In this paper, we establish a new class of dynamic inequalities involving Hardy–Steklov and Copson–Steklov type operators on time scales. The obtained results provide a unified treatment of the continuous and discrete settings, recovering known integral inequalities in the continuous case and yielding new inequalities in the discrete case. In this way, our results extend classical Hardy-type inequalities to a broader dynamic framework and demonstrate the versatility of the time-scales approach in the study of operator inequalities.
We introduce a new epidemic model that combines the Beddington-DeAngelis incidence mechanism with the Holling type II treatment response. The Beddington-DeAngelis formulation accounts for interactions between susceptible and infected individuals, capturing behavioral effects and inhibitory transmission controls. Meanwhile, the Holling type II treatment rate reflects healthcare saturation due to constrained medical resources during the disease outbreaks. We first establish the model’s well-posedness and then perform a thorough dynamical analysis, addressing both local and global stability of feasible equilibria. Codimension-one bifurcations-including transcritical, Hopf, and saddle-node are examined to identify qualitative shifts in system dynamics. A parameter sensitivity assessment highlights the key drivers influencing disease transmission. Using Pontryagin’s Maximum Principle, we evaluate optimal control strategies for disease mitigation. Our results reveal that rapid information dissemination through social media can destabilize the system, inducing limit cycle oscillations. We also study seasonal fluctuations in social media growth rate, finding that the nonautonomous system supports a globally attractive positive periodic solution when the growth rate remains within a critical interval. Exceeding this threshold breaks global attractivity and triggers transitions to one-periodic, multi-periodic, or chaotic behavior. Overall, the findings suggest that information-based interventions are highly effective and economical in the early phase of disease outbreak, whereas treatment-focused strategies are more impactful for long-term control. Notably, information-driven behavioral responses remain crucial, particularly when pharmaceutical options are limited.
A natural question is how changing the diffusion or reaction terms in the equation changes the structure of the traveling wave solution of the Fisher-KPP equation. This paper answers the question of how the addition of a nonlinear term expressing the Allee effect changes the classification of information about the existence and shape of the traveling wave. In addition to the details of the information for the front-type traveling wave solution, the existence and characterization of traveling waves that are unbounded at the endpoints of a finite interval are precisely analyzed. The results for this classification are obtained by the two-dimensional phase space including to infinity, which corresponds to the dynamics of a two-dimensional ordinary differential equation satisfied by the traveling wave. These are derived by the Poincaré compactification, one of the compactifications of phase space. Based on the results of the Fisher-KPP reaction-diffusion equation for traveling waves in the linear diffusion case, the author’s previous work can be integrated to discuss the effect of the Allee effect on traveling waves. In addition, a comparison and discussion of how the solution structure of the traveling wave changes with changes in the diffusion and reaction terms is given.
This study investigates the impact of individual movement and saturated treatment on the spatio-temporal dynamics of an infectious disease through an SIR reaction-diffusion model with self-diffusion under zero-flux boundary conditions. The well-posedness of the proposed model, including the existence, positivity, and boundedness of solutions, is established using partial differential equation techniques. The corresponding ODE system is analyzed for local stability, Hopf bifurcation, and Bautin bifurcation via normal form theory and Lyapunov coefficients. In addition, the spatial model is examined for diffusion-driven instabilities, including Hopf, Turing, and Turing-Hopf bifurcations. Numerical simulations performed using a finite difference scheme reveal the emergence of diverse spatial patterns such as holes, stripes, and mixed structures, which validate the theoretical findings. The results demonstrate how diffusion and saturated treatment influence disease persistence, oscillatory behavior, and spatial pattern formation in epidemic dynamics.
In this paper, we construct a discrete weighted h-Hartley-cosine convolution operator on the time scale 𝕋_h . We then establish the boundedness of this operator on the space l_1(𝕋_h) and prove the associated factorization identities. As an application, we derive l_1(𝕋_h) -solvability results for certain classes of integral equations and systems of integral equations on 𝕋_h . In addition, from a signal-processing viewpoint, we interpret the proposed generalized convolution as a discrete filtering mechanism and illustrate it on the ECG5000 dataset.
