Harvesting herbivores is a common practice for maintaining grassland stability, yet it can trigger counterintuitive ecological responses such as population rebounds and the sudden collapse of vegetation that may undermine management goals, highlighting the need to understand how harvesting can backfire. To understand these risks, we formulate a continuous time plant-herbivore model within a modified Leslie-Gower framework, explicitly incorporating herbivore harvesting and a Holling type II functional response. A complete local stability analysis of boundary and positive equilibria yields threshold conditions that distinguish a resilient, coexisting ecosystem from one destined for degradation. Using the harvesting rate as the central bifurcation parameter, we reveal several ecologically critical nonlinear phenomena. First, the system exhibits bistability, meaning that a healthy, coexisting state and a desertified state with extinct vegetation can both be stable under the same harvest pressure; stochastic events like drought could then push the system irreversibly into the degraded state, posing a clear warning for management. Second, a Hopf bifurcation gives rise to sustained oscillations, which elevate extinction risk during population lows and intensify overgrazing during peaks, thereby threatening long term vegetation recovery and complicating control efforts. Third, we identify a hydra effect, where increasing harvest mortality paradoxically increases herbivore abundance a biological consequence of overcompensation that challenges conventional pest control logic and implies that more aggressive harvesting can inadvertently worsen the problem. Numerical simulations support all analytical findings and further illustrate how initial conditions determine the system’s long term trajectory. These results provide a mechanistic framework for anticipating the nonlinear ecological consequences of herbivore removal and offer quantitative guidance for designing sustainable grazing and restoration strategies, underscoring that ignoring such counterintuitive effects risks management failure.
This paper proposes and analyzes an implicit-explicit (IMEX) difference scheme for solving a two-dimensional variable-coefficient partial integro-differential equation (PIDE). The equation arises from option pricing under a stochastic intensity jump model driven by mixed fractional Brownian motion. First, an IMEX scheme is developed to solve the two-dimensional main PIDE along with its associated boundary condition, which is governed by a one-dimensional PIDE. The second-order convergence rate of the scheme is rigorously established for the boundary PIDE in the discrete H^1-norm and for the main PIDE in the discrete L^2-norm. Finally, numerical experiments are conducted to validate the theoretical results.
We study optimal consumption and portfolio policies for an agent with a finite planning horizon and an irreversible consumption ratcheting constraint. During the planning horizon, the agent may increase consumption but cannot reduce it. After the terminal date, the consumption level reached by that time is permanently locked in, and the agent continues to consume at that level for the rest of life, where death occurs randomly. The problem is well defined whenever initial wealth is large enough to support the current consumption floor indefinitely. Using duality theory in complete markets, we decompose the problem into a continuum of optimal stopping problems through a layer-cake representation. The dual problem can then be interpreted in terms of American put option pricing on the shadow price process, with a strike determined by the relation between the risk-free rate and the effective discount rate. We characterize the optimal consumption policy through a free boundary driven by the shadow price, and show that the wealth-to-consumption ratio is reflected at an endogenous boundary. A central finding is that the lifetime lock-in effect makes the agent more cautious than in the standard finite-horizon ratcheting model: upward consumption adjustments occur less frequently, and the optimal risky share is uniformly lower for any given wealth-to-consumption ratio. This stronger precautionary behavior arises because any increase in consumption before the terminal date also raises the permanently committed consumption level afterward, thereby creating an additional lock-in cost.
This paper investigates the bipartite consensus problem for multi-agent systems via uncertain pinning control under directed signed switching topologies. The agent dynamics incorporate randomly varying nonlinearities and parameter uncertainties to reflect more realistic environmental influences. To handle the uncertain connectivity between the leader and followers induced by switching topology, a distributed pinning control protocol is developed. The main contributions are summarized in two theorems. A suitable multiple Lyapunov function is constructed, and M-matrix theory is employed for multi-agent systems without and with time-varying delays. Sufficient conditions for achieving bipartite consensus are established in the form of linear matrix inequalities (LMIs). These conditions ensure that all followers converge asymptotically to either the leader’s state or its opposite. The convergence direction depends on their subgroup affiliation. Finally, the effectiveness of the proposed control strategy is demonstrated through theoretical analysis and supported by numerical simulations.
