Search Results (23)

Many mathematical models using ordinary differential equations (ODEs) have been used to investigate what type of stressors cause honeybee colonies collapse. We propose a simple model of a delayed differential equation system (DDE) to describe the effect of insecticides over brood death rate and its influence over honeybee population dynamics. First, we remember some basic facts for the model with no delay. To analyze our model, we study the equilibria and perform stability and sensitivity analysis of the DDE system. Next, by using the delay time $\tau$ as a bifurcation parameter, we find that no Hopf bifurcation could arise as the time lag $\tau$ varies within biologically plausible ranges. Numerical simulations with real data are studied for the biological significance of the model. Full article

Impetigo is a highly contagious skin infection that primarily affects children and communities in low-income regions and has become a significant public health issue impacting both individuals and healthcare systems. A nonlinear deterministic model based on the transmission dynamics of skin sores (impetigo) is developed with a specific emphasis on the time delay effects in the infection and recovery processes. To address this complexity, we introduce a delay differential equation (DDE) to describe the dynamic process. We analyzed the existence of Hopf bifurcations associated with the two equilibrium points and examined the mechanisms underlying the occurrence of these bifurcations as delays exceeded certain critical values. To obtain more comprehensive insights into this phenomenon, we applied the center manifold theory and the normal form method to determine the direction and stability of Hopf bifurcations near bifurcation curves. This research not only offers a novel theoretical perspective on the transmission of impetigo but also lays a significant mathematical foundation for developing clinical intervention strategies. Specifically, it suggests that an increased time delay between infection and isolation could lead to more severe outbreaks, further supporting the development of more effective intervention approaches. Full article

This paper explores the linearized stability of nonlinear delay differential equations (DDEs) with impulses. The classical results on the existence of periodic solutions are extended from ordinary differential equations (ODEs) to DDEs with impulses. Furthermore, the classical results of linearized stability for nonlinear semigroups are generalized to periodic DDEs with impulses. A significant challenge arises from the need for a discontinuous initial function to obtain periodic solutions. To address this, first-kind discontinuous spaces $\mathcal{R}([a,b],{\mathbb{R}}^n)$ are introduced for defining DDEs with impulses, providing key existence and uniqueness results. This study also establishes linear stability results by linearizing the Poincaré operator for DDEs with impulses. Additionally, the stability properties of equilibrium solutions for these equations are analyzed, highlighting their importance due to the wide range of applications in various scientific fields. Full article

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work. Full article

Amid global climate change and population growth, the prevalence of saline–alkali lands significantly hampers sustainable agricultural development. This study employs theories of asymmetric information and bounded rationality to construct an evolutionary game model, analyzing the interactions among small farmers, family farms, and seed industry enterprises in the context of saline–alkali land management. It investigates the strategic choices and dynamics of these stakeholders under the influence of economic incentives and risk perceptions, with a focus on how government policies can foster green development. Utilizing Delay Differential Equations (DDEs) for simulations, this study highlights the risk of “market failure” without government intervention and underscores the need for government participation to stabilize and improve the efficiency of the green development process. The findings reveal that factors such as initial willingness to participate, the economic viability of salt-tolerant crops, seed pricing, research and development costs, and the design of incentive policies are crucial for sustainable land use. Accordingly, the paper proposes specific policy measures to enhance green development, including strengthening information dissemination and technical training, increasing the economic attractiveness of salt-tolerant crops, alleviating research and development pressures on seed companies, and optimizing economic incentives. This study provides a theoretical and policy framework for the sustainable management of saline–alkali lands, offering insights into the behavioral choices of agricultural stakeholders and supporting government strategies for agricultural and environmental protection. Full article

Chatter causes great damage to the machining process, and the selection of appropriate process parameters through chatter stability analysis is of great significance for achieving chatter-free machining. This article proposes a milling stability analysis method based on the barycentric rational interpolation differential quadrature method (DQM). The dynamics of the milling process considering the regeneration effect is first modelled as a time-delay differential equation (DDE). When adjacent pitch angles of the milling cutter are symmetric, the milling dynamic equation contains a single time delay. Otherwise, when adjacent pitch angles are asymmetric, the dynamic equation contains multiple time delays. The barycentric rational interpolation DQM is then used to approximate the differential and delay terms of the milling dynamics equation, and to construct a state transition matrix between adjacent milling periods. Finally, the chatter stability lobe diagram (SLD) is obtained based on the Floquet theory. According to the SLD, the appropriate spindle speed can be selected to obtain the maximum stable axial depth of cutting, thereby effectively improving the material removal rate. The accuracy and efficiency of the proposed method have been validated by two widely used milling models, and the results show that the proposed method has good accuracy and computational efficiency. Full article