In this paper, we have taken into account a model that incorporates two memory integrals with weak memory kernel and is delimited by a boundary consisting of Dirichlet boundary and no-flux boundary. We prove the existence of the global attractor and obtain conditions for the global stability of trivial steady-state via constructing the Lyapunov functional. Subsequently, the stability of the trivial steady-state solutions and associated bifurcation for this model are extensively investigated. It turns out that, for small parameter μ , the solutions of the equation will exhibit various dynamical behaviors when the parameters R and λ are altered. For the inhomogeneous steady-state solution bifurcated from the trivial solution, we also provided the conditions for its stability and instability.
Partial differential equations (PDEs) are essential in chemical engineering and other applied sciences for modeling complex systems, such as the glycolysis biochemical pathway. This research extends a well-established glycolysis model, originally formulated as a system of ordinary differential equations (ODEs) into a reaction-diffusion system described by PDEs. The study introduces and analyzes a nonlinear system of two PDEs representing autocatalytic glycolysis with spatial diffusion. Homogeneous Neumann boundary conditions are applied, and a discretization method approximates the system. The goal is to achieve finite-time stability and synchronization of the discrete glycolysis model. Finite-time stability ensures that the concentrations within the system converge to equilibrium within a finite period, providing rapid and precise control over the system’s behavior. The concept of finite-time synchronization is also explored, aiming for all system components to achieve synchronization within a defined time frame. The research includes stability analysis using Lyapunov functions, eigenvalue theory, and finite-time stability theorems, providing a comprehensive approach to understanding and controlling the dynamics of the glycolysis reaction-diffusion system.
In this paper, the initial boundary value problem of the wave equation with source term and logarithmic damping term is considered. The global existence of solutions is proved by means of semigroup theory combined with the stable set method. By Nakao’s inequality combined with some special algebra inequalities for logarithmic function, the exponential stability of solutions is obtained for two classes of logarithmic damping terms. These results improve upon those in the previous literature.
This study explores a predator-prey model operating in continuous time. The model incorporates a density-dependent mortality term for the predator population that reflects the influence of a double Allee effect. The proposed model is investigated analytically and numerically, with a particular focus on the impact of both the Allee effect and the density-dependent mortality. The analysis reveals that the system can exhibit bi-stability when coexistence equilibria exist. Interestingly, the model can even exhibit tri-stability, characterized by the presence of two stable coexistence equilibria, along with a stable predator-free equilibrium point. The ecological implications of bubbling and hydra phenomena within the system are thoroughly examined. Due to the double Allee effect, the emergence of bubbling cycles, characterized by increasing and decreasing amplitudes, is explored. All possible local bifurcations are identified and illustrated using one- and two-parametric bifurcation diagrams. A sensitivity analysis is performed to investigate how key parameters affect the biomass balance of the interacting species. Furthermore, diffusion-driven instability analysis is employed to derive pattern formation conditions for the predator-prey system. Numerical simulations are conducted to reveal rich Turing patterns with complex self-organized structures under the combined influence of diffusion and the Allee effect in the spatiotemporal domain. The simulations demonstrate a gradual dynamical shift from diffusion-driven patterns to Allee effect-driven patterns as the Allee parameters are adjusted. Additionally, the classification of irregular patches within the spatiotemporal dynamics allows for the identification of non-Turing patterns that may arise. Finally, the proposed reaction-diffusion framework provides a unified setting for investigating complex dynamical behaviours such as bubbling and hydra effects and offers new insights into their spatiotemporal manifestations.
