JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Dynamic analysis of a cholera epidemic model with Markov switching

LIAO Shu, XIA Nannan, DUAN Wenlong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 1-17. doi:10.6040/j.issn.1671-9352.0.2024.422

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A stochastic cholera epidemic model with Markov switching and affected by white noise is studied, with general transmission rates and isolation measures. Firstly, the existence and uniqueness of the global positive solution of the model are obtained. Secondly,by constructing appropriate stochastic Lyapunov functions,the sufficient conditions R_e for the extinction of the disease and R_c for the stationary distribution are obtained. When R_c>1, the model has a unique ergodic stationary distribution;when R_e<1, the disease dies out exponentially. Finally,numerical simulations are used to verify the above theoretical results. The results show that higher white noise intensity can lead to the extinction of infectious diseases. If the disease persists in one state and becomes extinct in another, whether the disease ultimately becomes extinct or persists depends on the probability distribution of the Markov chain across each state. Additionally,isolation measure is also an important method for epidemic prevention and control.

Dynamics analysis of infectious diseases transmission with vaccination on complex networks

ZHAO Yaqi, ZHANG Ruixia

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 18-32. doi:10.6040/j.issn.1671-9352.0.2024.369

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In this paper, based on complex networks, an SVAIR epidemic model with vaccination is established, in which the asymptomatic infected person(A)and the infected person (I)have different infectivity. The basic reproduction number R0 is calculated by the next generation matrix. By using the Lyapunov function and monotone iterative, it is proved that when R₀<1, the disease-free equilibrium is globally asymptotically stable. When R0>1, the endemic equilibrium point exists and is unique, the system is uniformly persistent and the endemic equilibrium point is globally attractive. Finally, we choose the scale-free network for sensitivity analysis and numerical simulation to verify the theoretical results, and results indicate that improving vaccine efficacy and vaccination rate can better control the spread of infectious diseases.

Pattern dynamics analysis for a predator-prey system considering prey refuge and schooling behavior

JIANG Yuan, LIU Hui, OUYANG Miao, SHEN Pei

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 33-44. doi:10.6040/j.issn.1671-9352.0.2025.070

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This paper considers a cross-diffusive predator-prey model incorporating refuge effects and schooling behavior, revealing the regulatory mechanism of refuge effects on both predator and prey populations. It is shown that as the refuge parameter increases, the schooling behavior of predator and prey populations becomes more pronounced. By employing a suitable refuge threshold parameter, the spatial dynamics of the system are investigated, and a series of spatiotemporal patterns are observed, including spot-like or hexagonal, stripe and mixed spot-stripe patterns. Specifically, linear stability analysis is applied to derive the conditions for Turing instability. Subsequently, multiple-scale analysis is utilized to derive amplitude equations near the Turing bifurcation critical point, and the mechanism and stability of pattern formation are studied. Theoretical results are verified through numerical simulations. Provides a theoretical basis for regulating population size and spatial patterns in species with collective behaviors.

Existence of martingale solution to stochastic cross-diffusion population-toxicant model

DU Yanyan, WANG Zong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 45-57. doi:10.6040/j.issn.1671-9352.0.2024.339

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In this paper, a random population-toxicant model with cross diffusion is established, and the approximate solution of the random cross diffusion model in finite dimensional space is obtained by using Galerkin finite element approximation method. The existence of the martingale solution in Hilbert space is proved by analyzing the existence and uniqueness, tightness criterion and weak convergence of the approximate solution.

Dynamics analysis of predator-prey model with fear effects and hunting cooperation

LI Zhiyuan, LI Zhaoxin, JIANG Yumei, ZHANG Daoxiang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 58-69. doi:10.6040/j.issn.1671-9352.0.2025.059

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In ecosystems, predator-induced fear can suppress prey reproduction, while cooperative behavior among species is a widespread phenomenon. This paper establishes a predator-prey model incorporating fear effects, hunting cooperation, and harvesting. For the non-spatial model, we prove the positivity and boundedness of solutions, derive sufficient conditions for the existence and local stability of all equilibrium points, and demonstrate the global stability of the interior equilibrium by constructing an appropriate Lyapunov function. Furthermore, we explore local bifurcations with respect to key parameters: trans-critical bifurcation is proven via Sotomayor's theorem, and Hopf bifurcation is analyzed under the hunting cooperation parameter. For the spatial model, we provide a detailed stability analysis, investigate the conditions for Turing instability, and identify various Turing patterns. The biological implications of these patterns in the two-dimensional spatial model are discussed. Finally, numerical simulations are conducted to validate the analytical findings for both non-spatial and spatial models.

