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In this paper, we introduce a stochastic two-strain epidemic model driven by Lévy noise describing the interaction between four compartments; susceptible, infected individuals by the first strain, infected ones by the second strain and the recovered individuals. The forces of infection, for both strains, are represented by saturated incidence rates. Our study begins with the investigation of unique global solution of the suggested mathematical model. Then, it moves to the determination of sufficient conditions of extinction and persistence in mean of the two-strain disease. In order to illustrate the theoretical findings, we give some numerical simulations.

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This paper concerns the dynamics of two stochastic hybrid delay Lotka-Volterra systems with harvesting and Lévy noise in a polluted environment (i.e., predator-prey system and competitive system). For every system, sufficient and necessary conditions for persistence in mean and extinction of each species are established. Then, sufficient conditions for global attractivity of the systems are obtained. Finally, sufficient and necessary conditions for the existence of optimal harvesting strategy are provided. The accurate expressions for the optimal harvesting effort (OHE) and the maximum of expectation of sustainable yield (MESY) are given. Our results show that the dynamic behaviors and optimal harvesting strategy are closely correlated with both time delays and three types of environmental noises (namely white Gaussian noises, telephone noises and Lévy noises).

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In this paper, we investigate the dynamical properties of a stochastic predator-prey model with a fear effect. We also introduce infectious disease factors into prey populations and distinguish prey populations into susceptible prey and infected prey populations. Then, we discuss the effect of Lévy noise on the population considering extreme environmental situations. First of all, we prove the existence of a unique global positive solution for this system. Second, we demonstrate the conditions for the extinction of three populations. Under the conditions that infectious diseases are effectively prevented, the conditions for the existence and extinction of susceptible prey populations and predator populations are explored. Third, the stochastic ultimate boundedness of system and the ergodic stationary distribution without Lévy noise are also demonstrated. Finally, we use numerical simulations to verify the conclusions obtained and summarize the work of the paper.

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Stochastic resonance is a remarkable phenomenon that can enhance signal processing by the addition of random noise. However, the effect of magnetic fields on stochastic resonance under channel noises has been inadequately studied. In this paper, the stochastic resonance in Hodgkin-Huxley neuronal network under Gaussian channel noises and non-Gaussian channel noise were studied, and the effects of electromagnetic field stimulation on stochastic resonance were considered. The results indicate that stochastic resonance in neuronal networks can be induced by Gaussian channel noise and non-Gaussian Levy channel noise, and stochastic resonance may occur more easily under Levy channel noise. The resonance amplitude was significantly improved by selecting appropriate parameters of the magnetic field, while, a too strong magnetic field can be detrimental to the resonance amplitude. Magnetic fields may induce the enhancement of the resonance amplitude by increasing the firing frequency and spiking regularity.

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This study concentrates on the analysis of a stochastic SIC epidemic system with an enhanced and general perturbation. Given the intricacy of some impulses caused by external disturbances, we integrate the quadratic Lévy noise into our model. We assort the long-run behavior of a perturbed SIC epidemic model presented in the form of a system of stochastic differential equations driven by second-order jumps. By ameliorating the hypotheses and using some new analytical techniques, we find the exact threshold value between extinction and ergodicity (persistence) of our system. The idea and analysis used in this paper generalize the work of N. T. Dieu et al. (2020), and offer an innovative approach to dealing with other random population models. Comparing our results with those of previous studies reveals that quadratic jump-diffusion has no impact on the threshold value, but it remarkably influences the dynamics of the infection and may worsen the pandemic situation. In order to illustrate this comparison and confirm our analysis, we perform numerical simulations with some real data of COVID-19 in Morocco. Furthermore, we arrive at the following results: (i) the time average of the different classes depends on the intensity of the noise (ii) the quadratic noise has a negative effect on disease duration (iii) the stationary density function of the population abruptly changes its shape at some values of the noise intensity. Mathematics Subject Classification 2020: 34A26; 34A12; 92D30; 37C10; 60H30; 60H10.

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The standard textbooks contain good explanations of how and why equilibrium thermodynamics emerges in a reservoir with particles that are subjected to Gaussian noise. However, in systems that convert or transport energy, the noise is often not Gaussian. Instead, displacements exhibit an α-stable distribution. Such noise is commonly called Lévy noise. With such noise, we see a thermodynamics that deviates from what traditional equilibrium theory stipulates. In addition, with particles that can propel themselves, so-called active particles, we find that the rules of equilibrium thermodynamics no longer apply. No general nonequilibrium thermodynamic theory is available and understanding is often ad hoc. We study a system with overdamped particles that are subjected to Lévy noise. We pick a system with a geometry that leads to concise formulae to describe the accumulation of particles in a cavity. The nonhomogeneous distribution of particles can be seen as a dissipative structure, i.e., a lower-entropy steady state that allows for throughput of energy and concurrent production of entropy. After the mechanism that maintains nonequilibrium is switched off, the relaxation back to homogeneity represents an increase in entropy and a decrease of free energy. For our setup we can analytically connect the nonequilibrium noise and active particle behavior to entropy decrease and energy buildup with simple and intuitive formulae.

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For a stochastic COVID-19 model with jump-diffusion, we prove the existence and uniqueness of the global positive solution. We also investigate some conditions for the extinction and persistence of the disease. We calculate the threshold of the stochastic epidemic system which determines the extinction or permanence of the disease at different intensities of the stochastic noises. This threshold is denoted by ξ which depends on white and jump noises. The effects of these noises on the dynamics of the model are studied. The numerical experiments show that the random perturbation introduced in the stochastic model suppresses disease outbreak as compared to its deterministic counterpart. In other words, the impact of the noises on the extinction and persistence is high. When the noise is large or small, our numerical findings show that COVID-19 vanishes from the population if ξ < 1 ; whereas the epidemic cannot go out of control if ξ > 1 . From this, we observe that white noise and jump noise have a significant effect on the spread of COVID-19 infection, i.e., we can conclude that the stochastic model is more realistic than the deterministic one. Finally, to illustrate this phenomenon, we put some numerical simulations.

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This paper addresses the realization of almost sure synchronization problem for a new array of stochastic networks associated with delay and Lévy noise via event-triggered control. The coupling structure of the network is governed by a continuous-time homogeneous Markov chain. The nodes in the networks communicate with each other and update their information only at discrete-time instants so that the network workload can be minimized. Under the framework of stochastic process including Markov chain and Lévy process, and the convergence theorem of non-negative semi-martingales, we show that the Markovian coupled networks can achieve the almost sure synchronization by event-triggered control methodology. The results are further extended to the directed topology, where the coupling structure can be asymmetric. Furthermore, we also proved that the Zeno behavior can be excluded under our proposed approach, indicating that our framework is practically feasible. Numerical simulations are provided to demonstrate the effectiveness of the obtained theoretical results.

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In this paper, we first investigate a stochastic two-predators one-prey model with Lévy noise and distributed delays. The global dynamical behaviour is discussed. The criteria on the existence of global positive solutions, stochastic boundedness, extinction and global asymptotic stability in the mean with probability one are established. And then, the harvesting for each species is introduced to the model. The optimal harvesting strategy and the maximum of expectation of sustainable yield (MESY, for short) are further established.

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The numerical solutions to a non-linear Fractional Fokker-Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where Lévy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable Lévy distribution as solutions to the FFP equation. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.