Narrow Results

This research rigorously investigates a stochastic Susceptible-Infectious-Susceptible (SIS) epidemic model incorporating a generalized nonlinear incidence rate. The study further expands the model scope by introducing white and Lévy noise components. Including jumps and noises enhances the likelihood of disease extinction within the population. Initially, we utilized a comprehensive mathematical analysis to establish the global stability of the equilibrium state devoid of the disease. Our findings demonstrate that the system maintains stability even in the presence of the disease. This serves as a fundamental insight into the dynamics of disease transmission. Moreover, we define a collection of adequate criteria that contribute to the continuous presence of the ailment, thereby guaranteeing its non-complete eradication. By identifying the critical factors that contribute to the sustained presence of the disease, we enhance our understanding of the long-term behavior of the epidemic. To validate our theoretical findings, we conduct numerical simulations. These simulations effectively demonstrate the accuracy and reliability of the presented results. By comparing the simulated data with our analytical predictions, we can confidently confirm the validity of our model and its applicability in practical scenarios.

Stochastic bifurcation of dynamical system induced by Lévy noise is investigated in this paper from the global viewpoint, which is demonstrated based on the global attractor obtained with the compatible cell mapping (CCM) method. Compared with the traditional cell mapping method, the efficiency of CCM method is greatly improved by concentrating the computational domain on the covering set of the global attractor rather than the whole state space. The computation time can be further reduced by a simple parallel procedure with the block matrix strategy. With the proposed method, the evolutionary process of global properties has been presented. It shows that the shape and size of stochastic attractors and stochastic saddles vary along the direction of unstable manifold with the parameters of Lévy noise. Stochastic bifurcation occurs when stochastic attractor collides with stochastic saddle. It can be also found that the global bifurcation under Lévy noise is discontinuous because of the jump property. The results are significantly different from those of Gaussian noise, which reveals a distinct global evolutionary mechanism of stochastic bifurcation. Furthermore, it also indicates that the CCM method with block matrix is an effective tool for global analysis.

Effects of Lévy noise on self-propelled particles in a two-dimensional potential is investigated. The current reversal phenomenon appears in the system. $V$ ($x$-direction average velocity) changes from negative to positive with increasing asymmetry parameter $\beta$, and changes from positive to negative with increasing self-propelled velocity $v_0$. $V$ has a maximum with increasing modulation constant $\lambda$.

In this paper, the Lévy noise-induced transition in an underdamped asymmetric bistable system is discussed. Lévy noise is generated  through the Janicki–Weron algorithm and the numerical solutions of system equation is obtained by the fourth-order Runge–Kutta method. Then the stationary probability density functions are obtained by solving the equation of system. The influence of the damped coefficient $\gamma$, asymmetric parameter r of system, stability index $\alpha$, skewness parameters $\beta$ and noise intensity D on the stationary probability density are analyzed. The numerical simulation results show that the asymmetric parameter r, stability index $\alpha$, skewness parameters $\beta$ and noise intensity D can induce the phase transition. However, the phase transition cannot be induced by the damped coefficient $\gamma$.

This paper primarily investigates the inverse stochastic resonance (ISR) of neuron network driven by Lévy noise with electrical autapse and chemical autapse, respectively. Firstly, the discharge of Hodgkin–Huxley (HH) neuron network under different noise parameters, autapse parameters and network coupling strength is shown. Then, the variation of average firing rate with Lévy noise in the case of electrical autapse and chemical autapse is presented. We find that there exists a minimum value of the average firing rate curve caused by stability index and noise intensity of Lévy noise across the whole network, which is the phenomenon of ISR. With the increase of autaptic intensity and coupling strength, the ISR inhibitory effect of neuron discharge is weakened. In addition, with the increase of coupling strength, the neuron discharge of neural network is more intense and regular. As a consequence, our work suggests that autaptic intensity and coupling efficient of neuronal network can regulate the neuronal firing activities and suppress the effect of ISR, and Lévy noise can induce ISR phenomenon in Newman–Watts neuronal network.

In this paper, the dynamical behavior of the FitzHugh–Nagumo (FHN) neural system with time delay driven by Lévy noise is studied from two aspects: the mean first-passage time (MFPT) and the probability density function (PDF) of the first-passage time (FPT). Using the Janicki–Weron algorithm to generate the Lévy noise, and through the order-4 Runge–Kutta algorithm to simulate the FHN system response, the time that the system needs from one stable state to the other one is tracked in the process. Using the MATLAB software to simulate the process above 20,000 times and recording the PFTs, the PDF of the FPT and the MFPT is obtained. Finally, the effects of the Lévy noise and time-delay on the FPT are discussed. It is found that the increase of both time-delay feedback intensity and Lévy noise intensity can promote the transition of the particle from the resting state to the excited state. However, the two parameters produce the opposite effects in the other direction.