This paper develops a heterogeneous Atangana–Baleanu fractional-order framework to model malware diffusion in multi-cloud environments, incorporating stochastic perturbations to represent ambient cyber noise. A twelve-compartment Malware Propagation in Multi-Cloud Environments (MMCE) topology is formulated to distinguish sector-specific infection classes, quarantine layers, traced nodes, and protection strata. By assigning distinct fractional orders to each compartment, the model captures compartment-dependent hereditary memory effects and anomalous temporal persistence. Within this framework, both deterministic and stochastic optimal control problems are formulated to minimize the cumulative infection burden and intervention costs in the presence of Brownian perturbations. The existence, uniqueness, and Ulam–Hyers stability of the resulting fractional dynamical systems are rigorously established under standard Lipschitz continuity conditions. Moreover, the associated Pontryagin-type optimality systems are analytically derived within the Atangana–Baleanu fractional setting, providing a mathematically consistent foundation for subsequent numerical implementation. Theoretical results indicate that heterogeneous memory effects and stochastic forcing play a significant role in shaping the structure and stability of optimal control strategies. Consequently, the proposed framework offers a rigorous and flexible platform for future computational investigations and for the development of memory-aware, resilience-oriented mitigation policies in multi-cloud cyber ecosystems.
Threshold interventions are essential for the targeted control of infectious diseases, but their effectiveness can be strongly influenced by vector preference, which is often neglected in existing non-smooth dynamic models. Accordingly, this study investigates the impact of vector preference on threshold control strategies. We develop a Filippov model with vector preference and a joint threshold policy, where interventions are triggered when the combined proportion of susceptible and infected hosts exceeds a critical level. We theoretically analyze the existence of sliding regions, the existence and stability of regular equilibria and pseudo-equilibria, as well as discontinuity-induced bifurcation phenomena. Our results reveal that vector preference can induce diverse equilibrium states. In particular, preference for susceptible hosts allows the coexistence of up to five equilibria, leading to various bistability and possible tristability. Numerical simulations further show that without preference the threshold level directly determines whether early intervention is triggered. Preference for susceptible hosts markedly increases threshold sensitivity and readily induces multistability, allowing disease persistence even when the basic reproduction number is below one; thus, effective control requires targeting the critical threshold rather than merely lowering it. In contrast, preference for infected hosts yields a more monotonic response and facilitates stable disease suppression over a broader range of threshold values. These results indicate that precise threshold setting is essential for effective disease control across different host preference patterns.
We propose and investigate a normalized time-fractional Keller–Segel (KS) model that incorporates logarithmic chemotactic sensitivity and a non-diffusive chemical response. The normalized time-fractional derivatives maintain a constant total memory effect across different fractional orders. This formulation enables clear interpretation of the effect of the fractional order and facilitates fair comparisons across varying fractional orders. An implicit-explicit finite difference scheme is applied to the density equation, and an analytical solution combined with the frozen coefficient technique is used to solve the chemoattractant equation. The numerical scheme achieves second-order accuracy in space and first-order accuracy in time, as demonstrated by convergence tests. Numerical experiments demonstrate that the fractional order significantly influences the blow-up behavior of the solution; specifically, smaller fractional orders result in stronger memory effects and lead to faster blow-up. Furthermore, the chemotactic sensitivity exponent and the fractional order play critical roles in determining the dynamics and the blow-up time of the system. The study confirms the validity and efficiency of the proposed algorithm and provides insights into the influence of the fractional process on chemotactic aggregation. The proposed method offers a promising direction for further investigation of biological systems influenced by nonlocal and memory-driven processes.