We develop a multi-variant, reaction–diffusion epidemic framework that couples mutation, spatial movement, vaccination, and symptomatic/asymptomatic transmission for SARS-CoV-2. The model tracks {S,E_i,I_i,1,I_i,2,R_i,V} over Ω⊆ℝ^d with strain-specific infection, recovery, and disease-induced mortality, and with mutation flows between exposed classes. Methodologically, we prove positivity and boundedness, characterize the disease-free equilibrium (DFE) with vaccination, and derive a next-generation reproduction number R_0 that explicitly accounts for mutation and spatial diffusion. Using Lyapunov techniques, we establish global stability of the DFE when R_0<1 , show existence and stability of endemic equilibria when R_0>1 , and provide conditions under which a higher- R_0 mutant becomes dominant. We also conduct sensitivity analysis and incorporate spatially varying vaccination ω (y) . Our results yield interpretable thresholds and dominance criteria that help: (i) anticipate strain replacement under given mutation profiles; (ii) quantify how mobility (diffusion) and immunity loss shift peak timing and size; and (iii) design spatially targeted vaccination strategies that lower effective R_0 and suppress mutant establishment. Simulations illustrate how tuning ω (y) and limiting opportunities for mutation (via faster case resolution or reduced contact) change epidemic trajectories and reduce severe disease burden. The framework thus provides a transferable tool for prioritizing interventions across space and variant landscapes.
This study develops and analyzes a mathematical model for the co-infection dynamics of malaria and typhoid fever in humans, explicitly incorporating waning immunity and enhanced susceptibility due to prior infection. Co-infection arises because both diseases are driven by similar environmental and socio-economic factors, increasing the likelihood that individuals exposed to one disease become susceptible to the other. We derive the basic reproduction numbers for malaria-only, typhoid-only, and the coupled co-infection systems and investigate their stability properties, including the possibility of backward bifurcation. An optimal control framework is formulated and analyzed using Pontryagin’s Maximum Principle to identify cost-effective intervention strategies. Model parameters are estimated using least-squares fitting to reported case data from Delhi, India, and sensitivity analysis is performed to quantify the influence of one disease on the other. Our results indicate that malaria infection significantly increases susceptibility to typhoid, whereas typhoid infection has a comparatively weaker effect on malaria acquisition.
In this study, we establish the existence of a renormalized solution for nonlinear parabolic problems characterized by measure data within the context of Musielak spaces. Our analysis involves the Leray-Lions operator, which maps from W^1,x_0 L_φ(Q) to its dual space, and includes two lower order terms. In this framework, φ represents a Musielak function. It is important to note that the two lower-order terms, G(x,t,ϖ ,∇ϖ ) and h(x,t,∇ϖ ) , satisfy only a growth condition. One for G is limited by φ , while the other for h is bounded by γ_x^-1γ _x(∇ϖ ) , where γ also denotes a Musielak function ( γ grows essentially less rapidly than φ at 0).
Based on the promoting effect of cannibalistic behavior on the egg-laying rate of adult individuals, this study constructs a dual-delay predator-prey system incorporating stage structure and Holling-II functional response, with both behavioral response delay and gestation delay. The objective is to reveal the influence mechanism of multi-delay coupling on population dynamics and ecosystem stability. Firstly, under a no-delay condition, the positive invariant set of the system is established, and the existence and local asymptotic stability of the positive equilibrium point within this set are analyzed, laying the foundation for subsequent delayed system studies. Secondly, using the cannibalism rate as a bifurcation parameter, the triggering conditions and dynamic evolution of Hopf bifurcation in the delay-free system are explored. Thirdly, for the dual-delay system, the conditions for Hopf bifurcation induced by either or both types of delays are derived. By employing the center manifold theorem and normal form theory, the key parameters determining the nature of Hopf bifurcation are obtained. Finally, numerical simulations are conducted to verify the correctness of the theoretical results.
This paper deals with the existence of normalized solutions for a Schrödinger system of Choquard type with Sobolev critical nonlinearities { -Δ u = λ _1 u + μ _1(I_α *|u|^p)|u|^p -2u+β r_1|u|^r_1-2u|v|^r_2, -Δ v = λ _2 v + μ _2(I_α *|v|^q)|v|^q -2v+β r_2|u|^r_1|v|^r_2 -2v . and the restrictions ∫ _ℝ^N|u|^2dx=a and ∫ _ℝ^N|v|^2dx=b , where a,b>0 are prescribed, N∈{3,4} , I_α (x) is the Riesz potential, α∈ (0,N) , μ _1 , μ _2 , β >0 , N+α/N1 and r_1+r_2=2^*:=2N/N-2 . The frequencies λ _1 and λ _2 appear as Lagrange multipliers. We prove that the above system has a normalized ground state solution for 0<β <β _0 , where β _0 is a constant.
