In this work, we focus on different statistical measures to uncover the complexity of extreme events in various nonlinear dynamical systems. Notably, the effectiveness of the variance-based measure, estimated from the mean recurrence time and entropy, reveals a clear distinction between extreme and non-extreme events. Additionally, the variance measure clearly highlights the critical transition point of unforeseen, rare, large-amplitude dynamics. Our investigation involved models from various disciplines, including the Brusselator chemical oscillator, CO2 laser, superconducting quantum interference devices, and a coupled neuron model. Deeper insights into the complexity of extreme events offer significant value for advancing early prediction methods.

This study examines a Caputo-type fractional-order food chain model, considering the Holling type II functional response with the vigilance effect. The model explores the interaction dynamics of the food chain model, which consists of prey, middle predators, and top predators. Additionally, habitat complexity is integrated into the model, which is assumed to reduce predation rates by lowering the encounter rates between predators and prey. All possible feasible equilibrium points are determined and the stability of our proposed model is explored near the equilibrium points. To support the analytical findings, numerical simulation results are given in terms of time series, phase portraits, and bifurcation diagrams. It is discovered that the proposed model can become more stable under a fractional-order derivative. Moreover, the interplay between the vigilance effect and habitat complexity is shown to influence the existence of stable and periodic dynamics.

In this work, a network of Morris–Lecar neurons with electromagnetic induction is imposed with nonlinear magnetic flux diffusion. We study wave propagation in a network of Morris–Lecar (ML) neurons with magnetic flux diffusion, connected to the local nodes of the nearest neighbors in a 110 × 110 lattice of neurons with periodic boundary conditions. First, we explore the effect of various initial conditions on the modified ML neuron network without imposing external stimuli. Subsequently, we apply external stimuli at different positions and study wave propagation by changing the amplitude and frequency of the stimuli. The effects of varying Nernst potential of potassium ions, coupling strengths, and flux constants are also analyzed. The resulting collective dynamics of the considered neuronal network are provided in snapshots with different model parameters. This study offers a novel perspective on wave propagation in networks of biological neurons.

This paper investigates the bifurcation problem in a fractional-order delayed food chain model that incorporates a fear effect. We observe that the fractional order significantly impacts the delayed system, influencing its stability in the presence of fear. Both the fractional order and the fear effect play crucial roles in determining the system’s stability. Furthermore, we observe stability switching induced by the fear effect while keeping the delay fixed. We identify the stability condition of the proposed model and precisely establish bifurcation points by utilizing delay as a bifurcation parameter. The system exhibits robust stability performance with smaller control parameters, and Hopf bifurcation arises as the control parameter surpasses a critical value. Additionally, through theoretical analysis and numerical simulations, we investigate the effects of fractional order, the fear effect, and time delay on the system’s stability.

This study delves into the examination of a network of adaptive synapse neurons characterized by a small-world network topology connected through electromagnetic flux and infused with randomness. First, this research extensively explores the existence of the global multi-stability of a single adaptive synapse-based neuron model with magnetic flux. The non-autonomous neuron model exhibits periodically switchable equilibrium states that are strongly related to the transitions between stable and unstable points in every whole periodic cycle, leading to the creation of global multi-stability. Various numerical measures, including bifurcation plots, phase plots, and basin of attraction, illustrate the intricate dynamics of diverse coexisting global firing activities. Moreover, the model is extended by coupling two neurons with a memristive synapse. The dynamics of the coupled neurons model are showcased with the help of largest Lyapunov exponents, and synchronized dynamics are viewed with the help of mean average error. Next, we consider a regular network of neurons connected to their nearest neighbors through the memristive synapse. We then reconstruct it into a small-world network by increasing the randomness in the rewiring links. Consequently, we observed collective behavior influenced by the number of neighborhood connections, coupling strength, and rewiring probability. We used spatio-temporal patterns, recurrence plots, as well as global-order parameters to verify the reported results.