Study on astochastic SIR model with noise interference in higher-order networks

LI Siyu, GUO Xiang, LIU Maoxing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 70-81. doi:10.6040/j.issn.1671-9352.0.2025.191

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To explore how random noise affects the spread of infectious diseases in higher-order networks, this study develops a stochastic SIR epidemic model rooted in simplicial complexes. Firstly, the model constructs a social network of node connections using simplicial complexes, leverages mean field theory to model the temporal evolution of network nodes, and integrates white noise perturbations into the transmission dynamics. Then, through dynamic analysis, the disease outbreak threshold is deduced, establishing the existence and uniqueness of the systems global positive solution under specific conditions, alongside proving disease extinction and asymptotic oscillations of the solution. Finally, simulations on two real-world networks confirm the role of noise intensity in disease spread. Critically, networks with differing topological properties exert distinct influences on transmission: higher average degrees of simplices expedite disease propagation.

Forced waves in a predator-prey lattice differential system in a shifting environment

TONG Maosen, ZHOU Rong

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 82-92. doi:10.6040/j.issn.1671-9352.0.2024.297

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In this paper, we study the existence and nonexistence of forced waves in a class of predator-prey lattice differential models in a shifting environment. Firstly, we prove the existence of forced waves for this model by constructing suitable upper and lower solutions and applying Schauder fixed point theorem. Then, we prove the nonexistence of forced wave by using the method of proof by contradiction.

Spreading speed of a class of Field-Road models with nonlocal dispersal

AN Da, ZHANG Xiao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 93-107. doi:10.6040/j.issn.1671-9352.0.2024.412

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To address the limitations of traditional local dispersal models in characterizing long-distance propagation, this paper studies a class of Field-Road models with nonlocal dispersal, building on the Field-Road model with local dispersal proposed by Berestycki et al. Firstly, we establish the threshold of fast diffusion roads to accelerate spreading speed of species and prove the existence of spreading speed along the road direction by constructing appropriate super-and sub-solutions and combining the comparison principle. Secondly, the influences of the diffusion rate of species on the road, the support radius of the nonlocal dispersal kernel function and the convection intensity on spreading speed are studied. The results show that the three parameters all accelerate spreading speed and it maintains the same order linear growth with the square root of the diffusion rate, the support radius and the convection intensity.

Optimal control for predator-prey system with size structure in a polluted environment

ZHANG Tainian

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 108-122. doi:10.6040/j.issn.1671-9352.0.2024.393

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We investigate the optimal control problem for a predator-prey system that depends on individual size in a polluted environment. The control variables include fertility and the input rate of exogenous toxicants. The optimality conditions for various problems-free terminal, infinite horizon, and constrained end point problem on fixed horizon-are derived using the theory of tangent-normal cones, the Dubovitskii-Milyutin theorem, and the adjoint system technique. These results offer theoretical underpinnings for controlling environmental pollution, protecting biodiversity, and scientifically exploiting biological resources.

Global regularity of weak solutions to multi-phase elliptic problems

MA Menglu, YAO Jiahui, TONG Yuxia

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 123-133. doi:10.6040/j.issn.1671-9352.0.2024.376

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Global regularity of weak solutions to the nonlinear elliptic equations corresponding to multi-phase functionals is considered. By using the Young inequality, the Hölder inequality, the Sobolev-Poincaré inequality and the Gehring lemma, the integrable exponent is improved. The global regularity of the weak solution of the equation is obtained.

Adaptive finite-time control of nonlinear systems based on time-varying powers and parametric uncertainties

ZHAO Ran, JIANG Mengmeng, NIU Xiao

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2026, 61(7): 134-144. doi:10.6040/j.issn.1671-9352.0.2024.428

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For a class of nonlinear systems with parameter uncertainty, the adaptive finite time control problem is studied. The powers of the system are time-varying functions instead of constants, and the nonlinear functions contain parametric uncertainties. In order to solve this problem, an adaptive state feedback controller is designed by cleverly combining the Lyapunov function, the symbolic function, adding a power integrator and adaptive control method, which can ensure that the system with parameter uncertainties and time-varying powers is semi-global practical finite time stability. Finally, the simulation results verify the feasibility and effectiveness of the control method proposed in this study.

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