This paper investigated the asymptotic behavior and dynamical bifurcation of a stochastic multi-strain epidemic model with jump diffusion. We defined a threshold parameter $\lambda$ as the Lyapunov exponent of the total infected population, which incorporates the stationary distribution of strain proportions on the disease-free boundary and a jump-induced correction term arising from the Lévy noise:

$$\lambda ={\int }_{\mathrm{\Delta }}[\sum _{i=1}^n({\beta }_i{y}_i-({\gamma }_i+{\mu }_i+{\eta }_i)y_i)-\frac{1}{2}(\sum _{i=1}^n{\sigma }_{2i}{y}_i{)}^2]{\mu }^{\ast }(dy)+{\int }_{\mathbb{Y}}[\ln (1+\sum _{i=1}^n{y}_i{f}_{2i}(u))-\sum _{i=1}^n{y}_i{f}_{2i}(u)]\nu (du).$$

This threshold provides a necessary and sufficient condition for overall disease persistence ( $\lambda >0$) versus extinction ( $\lambda \le 0$). Moreover, we introduced strain-specific thresholds ${\lambda }_i$ and established a competitive exclusion principle: The strain with the largest ${\lambda }_i$ dominates, while strains with smaller ${\lambda }_i$ go extinct; when two or more strains share the same maximal ${\lambda }_i$, they can coexist. Furthermore, $\lambda$ serves as a dynamical bifurcation point: When $\lambda \le 0$, the unique invariant measure is concentrated on the extinction set; when $\lambda >0$, this measure loses stability and a new invariant measure supported on the positive orthant emerges. Numerical simulations confirmed the critical role of $\lambda$ and illustrated competitive exclusion between strains under different noise intensities.

In this paper, we consider a class of Clifford-valued stochastic high-order Hopfield neural networks with time-varying delays whose coefficients are Clifford numbers except the time delays. Based on the Banach fixed point theorem and inequality techniques, we obtain the existence and global exponential stability of almost periodic solutions in distribution of this class of neural networks. Even if the considered neural networks degenerate into real-valued, complex-valued and quaternion-valued ones, our results are new. Finally, we use a numerical example and its computer simulation to illustrate the validity and feasibility of our theoretical results.

In this work, we first propose a concept of Weyl almost anti-periodic functions. Then, we make use of the contraction mapping principle and analysis techniques to research the existence of a unique Weyl almost anti-periodic solution to a neutral functional semilinear abstract differential equation. Finally, we give an example of a neutral functional partial differential equation to show the validity of the obtained results.

This paper investigated the almost periodic dynamics of a class of Clifford-valued high-order Hopfield fuzzy neural networks with time-varying delays and $D$ operators. Based on the Banach fixed point theorem, inequality techniques, and the definition of Besicovitch almost periodicity, we obtained the existence of Besicovitch almost periodic solutions for the considered neural network. The results of this paper are novel, and the method proposed in this paper can be used to study the existence of generalized almost periodic solutions and almost automorphic solutions to high-order neural networks. Finally, we provided a numerical example and computer simulation to demonstrate the effectiveness of the results obtained in this paper.

This article aimed to investigate the existence and stability of Besicovitch almost periodic ( $B_{ap}$) positive solutions for a stochastic generalized Mackey-Glass hematopoietic model. To begin with, we used stochastic analysis theory, inequality techniques, and fixed point theorems to prove the existence and uniqueness of ${\mathcal{L}}^p$-bounded and ${\mathcal{L}}^p$-uniformly continuous positive solutions for the model under consideration. Then, we used definitions to prove that this unique positive solution is also a $B_{ap}$ solution in finite-dimensional distributions. In addition, we established the global exponential stability of the $B_{ap}$ positive solution using reduction to absurdity. Finally, we provided a numerical example to verify the effectiveness of our conclusions.

This paper investigates the dynamics of high-order Hopfield neural networks incorporating fractional-order derivatives, stochastic disturbances, and time-varying delays. To address the more realistic scenario of discontinuous or weakly regular time-varying parameters, the analysis is conducted within the framework of Stepanov almost-periodicity. First, sufficient criteria for the existence and uniqueness of a Stepanov almost periodic solution in distribution for the considered network are established using Banach's fixed point theorem and inequality techniques. Subsequently, by treating the studied network as a drive system, a corresponding response system is constructed. Effective control strategies are designed to achieve finite-time synchronization between these two systems. Finally, a numerical example is provided to illustrate the validity of the theoretical results. This study offers new theoretical insights for analyzing almost periodic oscillations in complex fractional-order stochastic systems with delays and has potential applications in fields requiring precise temporal coordination, such as secure communication and cooperative control.

Octonion-valued neural networks (OVNNs) provide a powerful framework for modeling and processing high-dimensional data due to their eight-dimensional normed division algebraic structure. However, their inherent non-commutativity and non-associativity, coupled with ubiquitous time delays and stochastic disturbances in real-world systems, make the analysis of their dynamical behavior a significant challenge. This paper focuses on the complex oscillatory dynamics, specifically pseudo almost automorphy, within a class of stochastic higher-order Hopfield neural networks (NNs) based on octonions. To adequately characterize the stochastic processes involved, a novel concept of pseudo almost automorphic stochastic processes in finite-dimensional distributions is first proposed. Subsequently, by fixed point theorems and employing inequality techniques, sufficient criteria are established for the existence and global exponential stability of pseudo almost automorphic solutions in finite-dimensional distributions for the considered octonion-valued stochastic higher-order Hopfield NNs with time-varying delays. The obtained results are not only new for the octonionic system, but also remain novel even when the system degenerates to its real-valued counterpart. Furthermore, the analytical framework developed herein offers a general methodology applicable to studying pseudo almost automorphic dynamics in other types of complex-valued NNs.