Search Results (60)

Within a circular symmetric domain, we investigate the corresponding dynamic behaviors of a reaction-diffusion predator-prey system with nonlocal competition. First, sufficient conditions for the local stability of the steady state are established and criteria for the occurrence of various bifurcations are derived. Furthermore, based on the center manifold method and normal form theory, the third order normal form formula for the Turing-Hopf bifurcation is refined, which adapts to the systems with nonlocal competing terms of a wide class of kernel functions in principle in a circular domain. Finally, a series of numerical simulations are performed to validate the analytical results and visually illustrate the rich spatiotemporal patterns by nonlocal competition. Full article

In this paper, we develop a two-strain SAIR epidemic model with asymptomatic transmission to investigate the mechanisms governing strain competition and coexistence. The basic reproduction numbers are derived, and threshold conditions for disease extinction and persistence are established. When the reproduction numbers differ, the strain with the larger value dominates, leading to competitive exclusion. In contrast, when the two strains have identical transmission potential, the model admits a continuum of endemic equilibria, representing a regime of neutral competition. The global dynamics of the system are rigorously characterized using Lyapunov functions and center manifold theory, including the stability of the disease-free equilibrium, boundary equilibria, and the coexistence equilibrium set. Numerical simulations are performed to support the analytical results and to illustrate the effects of key parameters on the system dynamics. These findings reveal how asymptomatic transmission and parameter balance shape multi-strain interactions, providing new insights into the persistence and coexistence of competing pathogens. Full article

To investigate the underlying mechanisms of talent mobility in higher-education institutions influenced by factors such as the development environment, macroeconomic policies, and evaluation mechanisms, this paper proposes a nonlinear stochastic differential equation (SDE) dynamical model that incorporates time delays and multiplicative noise. We analyze the dynamic processes of talent mobility under varying conditions regarding the number of nodes, policy implementation cycles, and noise intensity. First, we employ central manifold theory and stochastic averaging methods to reduce the system to a one-dimensional averaged $It\widehat{o}$ equation. Subsequently, with $\tau$ as a parameter, we conduct an in-depth study of the system’s stochastic bifurcation behavior using the corresponding Fok–Planck–Kolmogorov equations. Finally, we validate the theoretical conclusions through numerical simulations. The results indicate that the number of nodes, policy delay, and noise intensity all have significant effects on system stability; an increasing delay induces random P-bifurcation in the system, and when $N\le 3$ and $N>3$, the system exhibits distinctly different steady-state behaviors. We also found that excessively high noise intensity disrupts system stability, whereas moderate noise intensity has a positive effect on stability. This study not only provides theoretical insights into the dynamic evolution mechanisms of talent mobility in regional universities but also offers valuable guidance for universities in formulating talent recruitment and evaluation policies. The methodology employed in this study opens up a promising avenue for analyzing complex dynamic problems in the field of sociology. Full article

In this study, we examine the effectiveness of combining interleukin-2 (IL-2) with highly active antiretroviral therapy (HAART) in controlling HIV replication. A mathematical model of the immune system is developed to analyze immune recovery when IL-2 is administered alongside HAART. We investigate the stability of the endemic equilibrium and Hopf bifurcation and determine the direction and stability of periodic solutions using center manifold theory. Numerical simulations are conducted to support the theoretical findings. The results show that the disease-free equilibrium is stable when the basic reproduction number $R_0<1$, while the endemic equilibrium exists when $R_0>1$. Our results also reveal the presence of a subcritical Hopf bifurcation in the system. An optimal control problem is also studied, showing that the combined therapy of IL-2 and HAART improves treatment outcomes, reduces side effects, and has a unique optimal control pair. Sensitivity analysis further highlights the importance of system parameters in influencing treatment effectiveness. Full article

The aim of this research is to formulate and analyze a modified $S{I}_{p}IVR$ mathematical model to study the transmission dynamics of poliovirus and assess the impact of vaccination on disease control. The proposed model extends classical SEIV-type frameworks by incorporating a recovered compartment with long-term immunity and by replacing the traditional exposed class with a pre-infectious compartment ($I_p$) that captures silent viral shedding during the incubation phase of poliovirus. This modification addresses the critical epidemiological feature that individuals can transmit the virus before showing symptoms while maintaining biological accuracy in compartment definition. Several fundamental analytical properties are rigorously established, including positivity, boundedness, and the existence of a biologically meaningful invariant region. The basic reproduction number $R_0$ is derived using the next-generation matrix approach, and comprehensive stability analysis is carried out. The analysis shows that the DFE is locally and globally asymptotically stable whenever $R_0<1$. Using center manifold theory, a forward bifurcation is rigorously demonstrated, indicating that disease persistence emerges smoothly as $R_0$ crosses unity. Local and global sensitivity analyses of the basic reproduction number $R_0$ identify critical epidemiological parameters, and points to vaccination coverage and transmission rates as key drivers of outbreak dynamics. Numerical simulations confirm the analytical results and illustrates two different epidemiological scenarios, one with $R_0<1$, and another with $R_0>1$ along with neural network analysis by using the same data from both cases in a built-in function package in MATLAB-2020 software. It utilizes all of its hidden layers to check the data used by the model for validation performance and training and to find the absolute and mean squared errors. It also shows how vaccination suppresses the spread of infection. These findings provide a strong mathematical basis for public health policy, offering strategic insight into how vaccination campaigns might be optimized to accelerate progress toward global polio eradication. Full article