The dynamical behavior of tumor growth model driven by Lévy noise terms is investigated. For α = 2 and β = 0, the process driven by white Lévy noise approach to the standard Gaussian white noise can be viewed in the analysis of the steady-state probability distribution and the mean first-passage time. When β → 0, the index α would increase the mean first-passage time as scale σ < 0 and shorten the mean first-passage time as scale σ > 0. A nonzero β parameter induces α to decrease the mean first-passage time. Thus analyzing the initial situation of tumor is very important to obtain more therapy time.

In this paper, the stochastic resonance phenomenon in a tumor growth model under subthreshold periodic therapy and Lévy noise excitation is investigated. The possible reoccurrence of tumor due to stochastic resonance is discussed. The signal-to-noise ratio (SNR) is calculated numerically to measure the stochastic resonance. It is found that smaller stability index is better for avoiding tumor reappearance. Besides, the effect of the skewness parameter on the tumor regrowth is related to the stability index. Furthermore, increasing the intensity of periodic treatment does not always facilitate tumor therapy. These results are beneficial to the optimization of periodic tumor therapy.

In this paper, the steady state characteristics and stochastic resonance (SR) in two-dimensional FitzHugh–Nagumo (FHN) neuron system driven by Lévy noise are studied. The system is simulated by Janicki–Weron algorithm and fourth-order Runge–Kutta method, and the steady state characteristics of the system are analyzed by stationary probability density (SPD) functions. Then, the SR is determined by the classical measure of signal-to-noise ratio (SNR). Through numerical simulation, it is found that the Lévy noise can induce the transition of the system. In addition, the effects of different parameters on the SR are analyzed by SNR.

In this paper, the first-passage behavior of under-damped asymmetric bistable system driven by Lévy noise is studied. The two aspects considered are the mean first-passage time (MFPT) and the distribution of first-passage time in two opposite directions. To begin with, using the Janicki–Weron algorithm to generate Lévy noise, the system driven by Lévy noise is simulated through the fourth-order Runge–Kutta algorithm. Then the first-passage time of $2\times 10^4$ response tracks is calculated, and the MFPT and the distribution of first-passage time are obtained. Finally, the influence of Lévy noise and system parameters on MFPT and the distribution of first-passage time are analyzed. Moreover, the noise enhanced stability (NES) effect is found.

In this paper, based on the Hodgkin–Huxley (H-H) neuron model, three distinct types of noises are introduced into the input of the neuron. The noises are Gaussian white noise (GWN), Lévy noise and Colored noise, separately. The phenomenon of stochastic resonance under and above threshold in one-dimensional H-H neuron is discussed. First, the responses of H-H neuron to different input signal frequency, amplitude and constant offset stimulation are researched. The neuron membrane potential diagram and spectrum diagram are given. Then, the mutual information rate (MIR) is taken as the evaluation index. The results show that adding a noise of certain intensity can increase the response of neuron to the orignal signal and the MIR between the output and input of neuron has unimodal property, which proves the existence of stochastic resonance under and above the threshold. We also find that neurons have frequency selectivity and different transmission performance for different amplitude of input signals. In addition, the constant offset stimulus should not be too large. Finally, it is discovered that the adaptability of subthreshold stochastic resonance and suprathreshold stochastic resonance (SSR) characteristics characteristics of one-dimensional H-H neurons induced by three distinct types of noises is different. What is more important is that the results of this paper will be instrumental in the future researching on the affect of these three noises in the stochastic resonance of neuron networks.

The birhythmic oscillation is of great significance in biology and engineering, and this paper presents a bifurcation analysis in a time-delayed birhythmic oscillator containing fractional derivative and Lévy noise. The numerical method is used to explore the influence of various parameters on the bifurcation of the birhythmic system, and the role of fractional derivative and Lévy noise in inducing or inhibiting birhythmicity in a time-delayed birhythmic biological system is examined in this work. First, we use a numerical method to calculate the fractional derivative, which has a fast calculation speed. Then the McCulloch algorithm is employed to generate Lévy random numbers. Finally, the stationary probability density function graph of the amplitude is obtained by Monte Carlo simulation. The results show that the fractional damping and Lévy noise can effectively control the characteristics of the birhythmic oscillator, and the change of the parameters (except the skewness parameter) can cause the system bifurcation. In addition, this article further discusses the interaction of fractional derivative and time delay in a birhythmic system with Lévy noise, proving that adjusting parameters of time delay can lead to abundant bifurcations. Our research may help to further explore the bifurcation phenomenon of birhythmic biological system, and has a practical significance.