Finite-time stability analysis is a powerful tool for understanding the long-term behavior of epidemiological models and has been widely used to study the spread of infectious diseases such as COVID-19. In this paper, we present a finite-time stability analysis of a stochastic susceptible-infected-recovered (SIR) epidemic compartmental model with switching signals. The model includes a linear parameter variation (LPV) and switching system that represents the impact of external factors, such as changes in public health policies or seasonal variations, on the transmission rate of the disease. We use the Lyapunov stability theory to examine the long-term behavior of the model and determine conditions under which the disease is likely to die out or persist in the population. By taking advantage of the average dwell time method and Lyapunov functional (LF) method, and using novel inequality techniques the finite-time stability (FTS) criterion in linear matrix inequalities (LMIs) is developed. The finite-time stability of the resultant closed-loop system, with interval and linear parameter variation (LPV), is then guaranteed by state feedback controllers. By analyzing the modified SIR model with these interventions, we are able to examine the efficiency of different control measures and determine the most appropriate response to the COVID-19 pandemic and demonstrate the efficacy of the suggested strategy through simulation results.

Fear and prey refuges are two significant topics in the ecological community because they are closely associated with the connectivity of natural resources. The effect of fear on prey populations and prey refuges (proportional to both the prey and predator) is investigated in the nonlinear-type predator-harvested Leslie–Gower model. This type of prey refuge is much more sensible and realistic than the constant prey refuge model. Because there is less research on the dynamics of this type of prey refuge, the current study has been considered to strengthen the existing literature. The number and stability properties of all positive equilibria are examined. Since the calculations for the determinant and trace of the Jacobian matrix are quite complicated at these equilibria, the stability of certain positive equilibria is evaluated using a numerical simulation process. Sotomayor’s theorem is used to derive a precise mathematical confirmation of the appearance of saddle-node bifurcation and transcritical bifurcation. Furthermore, numerical simulations are provided to visually demonstrate the dynamics of the system and the stability of the limit cycle is discussed with the help of the first Lyapunov number. We perform some sensitivity investigations on our model solutions in relation to three key model parameters: the fear impact, prey refuges, and harvesting. Our findings could facilitate some biological understanding of the interactions between predators and prey.

This work investigates the dynamic complications of the Ricker type predator-prey model in the presence of the additive type Allee effect in the prey population. In the modeling of discrete-time models, Euler forward approximations and piecewise constant arguments are the most frequently used schemes. In Euler forward approximations, the model may undergo period-doubled orbits and invariant circle orbits, even while varying the step size. In this way, differential equations with piecewise constant arguments (Ricker-type models) are a better choice for the discretization of a continuous-time model because they do not involve any step size. First, the interaction between prey and predator in the form of the Holling-Ⅱ type is considered. The essential mathematical features are discussed in terms of local stability and the bifurcation phenomenon as well. Next, we apply the center manifold theorem and normal form theory to achieve the existence and directions of flip bifurcation and Neimark-Sacker bifurcation. Moreover, this paper demonstrates that the outbreak of chaos can stabilize in the considered model with a higher value of the Allee parameter. The existence of chaotic orbits is verified with the help of a one-parameter bifurcation diagram and the largest Lyapunov exponents, respectively. Furthermore, different control methods are applied to control the bifurcation and fluctuating phenomena, i.e., state feedback, the Ott-Grebogi-Yorke, and hybrid control methods. Finally, to ensure our analytical results, numerical simulations have been carried out using MATLAB software.

The discrete-time system has more complex and chaotic dynamical behaviors as compared to the continuous-time system. This paper extends a discrete Leslie–Gower predator-prey system with the Allee effect in the predator’s population, whose dynamics are analyzed and explored. We have determined the equilibrium points and studied their local stability properties. We find that the system undergoes flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point by choosing the Allee parameter as a bifurcation parameter. We discuss the stability and direction of both bifurcations with the help of the normal form theory and center manifold theorem. The flip bifurcation and Neimark–Sacker bifurcation are the most common routes to the chaotic orbit in the discrete system. Moreover, we utilize state feedback, pole placement, and hybrid control methods to control the chaos in the system. The work is complete with the numerical simulations to confirm the analytical findings.

In this article, we propose the Holling type-II predator-prey model involving cannibalism and gestation delay in predators. We study the existence of all possible equilibrium points of the proposed model. We give the condition for the local stability and Hopf bifurcation analysis for the nondelayed model. Next, we also establish the local stability and Hopf bifurcation analysis for the corresponding delayed model. Finally, we discuss how cannibalism and delay play an important role in stabilizing and destabilizing the proposed system both theoretically and numerically.

This study presents an extended dissipative analysis of fractional order fuzzy networked control system with uncertain parameters. First, we designed the network-based fuzzy controller for the considered model. Second, a novel Lyapunov-Krasovskii functional (LKF) approach, inequality techniques, and some sufficient conditions are established, which make the proposed system quadratically stable under the extended dissipative criteria. Subsequently, the resultant conditions are expressed with respect to linear matrix inequalities (LMIs). Meanwhile, the corresponding controller gains are designed under the larger sampling interval. Finally, two numerical examples are presented to illustrate the viability of the obtained criteria.