We analyze the population dynamics of a microbial cross-protection model and derive the exact conditions under which a Fold–Hopf bifurcation emerges. By applying center-manifold reduction and normal-form theory, we reduce the infinite-dimensional delay differential system to a finite-dimensional ordinary differential system, enabling rigorous bifurcation analysis. Numerical simulations reveal a rich repertoire of dynamical behaviors, including stable equilibria, sustained oscillations, and noise-induced irregularities. Our findings identify time-delay-induced Fold–Hopf bifurcation as a fundamental mechanism driving oscillatory coexistence in cross-protection mutualisms, for previously reported experimental observations. Full article

Dysbiosis of the gut microbiota and dysregulated immune responses are key pathological features in both the onset and progression of Crohn’s disease. We propose a phagocyte–bacteria diffusion model with a time delay to explore their dynamic interactions and impact on the progression of Crohn’s disease. We first supplement the proof of the positivity, boundedness, existence, uniqueness, and global stability of the solutions for the ordinary differential system without time delay. Then we examine the stability of the positive equilibrium point and the occurrence of a Hopf bifurcation. By applying normal form and center manifold theory, we determine the direction of the bifurcation and the stability of the bifurcating periodic solution. Numerical simulations are used to verify the theoretical results. We find that the time delay significantly slows the system’s approach to a steady state. With a fixed delay, increased intestinal permeability prolongs the stabilization time. Conversely, with fixed intestinal permeability, a larger delay renders the system more prone to oscillations. Furthermore, a higher maximum engulfment rate by phagocytes reduces bacterial biomass but prolongs stabilization, whereas an increased phagocyte death rate shortens it. Additionally, an elevated bacterial growth rate increases both the bacterial biomass and the stabilization time. These results enhance our understanding of the dynamic equilibrium in immune systems. Full article

This study examines a discrete-time predator–prey model constructed via piecewise constant discretization of its continuous counterpart. Through comprehensive qualitative and dynamical analyses, we reveal a rich set of nonlinear phenomena, encompassing Neimark–Sacker bifurcation, flip bifurcation, and codimension-two bifurcations corresponding to 1:2, 1:3, and 1:4 resonances. Rigorous analysis of these bifurcation scenarios, conducted via center manifold theory and bifurcation methods, establishes a robust mathematical framework for their characterization. Numerical simulations corroborate the theoretical predictions, exposing intricate dynamical phenomena such as quasiperiodic oscillations and chaotic attractors. Our results demonstrate that resonance-driven bifurcations are potent drivers of ecological complexity in discrete systems, acting as key determinants that orchestrate the emergent dynamics of populations—a finding with profound implications for interpreting patterns in real-world ecosystems subject to discrete generations or seasonal pulses. Full article

This study investigates the dynamic behaviors of the discrete epidemic model influenced by media coverage through integrated analytical and numerical approaches. The primary objective is to quantitatively assess the impact of media coverage on disease outbreak patterns using mathematical modeling. Firstly, the Euler method is used to discretize the model (2), and the periodic solution is strictly analyzed. Secondly, the coefficients and conditions of restricted flip and Neimark–Sacker bifurcation are studied by using the center manifold theorem and bifurcation theory. By calculating the largest Lyapunov exponent near the critical bifurcation point, the occurrence of chaos and limit cycles is proved. On this basis, the chaotic control of the system is carried out by using state feedback and hybrid control. Under certain conditions, the chaos and bifurcation of the system can be stabilized by control strategies. Numerical simulations further reveal bifurcation dynamics, chaotic behaviors, and control technologies. Our results show that media coverage is a key factor in regulating the intensity of disease transmission and chaos. The control technology can effectively prevent the large-scale outbreak of epidemic diseases. Importantly, enhanced media coverage can effectively promote public awareness and defensive behaviors, thereby contributing to the mitigation of disease transmission. Full article

In this paper, a predator–prey model with hunting cooperation and maturation delay is studied. Through theoretical analysis, we investigate the existence of multiple stability switches of the positive equilibrium. By applying Hopf bifurcation theory, the conditions for Hopf bifurcation are derived, indicating the emergence of periodic solutions as the maturation delay passes through critical values. Utilizing center manifold theory and normal form analysis, we determine the stability and direction of the bifurcating orbits. Numerical simulations are performed to validate the theoretical results. Furthermore, the simulations vividly demonstrate the appearance of period-doubling bifurcations, which is the onset of chaotic behavior. Bifurcation diagrams and phase portraits are employed to precisely characterize the transition processes from a stable equilibrium to periodic, period-doubling solutions and chaotic states under different maturation delay values. The study reveals the significant influence of maturation delay on the stability and complex dynamics of predator–prey systems with hunting cooperation. Full article

