We explore the most probable phase portrait (MPPP) of a stochastic single-species model incorporating the Allee effect by utilizing the nonlocal Fokker–Planck equation (FPE). This stochastic model incorporates both non-Gaussian and Gaussian noise sources. It has three fixed points in the deterministic case. One is the unstable state, which lies between the two stable equilibria. Our primary focus is on elucidating the transition pathways from extinction to the upper stable state in this single-species model, particularly under the influence of jump-diffusion noise. This helps us to study the biological behavior of species. The identification of the most probable path relies on solving the nonlocal FPE tailored to the population dynamics of the single-species model. This enables us to pinpoint the corresponding maximum possible stable equilibrium state. Additionally, we derive the Onsager–Machlup function for the stochastic model and employ it to determine the corresponding most probable paths. Numerical simulations manifest three key insights: (i) when non-Gaussian noise is present in the system, the peak of the stationary density function aligns with the most probable stable equilibrium state; (ii) if the initial value rises from extinction to the upper stable state, then the most probable trajectory converges towards the maximally probable equilibrium state, situated approximately between 9 and 10; and (iii) the most probable paths exhibit a rapid ascent towards the stable state, then maintain a sustained near-constant level, gradually approaching the upper stable equilibrium as time goes on. These numerical findings pave the way for further experimental investigations aiming to deepen our comprehension of dynamical systems within the context of biological modeling.

In the present work, we investigate the dynamics of the infinite-dimensional stochastic partial differential equation (SPDE) with multiplicative white noise. We derive the effective equation on the approximate slow manifold in detail by utilizing a finite-dimensional stochastic differential equation (SDE) describing the motion of interfaces. In particular, we verify the equivalence between the full SPDE and the coupled system under small stochastic perturbations. Moreover, we apply our results to effective dynamics of stochastic models with multiplicative white noise, illustrated with four examples on the stochastic damped wave equation, the stochastic Allen-Cahn equation, the stochastic nonlinear Schrödinger equation and the stochastic Swift-Hohenberg equation.

Stochastic vegetation-water dynamical systems play a pivotal role in ecological stability, biodiversity, water resource management, and adaptation to climate change. This research proposes a machine learning-based method for analyzing rare events in stochastic vegetation-water dynamical systems with multiplicative Gaussian noise. Utilizing the Freidlin-Wentzell large deviation theory, we derive the asymptotic expressions for the quasipotential and the mean first exit time. Based on the decomposition of vector field, we design a neural network architecture to compute the most probable transition paths and the mean first exit time for both non-characteristic and characteristic boundary scenarios. The results indicate that this method can effectively predict early warnings of vegetation degradation, providing new theoretical foundations and mathematical tools for ecological management and conservation. Moreover, the method offers new possibilities for exploring more complex and higher-dimensional stochastic dynamical systems.

This study aims to examine the effect of Lévy noise on the solutions of the nonlinear Schrödinger equation. An improved diversity of stochastic solutions is instinctively located discretely on certain conditions by applying the generalized Kudryashov method. Moreover, the dynamical behaviors of these exact results of the nonlinear Schrödinger equation are interpreted in the context of the effect of Lévy noise. Even mathematical evaluations have been conducted and presented.

Lévy noise influences diverse non-equilibrium systems across scales, including quantum devices, active biological matter, and financial markets. While such noise is pervasive, its overall impact on activated transitions between metastable states remains unclear, despite prior studies of specific noise forms and scaling limits. In this work, we introduce a unified framework for Lévy noise defined by its finite intensity and independent stationary increments. By identifying the most probable transition paths as minimizers of a stochastic action functional, we derive analytical scaling laws for escape rates under weak noise, thereby extending the classical Arrhenius law. Our results demonstrate that Lévy noise universally enhances escape efficiency by reducing the effective potential barrier compared to Gaussian noise with equivalent intensity. Strikingly, even vanishingly weak Lévy noise can exponentially increase escape rates across a broad range of amplitude distributions. This phenomenon arises from discontinuous most probable transition paths, where escape occurs via finite jumps. We validate these paths through the cumulant-generating function, a path integral representation, the mean first passage time and numerical simulations. Our findings reveal fundamental distinctions in escape dynamics under thermal and athermal fluctuations, suggesting new strategies to optimize switching processes in metastable systems through engineering noise properties.

In this work, a Josephson junction consisting of two superconducting layers sandwiching an insulating layer is explored, which is subject to the effects of thermal fluctuations. The precise expressions for the evolution of Josephson phase and the supercurrent through the junction are derived. A clockwise hysteresis cycle in the current-voltage characteristic curve is demonstrated mathematically. Additionally, the bifurcation of a planar limit cycle is established. The numerous stochastic thermodynamic properties of the resistively shunted superconducting tunnel junction system are described, considering the influence of three specific parameters: the conductance, the current bias and the noise intensity. Moreover, the probability density is characterized using the Fokker-Planck equation.