The spatial spread of Mpox is influenced by human mobility, contact patterns, and zoonotic spillover. However, the mechanisms that drive the formation and persistence of localized hotspots remain poorly understood. In this study, we develop a network-based reaction–diffusion model that integrates adaptive human mobility, heterogeneous transmission, and spatially varying spillover. We first establish key analytical properties of the model, including positivity, boundedness, and a network-based basic reproduction number. Our analysis shows that mobility adapts to infection prevalence, effectively modifying diffusion across the network. As prevalence increases, mobility is reduced, limiting spatial spread regardless of epidemic timing. Numerical simulations reveal that, unlike classical models with constant diffusion, prevalence-dependent mobility slows spatial invasion, suppresses diffusion-driven instability, and promotes persistent localized hotspots. Higher-order transmission amplifies local outbreaks without changing invasion thresholds, while heterogeneous spillover increases transient infection levels but does not sustain spatial instability. Overall, the results demonstrate that feedback between infection prevalence and human mobility plays a central role in shaping spatial Mpox dynamics. The proposed framework provides a mechanistic explanation for persistent spatial heterogeneity and localized endemicity in connected populations.
This paper investigates a reaction-diffusion chemostat model involving two species that compete for a single limited resource. The study focuses on the combined effects of diffusion and growth on the extinction and survival of species under different competitive scenarios. For the weak-strong competition case, there exist two critical diffusion rates, which classify the global dynamics of this system into two outcomes: (i) persistence of the species with a strong growth capacity; (ii) extinction of both species. For the evenly matched competition case, the analysis reveals that the existence of two critical curves associated with growth rates will separate competition outcomes into competitive exclusion and coexistence. The study further provides the numerical approaches that not only confirm the theoretical results, but also illustrate the geometry of the critical curves within the diffusion-growth rates plane. These theoretical and numerical results contribute valuable insights into the dynamics of species competition in resource-limited environments.
This study presents a comprehensive numerical investigation of thermo-hydrodynamic behaviour and entropy generation in a two-dimensional square cavity filled with a ternary hybrid nanofluid (Fe3O4–Ag–TiO2/H2O) under magnetohydrodynamic (MHD) mixed convection. The system features internally heated circular fins and is driven by sinusoidally varying velocities along the left and bottom walls, introducing periodic momentum forcing. Key governing parameters are systematically examined, Hartmann number (0 ≤ Ha ≤ 80), Richardson number (Ri = 0.01–1), nanoparticle volume fraction (0≤φ≤ 0.08), sinusoidal wavelength (0.2 ≤ Lx = Ly = LDriven ≤ 0.8), and phase deviation between moving walls. High-fidelity simulations are conducted using a custom FORTRAN solver combining the Finite Volume Method with Full Multigrid Acceleration to resolve coupled continuity, momentum, energy, and entropy generation equations. Rigorous parameterization ensures accurate representation of interactions among magnetic damping, buoyancy, and nanoparticle-enhanced thermal transport. Results show that sinusoidal velocity modulation effectively controls flow coherence and entropy production, with certain phase relationships minimizing irreversibility while maintaining favorable heat transfer. Optimal operation is achieved under moderate magnetic fields (Ha = 20), Richardson number (0.1 ≤ Ri ≤ 0.5), nanoparticle concentration (φ =0.06), and wavelength (LDriven = 0.6), illustrating the trade-off between thermal performance and entropy generation. These findings provide new physical insights into periodically driven MHD convection systems and practical guidelines for designing energy-efficient thermal systems utilizing hybrid nanofluids.
Dengue virus (DENV) and Zika virus (ZIKV) are primarily transmitted by Aedes aegypti and Aedes albopictus. Because they share the same mosquito vectors, co-infections with both viruses have been reported worldwide. These pathogens are among the most significant mosquito-borne viruses, responsible for considerable morbidity and mortality. Simultaneous infection may affect viral activity as well as the host immune response, which could influence clinical outcomes. Yet, the within-host mechanisms governing DENV-ZIKV interactions remain poorly understood. In this work, we develop a within-host model of DENV-ZIKV co-infection incorporating cytotoxic T lymphocyte (CTL) immunity, including cross-reactive CTL responses. The system tracks uninfected target cells, infected cells, free viruses and CTLs. All solutions remain non-negative and bounded. The analysis identifies four equilibria: disease-free, DENV mono-infection, ZIKV mono-infection, and viral coexistence. Using the next-generation matrix, we derive the reproduction numbers for the DENV submodel, the ZIKV submodel, and the co-infection system (R_D, R_Z and R_0 =max{R_D,R_Z}). In addition, we compute the invasion reproduction numbers for the DENV and ZIKV submodels, denoted by R_D^inv and R_Z^inv, respectively. Global stability is established and verified via Lyapunov functions. Sensitivity analysis of R_D and R_Z highlights parameters most strongly influencing viral clearance. The model further examines three therapeutic strategies: (i) entry-blocking antiviral, (ii) agents reducing viral output, and (iii) interleukin-2 immunotherapy (IL-2) therapy enhancing CTL activity. The influence of CTLs cross-reactivity on co-infection dynamics is also established. Results show that cross-reactive CTLs can reduce R_D and R_Z similarly to treatment interventions. Numerical simulations confirm theoretical predictions, demonstrating that combining antiviral and immune-based strategies enhances viral control by limiting replication and boosting immune-mediated clearance. Moreover, ignoring cross-reactive CTL responses could theoretically lead to an overestimation of the antiviral treatment intensity required for viral elimination. These findings reflect the interactions captured by the model dynamics and are not intended to provide direct clinical predictions.