This study investigates the dynamical properties of a generalized extended (3+1)-dimensional nonlinear evolution equation. A variety of ansatz strategies will be employed to accomplish this. The initial solution will be the soliton solution or shock wave soliton solution. Furthermore, the singular soliton solution of the extended (3+1)-dimensional nonlinear evolution problem will be presented. Based on the invariance surface condition, we will derive more analytical solutions. Subsequently, we employ the multiplier method to construct conservation laws. Graphical representations of the obtained solutions will be shown based on suitable selections of the arbitrary parameters included in the solutions to enhance comprehension of the underlying physics.
This paper presents numerical methods based on non-polynomial spline functions for solving singularly perturbed parabolic partial differential equations (SPPPDEs) with small shift terms. These equations frequently arise in applications such as neural signal transmission and tumor growth modeling, where delay effects and sharp gradients pose significant computational challenges. In the numerical schemes proposed in this paper, Taylor series expansion is used for handling the shift terms, backward Euler method is used for discretizing the time domain, and a generalized Shishkin mesh for spatial discretization. This combination ensures accurate resolution of boundary and interior layers. Representative numerical experiments demonstrate the methods’ second-order convergence in both time and space, with uniform accuracy across a range of perturbation parameters ( ϵ ). Compared to traditional finite difference methods and spline-based methods, the proposed approaches have shown an improved ϵ -uniform convergence of the solution. The maximum pointwise errors, graphs of solutions of the test problems and the error graphs further validate the method’s effectiveness in capturing layer behavior and delay effects.
We deduce different conditions to ensure positive solutions for a class of nonlinear two-term Riemann-Liouville fractional q-derivative boundary value problems which depends on a parameter and an integral condition. As a first step, we start by bringing this class of problems through a q-integral equation where its kernel is explicitly identified and its most characteristic properties are analysed. We then use an upper and lower solutions method, an appropriate cone and other techniques to obtain different types of conditions that ensure the existence of a positive solution to the class problems under study. Specific Lipschitz conditions are also identified to ensure the uniqueness of a continuous and positive solution. Finally, using a non-negative continuous concave functional on a cone and comparing the images it produces with the norm of potential solutions to the class of problems, other conditions are obtained that guarantee the existence of three (different) positive solutions to the class of problems under analysis.
This paper investigates the stability of the two-dimensional (2D) micropolar Rayleigh-Bénard convection system with fractional horizontal dissipation near its hydrostatic equilibrium, focusing on deriving anisotropic stability estimates for the system. We extend the results of Luo et al. (J. Math. Phys., 65 (2024), 051510.) on the integer-order horizontal dissipation to a fractional framework. Due to the non-local nature of fractional operators, the standard energy estimate techniques become inapplicable. To resolve this issue, we establish new fractional anisotropic interpolation inequalities and the very general strong Poincaré type inequality involving fractional derivative. When the spatial domain is Ω =𝕋×ℝ (where 𝕋 = [0, 1] is a 1D periodic box and ℝ is the real line), we solve the stability problem in the Sobolev space H^2(Ω ) . Furthermore, we prove that the oscillatory part (u,ω,θ) of the solution in H^1(Ω ) decays to zero exponentially in time.
We analyze the polynomial vector fields in ℝ^3 that have the Whitney umbrella as an invariant surface. We characterize the maximum number of invariant meridians and parallels that such polynomial vector fields in the Whitney umbrella can have in function of the degree of the polynomial vector field. Some algebraic computations have been done with the algebraic manipulator Mathematica.
We establish two comparison theorems for modified Euler type linear and half-linear differential equations with perturbations in both terms. The presented comparison theorems imply the oscillation and non-oscillation of equations in the studied form applying previously known oscillation and non-oscillation criteria, respectively. Thus, we enlarge the set of analyzed equations whose oscillation behavior is known.
