequality techniques. Sufficient conditions on the existence and ergodicity of the stationary distribution of solutions for the autonomous and non-impulsive case are given. Finally, numerical simulations are performed to illustrate the main results.

On Nonlinear Forced Impulsive Differential Equations under Canonical and Non-Canonical Conditions

This study is connected with the nonoscillatory and oscillatory behaviour to the solutions of nonlinear neutral impulsive systems with forcing term which is studied for various ranges of of the neutral coefficient. Furthermore, sufficient conditions are obtained for the existence of positive bounded solutions of the impulsive system. The mentioned example shows the feasibility and efficiency of the main results.

Periodic Solution of A Delayed Intraguild Predation Impulsive System with Strong Allee Effect

With periodic coefficients and strong Allee effects, we establish a delayed intraguild predation impulsive model. We obtain a set of sufficient conditions for the existence of positive periodic solution of the model using Mawhin’s continuation theorem and analysis techniques. Finally, we identify the effectiveness of the theoretical results through some numerical simulations.

Functional Coupled Systems with Generalized Impulsive Conditions and Application to a SIRS-Type Model

In this paper, we consider a first-order coupled impulsive system of equations with functional boundary conditions, subject to the generalized impulsive effects. It is pointed out that this problem generalizes the classical boundary assumptions, allowing two-point or multipoint conditions, nonlocal and integrodifferential ones, or global arguments, as maxima or minima, among others. Our method is based on lower and upper solution technique together with the fixed point theory. The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and the asymptotic behavior of the unknown functions.

Deliberative Decision-Making in Macaques Removes Reward-Driven Response Vigor

Most of our daily decisions are governed by one of two systems: an impulsive system driving instantaneous decisions and a deliberative system driving thoughtful ones. The impulsive system reacts to immediately available concrete rewards. In contrast, the deliberative system reacts to more delayed rewards and/or punishments, which imposes consideration of longer-term choice consequences. Contingency management for addiction treatment is hypothesized to engage deliberative processes. Ultimately, in both decision-making situations, an action is needed to enact the decision. Whether those actions differ in implementation is an open question whose answer could inform as to whether distinct neural systems are engaged. To explore whether there is evidence of separate mechanisms between deliberated and immediate choices, we trained monkeys to perform a decision-making task where they made a choice on a touch screen between two visual cues predicting different amounts of reward. In immediate choice (IC) trials, the cues appeared at the final response locations where subjects could immediately touch the chosen cue. In deliberated choice (DC) trials, compound cues appeared orthogonally to the response locations. After a delay, allowing for decision formation, an identifying cue component was displaced to the randomly assigned response locations, permitting subjects to reach for the chosen cue. Both trial types showed an effect of cue value on cue selection time. However, only IC trials showed an effect of the competing cue on response vigor (measured by movement duration) and a reach trajectory that deviated in the direction of the competing cue, suggesting a decision reexamination process. Reward modulation of response vigor implicates dopaminergic mechanisms. In DC trials, reach trajectories revealed a commitment to the chosen choice target, and reach vigor was not modulated by the value of the competing cue. Our results suggest that choice–action dynamics are shaped by competing offers only during instantaneous, impulsive choice. After a deliberated decision, choice–action dynamics are unaffected by the alternative offer cue, demonstrating a commitment to the choice. The potential relevance to contingency management is discussed.