In this work, we present a periodic switched pest control model with predation-induced fear, intraspecific cooperative hunting and seasonally migrating in pest population. The aims of this paper are to acquire the sufficient conditions for the permanence and pest-eradication boundary periodic solution of system (2.1) being globally asymptotically stable (GAS). Simulations are employed to validate our conclusions. Through numerical analysis, intraspecific cooperative hunting is found to suppress the pest population effectively. Our conclusion also indicates that the release rate of natural enemies should be dynamically adjusted according to the migration period of the pests. Moreover, under a certain level of fear effect, the pests suppressed by the fear effect are stronger than that by intraspecific cooperative hunting. The optimal release rate of natural enemies under different scenarios is further discussed. Finally, through bifurcation diagrams, we discover complex dynamical properties of the system, such as periodic bifurcations and chaos.
This paper investigates a class of predator-prey systems with time delays, stochastic perturbations, and patch diffusion, which incorporate the Beddington–DeAngelis (B-D) functional response. The proposed model integrates environmental noise, dispersal processes, and time delays, forming a comprehensive stochastic delayed diffusion framework that better reflects realistic ecological scenarios. By employing appropriate Lyapunov functions and the comparative principle of stochastic differential equations, we prove the existence of a unique global positive solution for any given positive initial value. Sufficient conditions are also established for uniform persistence of the system. Furthermore, through the construction of a novel Lyapunov functional, we derive sufficient conditions for the global attractivity of the system’s solutions, providing theoretical insights into the long-term dynamical behavior under the combined effects of stochasticity, delays, and diffusion. Finally, numerical simulations are conducted using MATLAB along with the Milstein discretization method, which corroborate the correctness and effectiveness of the theoretical results. This study offers a new theoretical framework and methodology for analyzing persistence and stability in stochastic delayed diffusive predator-prey systems.
Constructing efficient integrated pest management (IPM) strategies requires an in-depth understanding of the interactions among biological populations and their density-dependent characteristics. In resource-limited ecosystems, resource competition and mutual interference often occur among predator populations. Given this trait, this study establishes a type of predator-prey IPM model with Beddington-DeAngelis functional response and impulsive effects. Through qualitative analysis, we establish the uniform boundedness of solutions for the ordinary differential equation model, and prove the existence and stability of periodic solutions as well as the bifurcation behavior. For the impulsive control model, we verify the boundedness of solutions and the conditions for the permanence of the system, derive the existence of the pest-eradication periodic solution and the key threshold for its global asymptotic stability, and conduct a bifurcation analysis. Numerical simulations reveal the rich dynamical behaviors of the model, confirming that the impulsive control model is more consistent with realistic resource-limited scenarios, and that parameter variations significantly regulate the dynamic evolution of pest and natural enemy populations. The research conclusions indicate that combining pesticide spraying with natural enemy release, optimizing the timing of impulsive control based on real-time population monitoring, and prioritizing natural enemy species with high predation efficiency and low intraspecific competition are conducive to constructing an efficient, synergistic and environmentally sustainable IPM system.