We explore chaos, local dynamics, codimension-one, and codimension-two bifurcations of an asymmetric discrete predator–prey model. More precisely, for all the model’s parameters, it is proved that the model has two boundary fixed points and a trivial fixed point, and also under parametric conditions, it has an interior fixed point. We then constructed the linearized system at these fixed points. We explored the local behavior at equilibria by the linear stability theory. By the series of affine transformations, the center manifold theorem, and bifurcation theory, we investigated the detailed codimensions-one and two bifurcations at equilibria and examined that at boundary fixed points, no flip bifurcation exists. Furthermore, at the interior fixed point, it is proved that the discrete model exhibits codimension-one bifurcations like Neimark–Sacker and flip bifurcations, but fold bifurcation does not exist at this point. Next, for deeper understanding of the complex dynamics of the model, we also studied the codimension-two bifurcation at an interior fixed point and proved that the model exhibits the codimension-two 1:2, 1:3, and 1:4 strong resonances bifurcations. We then investigated the existence of chaos due to the appearance of codimension-one bifurcations like Neimark–Sacker and flip bifurcations by OGY and hybrid control strategies, respectively. The theoretical results are also interpreted biologically. Finally, theoretical findings are confirmed numerically. Full article

In this paper, we investigate the nonlinear dynamic behavior of a cart–double pendulum system with single time delay feedback control. Based on the center manifold theorem and stochastic averaging method, a reduced-order dynamic model of the system is established, with a focus on analyzing the influence of time delay parameters and stochastic excitation on the system’s Hopf bifurcation characteristics. By introducing stochastic differential equation theory, the system is transformed into the form of an Itô equation, revealing bifurcation phenomena in the parameter space. Numerical simulation results demonstrate that decreasing the time delay and increasing the time delay feedback gain can effectively enhance system stability, whereas increasing the time delay and decreasing the time delay feedback gain significantly improves dynamic performance. Additionally, it is observed that Gaussian white noise intensity modulates the bifurcation threshold. This study provides a novel theoretical framework for the stochastic stability analysis of time delay-controlled multibody systems and offers a theoretical basis for subsequent research. Full article

This paper studies the effects of resource limitations, immunity decay, and delays on an Ebola epidemic model and an optimal control strategy. The model includes two types of delays: a delay in the incubation period of infected individuals and a delay in treatment. Conditions for a Hopf bifurcation at the endemic equilibrium are verified, with its direction and stability analyzed via normal form theory and the center manifold theorem. We also studied the optimal control problem for the SIRD delay model using educational campaigns and Ebola survivors’ immunity as control variables. Furthermore, we formulate an optimization problem based on Pontryagin’s maximum principle. This problem uses a modified Runge-Kutta approach with delays to discover the best control strategy to reduce infections and intervention costs. Finally, simulation results confirm analytical conclusions and show the practical implications of the optimum Ebola control plan using the dde23 MATLAB R2024a built-in solver and DDE-Biftool. Full article

The Basener and Ross mathematical model is widely recognized for its ability to characterize the interaction between the population dynamics and resource utilization of Easter Island. In this study, we develop and investigate a discrete-time version of the Basener and Ross model. First, the existence and the stability of fixed points for the present model are investigated. Next, we investigated various bifurcation scenarios by establishing criteria for the occurrence of different types of codimension-one bifurcations, including flip and Neimark–Sacker bifurcations. These criteria are derived using the center manifold theorem and bifurcation theory. Furthermore, we demonstrated the existence of codimension-two bifurcations characterized by 1:2, 1:3, and 1:4 resonances, emphasizing the model’s complex dynamical structure. Numerical simulations are employed to validate and illustrate the theoretical predictions. Finally, through bifurcation diagrams, maximal Lyapunov exponents, and phase portraits, we uncover a diversity of dynamical characteristics, including limit cycles, periodic solutions, and chaotic attractors. Full article

This paper investigates the dynamical behavior of a discrete fractional-order modified Leslie–Gower model with a Michaelis–Menten-type harvesting mechanism and a Holling-II functional response. We analyze the existence and stability of the nonnegative equilibrium points. For the interior equilibrium points, we study the conditions for period-doubling and Neimark–Sacker bifurcations using the center manifold theorem and bifurcation theory. To control the chaos arising from these bifurcations, two chaos control strategies are proposed. Numerical simulations are performed to validate the theoretical results. The findings provide valuable insights into the sustainable management and conservation of ecological systems. Full article