In the context of global food security and the pursuit of sustainable agricultural development, fostering synergistic innovation in the seed industry is of strategic importance. However, the collaborative innovation process between seed companies, research institutions, and governments is fraught with challenges due to information asymmetry and bounded rationality within the research and development phase. This paper establishes a multi-agent evolutionary game framework, taking the breeding of salt-tolerant rice as a case study. This study, grounded in the theories of information asymmetry and bounded rationality, constructs a two-party evolutionary game model for the interaction between enterprises and research institutions under market mechanisms. It further extends this model to include government participation, forming a three-party evolutionary game model. The aim is to uncover the evolutionary trends in collaborative behavior under various policy interventions and to understand how governments can foster collaborative innovation in salt-tolerant rice breeding through policy measures. To integrate the impact of historical decisions on the evolution of collaborative innovation, this research employs a delay differential equation (DDE) algorithm that takes historical lags into account within the numerical simulation. The stability analysis and numerical simulation using the DDE algorithm reveal the risk of market failure within the collaborative innovation system for salt-tolerant rice breeding operating under market mechanisms. Government involvement can mitigate this risk by adjusting incentive and restraint mechanisms to promote the system’s stability and efficiency. Simulation results further identify that the initial willingness to participate, the coefficient for the distribution of benefits, the coefficient for cost sharing, and the government’s punitive and incentivizing intensities are crucial factors affecting the stability of collaborative innovation. Based on these findings, the study suggests a series of policy recommendations including enhancing the initial motivation for participation in collaborative innovation, refining mechanisms for benefit distribution and cost sharing, strengthening regulatory compliance systems, constructing incentive frameworks, and encouraging information sharing and technology exchange. These strategies aim to establish a healthy and effective ecosystem for collaborative innovation in salt-tolerant rice breeding. While this research uses salt-tolerant rice breeding as a case study, the proposed cooperative mechanisms and policy suggestions have universal applicability in various agricultural science and technology innovation scenarios, especially when research meets widespread social needs but lacks commercial profit drivers, underscoring the essential role of government incentives and support. Consequently, this research not only contributes a new perspective to the application of evolutionary game theory in agricultural science and technology innovation but also offers empirical backing for policymakers in advancing similar collaborative innovation endeavors. Full article

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases. Full article

Chatter stability analysis is an effective way to optimize the cutting parameters and achieve chatter-free machining. This paper proposes a milling chatter stability analysis method based on the localized differential quadrature method (LDQM), which has the advantages of simple principle, easy application, and high computational efficiency. The milling process, considering the regeneration effect, is modeled using linear periodic delay differential equations (DDE), then the state transition matrix during the adjacent cutting period is constructed based on the LDQM, and finally, the stability of the milling process is obtained based on the Floquet theory. The accuracy and computation efficiency of the proposed method are verified through two benchmark milling models. The simulation results demonstrate that the proposed method in this paper can accurately and quickly obtain the chatter stability lobe diagram (SLD) of the milling process, which will provide guidance for optimizing the process parameters. Full article

Ordinary differential equations (ODE) have long been an important tool for modelling and understanding the dynamics of many real systems. However, mathematical modelling in several areas of the life sciences requires the use of time-delayed differential models (DDEs). The time delays in these models refer to the time required for the manifestation of certain hidden processes, such as the time between the onset of cell infection and the production of new viruses (incubation periods), the infection period, or the immune period. Since real biological systems are always subject to perturbations that are not fully understood or cannot be explicitly modeled, stochastic delay differential systems (SDDEs) provide a more realistic approximation to these models. In this work, we study the predator–prey system considering three time-delay models: one deterministic and two types of stochastic models. Our numerical results allow us to distinguish between different asymptotic behaviours depending on whether the system is deterministic or stochastic, and in particular, when considering stochasticity, we see that both the nature of the stochastic systems and the magnitude of the delay play a crucial role in determining the dynamics of the system. Full article