This paper deals with the investigation of the three species food-web model. This model includes two logistically growing interaction species, namely [Formula: see text] and [Formula: see text], and the third species [Formula: see text] behaves as the predator and also host for [Formula: see text]. The species [Formula: see text] predating on the species [Formula: see text] with the Holling type-II functional response, while the first species [Formula: see text] is benefited from the third species [Formula: see text]. Further, the effect of fear is incorporated in the growth rate of species [Formula: see text] due to the predator [Formula: see text] and time lag in [Formula: see text] due to the gestation process. We explore all the biologically possible equilibrium points, and their local stability is analyzed based on the sample parameters. Next, we investigate the occurrence of Hopf-bifurcation around the interior equilibrium point by taking the value of the fear parameter as a bifurcation parameter for the non-delayed system. Moreover, we verify the local stability and existence of Hopf-bifurcation for the corresponding delayed system. Also, the direction and stability of the bifurcating periodic solutions are determined using the normal form theory and the center manifold theorem. Finally, we perform extensive numerical simulations to support the evidence of our analytical findings.

In this paper we are defining the fuzzy type of mixed delay differential equations. The necessity of studying the mixed delay differential equation in terms of fuzzy is that the single real valued solution can be ordered as the set of fuzzy valued solution. So it is very important in establishing the existence of solution for the fuzzy mixed type of delay differential equations using necessary theorems and lemmas. In addition we are also proving that the existing solution is unique. (C)2021 L&H Scienti fic Publishing, LLC. All rights reserved.

In the present work, we attempt to analyze the discrete form of Leslie type predator-prey system with the extension of nonlinear type harvesting of predators. The proposed model is obtained with the help of theory of piecewise constant argument for differential equations. We give the local stability properties of all possible non-negative fixed points. Also, our study reveals that the discrete system admits two bifurcations which are flip and Neimark-Sacker by making the use of center manifold argument and bifurcation theory, where the harvesting parameter is varied in order to take place of such bifurcations. Some simulations are carried out to depicts the obtained analytical results such as bifurcation plots and phase portraits. Also, it can be confirmed from numerical simulations that the considered system exhibits chaotic behavior for smaller values of harvesting parameter and becomes stable for larger values of same parameter. The largest Lyapunov exponents are plotted to show the sensitivity of chaotic regime.

Dynamical analysis of a delayed tri-trophic food chain consisting of prey, an intermediate, and a top predator is investigated in this paper. The additive Allee effect is introduced in the prey population, and it is assumed that there is a time lag due to the gestation effect in the intermediate predator. The interference among the prey and the intermediate predator is according to Holling type II, while the interaction between the intermediate and top predators follows the Crowley–Martin functional response. The local stability and bifurcation analysis of the proposed model at the interior equilibrium point are studied. Numerical simulations are provided to ensure the mathematical results.

In this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

This paper explores on the qualitative analysis of a modified Leslie-Gower prey-predator model where the consumption rate of prey is according to Beddington-DeAngelis functional response and Allee effect on prey population. Moreover time-lag (τ) is established to exploit gestation period of predations. The permanence analysis of proposed system is investigated. Then we study the local stability of non-delayedmodel at all possible equilibriumpoints and it is demonstrated that the given model experiences Hopf bifurcation about interior equilibrium point with respect to delay τ . Thereafter the stability and direction of Hopf bifurcation are formulated through normal and centermanifold theorems. The derived criteria are justified with the help of numerical simulations.

This article verifies the accuracy of Runge Kutta method of order four for Fuzzy multiple retarded delay differential equations to solve Fuzzy multiple neutral delay differential equations and to implement the technique to solve Fuzzy pure multiple neutral delay differential equations. The idea is to analyze the adoption of the method found for Retarded delay system to solve neutral delay systems where the two systems contains multiple delay. An numerical example is presented to verify the theory

In this Paper, We are combining so many mathematical-cum-engineering topics such as Fuzzy systems, Delay systems and Hybrid Systems under one roof called Numerical Solutions. The fuzzy valued problem was solved numerically and that approximate solution was compared with that of exact solutions. The non fuzzy and fuzzy valued numerical solutions and their graphical illustrations are also provided for the better understanding of the multiple hybrid single retarded delay problems.