This study explores the mechanism of pattern formation in a hyperbolic reaction-diffusion system that incorporates fitness taxis of species. Through rigorous mathematical derivation, we have established the conditions for Turing instability and wave instability, where the inertial motion of species and fitness taxis collectively regulate complex spatiotemporal behavior. To confirm the theoretical findings, we revealed the hierarchical structure of non-equilibrium dynamics through numerical simulations and demonstrated how inertial effects and fitness taxis synergistically drive different dynamics in the reaction-diffusion framework.
Tuberculosis (TB) remains a major public health concern, particularly in resource-limited settings where sustained funding and psychosocial factors critically influence disease dynamics. This study presents a novel mathematical model that integrates fear, stigma, and funding to assess the sustainability of TB-funded prevention programs. The model, formulated as a system of nonlinear differential equations, analyses disease-free and endemic equilibria and investigates the conditions under which TB transmission can be effectively reduced. Our results show that consistent financial investment is essential for long-term TB control, eliminating backward bifurcation and ensuring that reductions in the basic reproduction number (ℛ_0) translate into sustained declines in TB incidence. We introduce a funding sustainability ratio that provides a quantitative threshold for maintaining effective program outcomes. Simulations reveal that enhanced prevention efforts and health education, supported by stable funding, significantly lower TB prevalence and the size of the stigmatised infectious population. The study further reveals a counterintuitive effect of fear, as moderate levels encourage preventive behaviours, whereas excessive fear intensifies stigma and discourages timely treatment seeking, thereby prolonging transmission. Our findings emphasise that health education alone is insufficient without sustained financial support. Moreover, fear-driven isolation may reduce short-term transmission but imposes substantial social costs. These results underscore the importance of integrating biomedical interventions with targeted psychosocial strategies, underpinned by long-term, stable funding, to achieve sustainable TB control in high-burden communities.
Based on ecology theory, a enterprise cluster model has been modeled which describes the competition and cooperation of enterprises cluster in real economic environments. Initially, we deal with the existence, uniqueness and stochastically ultimate boundedness of the positive solution to the model. Subsequently, we obtain the sufficient conditions which lead to the extinction of the enterprise. The presence of a stationary distribution is confirmed by the use of Lyapunov function. For enterprise cluster model, the stationary distribution indicates that the enterprises will persist over the long term. Finally, three numerical examples are provided to support and illustrate our main results, offering practical insights into the dynamical properties of the model. This work contributes to the development of more accurate competition and cooperation models that can support the healthy and sustainable development of enterprises.
Currently, the pathogenesis of long COVID and viral reactivation remains unclear. The study proposes a dynamic model of SARS-CoV-2 transmission from the vascular to the tissue. The stability of the equilibrium points is discussed in the model. Clinical pulmonary viral data from eight patients are used to fit parameters, and parameter robustness is validated through practical identifiability. For most patients, pulmonary viral loads exhibit periodic oscillations, whereas vascular viruses decay asymptotically. Curiously, numerical simulation shows multistability phenomena, such as coexisting periodic and chaotic attractors or multiple periodic attractors under potent drug administration. The periodic orbits explain virus persistence and reactivation in tissues. Coexisting attractors, particularly chaotic attractors, indicate the complexity of SARS-CoV-2 infection and reveal the pathogenesis of long COVID and viral reactivation. Therefore, this study applies classic chaos control methods to reduce the probability of chaos occurring. The controlled results are a disease-free state and a desirable immune status with clinical significance (protective immunity). Notably, this desirable state is not the equilibrium point of the model. By comparing different control methods, some recommendations are provided on therapeutic aspects. In summary, this study establishes a novel mathematical model that provides strong numerical support for medical hypotheses about long COVID and viral reactivation, as well as practical guidance for clinicians.
This paper investigates the Turing bifurcation of Hopf bifurcating periodic solutions in a Holling-Tanner type population system with nonlinear diffusion. It focuses on how diffusion (including self-diffusion, nonlinear diffusion and cross-diffusion) destabilizes periodic solutions and induces the generation of new and abundant spatially ordered patterns. By employing the local Hopf bifurcation theorem, perturbation theory, implicit function theorem and Floquet theory, a diffusion rate formula is derived to determine the conditions for Turing bifurcation of the stable periodic solutions induced by diffusion. Finally, numerical simulations are carried out to verify the theoretical analysis results, and new phenomena of spatial Turing patterns in populations are revealed.