This paper proposes a difference discretization method (DDM) under a difference frame for the prediction of milling stability. In this method, the dynamic milling process is described by a delay differential equation (DDE) with two degrees of freedom rather than the traditional state-space form with a single discrete time delay. After discretization, only the velocity and acceleration in the DDE are approximated by the first- and second-order central difference for each smaller time interval, while the other items are kept unchanged. Then, the criterion for the optimal discretization interval number is put forward and derived based on the largest effective time interval (also called the critical time interval). The use of the critical time interval cannot only obtain sufficient accuracy, but also promotes as much efficiency as possible. Subsequently, a new DDM (NDDM) with varied discretization interval numbers as the milling rotating speeds is developed. Finally, the effectiveness of the proposed algorithm is demonstrated by using a benchmark example for a two-degrees-of- freedom milling model compared to the full discretization method (FDM) and the Hermite-interpolation full discretization method (HFDM). The results show that the proposed method has satisfactory stability charts and is able to increase the efficiency by 100% or more. Full article

This study aims to investigate the asymptotic behavior of a class of third-order delay differential equations. Here, we consider an equation with a middle term and several delays. We obtain an iterative relationship between the positive solution of the studied equation and the corresponding function. Using this new relationship, we derive new criteria that ensure that all non-oscillatory solutions converge to zero. The new findings are an extension and expansion of relevant findings in the literature. We apply our results to a special case of the equation under study to clarify the importance of the new criteria. Full article

In this paper, we focus on investigating the performance of the mathematical software program Maple and the programming language MATLAB when using these respective platforms to compute the method of steps (MoS) and the Laplace transform (LT) solutions for neutral and retarded linear delay differential equations (DDEs). We computed the analytical solutions that are obtained by using the Laplace transform method and the method of steps. The accuracy of the Laplace method solutions was determined (or assessed) by comparing them with those obtained by the method of steps. The Laplace transform method requires, among other mathematical tools, the use of the Cauchy residue theorem and the computation of an infinite series. Symbolic computation facilitates the whole process, providing solutions that would be unmanageable by hand. The results obtained here emphasize the fact that symbolic computation is a powerful tool for computing analytical solutions for linear delay differential equations. From a computational viewpoint, we found that the computation time is dependent on the complexity of the history function, the number of terms used in the LT solution, the number of intervals used in the MoS solution, and the parameters of the DDE. Finally, we found that, for linear non-neutral DDEs, MATLAB symbolic computations were faster than Maple. However, for linear neutral DDEs, which are often more complex to solve, Maple was faster. Regarding the accuracy of the LT solutions, Maple was, in a few cases, slightly better than MATLAB, but both were highly reliable. Full article

The literature suggests that effective defence against tumour cells requires contributions from both Natural Killer (NK) cells and CD$8^{+}$ T cells. NK cells are spontaneously active against infected target cells, whereas CD$8^{+}$ T cells take some times to activate cell called as cell-specific targeting, to kill the virus. The interaction between NK cells and tumour cells has produced the other CD$8^{+}$ T cell, called tumour-specific CD$8^{+}$ T cells. We illustrate the tumour–immune interaction through mathematical modelling by considering the cell cycle. The interaction of the cells is described by a system of delay differential equations, and the delay, $\tau$ represent time taken for tumour cell reside interphase. The stability analysis and the bifurcation behaviour of the system are analysed. We established the stability of the model by analysing the characteristic equation to produce a stability region. The stability region is split into two regions, tumour decay and tumour growth. By applying the Routh–Hurwitz Criteria, the analysis of the trivial and interior equilibrium point of the model provides conditions for stability and is illustrated in the stability map. Numerical simulation is carried out to show oscillations through Hopf Bifurcation, and stability switching is found for the delay system. The result also showed that the interaction of NK cells with tumour cells could suppress tumour cells since it can increase the population of CD$8^{+}$ T cells. This concluded that the inclusion of delay and immune responses (NK-CD$8^{+}$ T cells) into consideration gives us a deep insight into the tumour growth and helps us understand how their interactions contribute to kill tumour cells. Full article

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications. Full article