In this paper, we study the stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise in three dimensions with a hereditary viscous term which depends on the past history. We establish the local solvability of the Cauchy problem for such systems. The local monotonicity property of the nonlinear term of the cutoff problem and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. Finally, we show that the global solvability results hold under smallness condition on the initial data and suitable assumptions on the noise coefficients.

We investigate the effect of Lévy noise on the noise-induced residual aperiodic stochastic resonance (ASR). It is shown that, the residual ASR still exists when the noise is Lévy distributed. However, it is reduced by Lévy noise compared to that in the Gaussian noise with the same intensity. It is also shown that, the minimum bit error rate (BER) of the Lévy-noise-induced ASR is slightly higher than that of the Gaussian-noise-induced ASR, which indicates that tuning system parameters is a better approach to get the optimal performance.

This paper aims to explore the Lévy noise-induced effects in underdamped asymmetric bistable system. Lévy noise is generated by Janicki–Weron algorithm which is different from the usual Gaussian noise. The numerical solutions of system equation are obtained by the fourth-order stochastic Runge–Kutta algorithm. Then the quasi-steady-state probability density (QSPD) is obtained by solving the equation of system, and the stochastic resonance (SR) is determined by the classical measure of signal-to-noise ratio (SNR). The influence of various parameters of the Lévy noise and the system parameters on QSPD and SNR is discussed. Noise-induced transitions occur by varying the parameters of the Lévy noise and the driven system. Moreover, within certain limits, the larger value of the stability index $\alpha$ of Lévy noise, signal amplitude $A$, and the absolute values of asymmetric parameter $r$ can give rise to the SR phenomenon. On the contrary, the larger values of skewness parameters $\beta$ of Lévy noise and damping parameter $\gamma$ further weaken the occurrence of the SR phenomenon in the given system.

In this paper, we establish a stochastic dynamical grazing ecosystem with time delays and fluctuations. The effects of time delay, Gaussian noise and Lévy noise on the stationary probability distribution (SPD), the mean first passage time (MFPT) and stochastic resonance (SR) are analyzed. Our research results show the following: (i) For small time delay, the increasing Gaussian noise intensity leads to catastrophic regime shift (CRS) from high vegetation state to low vegetation state, while the increasing Lévy noise intensity contributes to the recovery of these shifts. For large time delay, the increasing Gaussian noise intensity or Lévy noise intensity causes the CRS phenomenon, the larger the time delay, the more frequent the CRS phenomenon. (ii) The increasing Gaussian noise intensity can diminish the stability of the high vegetation state and low vegetation state, the increasing Lévy noise intensity and Lévy stability index can enhance the stability of the high vegetation state and diminish the stability of the low vegetation state. The increasing Lévy noise intensity can lead to noise-enhanced stability (NES) of the high vegetation state, and the larger the time delay, the more pronounced the NES phenomenon. (iii) The increase of time delay can weaken SR phenomenon when signal-to-noise ratio (SNR) is a function of Gaussian noise intensity and Lévy noise intensity.

We investigate the perturbation of the nonlinear differential equation by random noise terms consisting of Brownian motion and an independent Poisson random measure. We find conditions under which the perturbed system is almost surely exponentially stable and estimate the corresponding Lyapunov exponents.

This paper is devoted to stochastic non-Newtonian fluid driven by Lévy noise. By the tight compactness of distribution of the solution for finite-dimensional approximate in a Hilbert space, Skorohod embedding theorem and representation of martingale, the existence of the martingale solution is shown. Moreover, the procedure of the proof for the Markov selection in [J. Differential Equations250 (2011) 2737–2778] are simplified to show the existence of Markov selection for the martingale solution.

We establish the existence, uniqueness and approximation of the strong solutions for the stochastic 3D LANS-α model driven by a non-Gaussian Lévy noise. Moreover, we also study the stability of solutions. In particular, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution.

We propose and analyze a numerical scheme for the approximation of the solution for the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise. We prove the convergence of the scheme which is a linear evolution equation with additive noise.