Narrow Results

An expansion of the periodic solution of the van der Pol oscillator for small values of the damping parameter $𝜖$ is shown via a frequency domain method and harmonic balancing technique. An approximation of the amplitude of the first harmonic and the frequency $\omega$ in terms of $𝜖$ is obtained by an analytical form instead of the conventional algorithmic approach used before with this methodology. Its decomposition under different harmonics $sink𝜗$ and $cosk𝜗$ where $k=3,5,7$ is written in terms of their leading coefficients in power series of $𝜖$ to make easy comparisons with other methods in the literature.

In this paper, we propose a delayed viral infection model to incorporate a logistic proliferation, a cell pyroptosis effect and the three intracellular time delays. We present the basic reproduction number and investigate the existence and the global stability of equilibria: infection-free equilibrium $p_0$ and infection equilibrium $p^{\ast }$, respectively. By considering different combinations of the time delays, we investigate the existence and the properties of Hopf bifurcation from $p^{\ast }$ when it is unstable. We also numerically explore the viral dynamics beyond stability. Bifurcation diagrams are used to show the stability switches, multiperiod solutions and irregular sustained oscillations with the variation of time delays. The results reveal that both the logistic proliferation term and time delays are responsible for the rich dynamics, but the logistic proliferation term may be the main factor for the occurrence of the Hopf bifurcation. Moreover, we show that ignoring the cell pyroptosis effect may underevaluate the viral infection risk and the sensitivity analysis implies that taking effective strategies for reducing the impact of cell pyroptosis is beneficial for decreasing the viral infection risk.

How to secure the virtual environment in the cloud computing is a crucial issue since the cloud computing has been providing various services in many areas in the globe. The main aim of this paper is to investigate a delayed malware propagation model for cloud computing security. Time delay due to time interval that the infected virtual machines need to reinstall system and time delay due to the temporary immunization period of the protected virtual machines are introduced into the model. First, sufficient conditions for local stability and existence of Hopf bifurcation are derived by choosing different combinations of the two time delays as bifurcating parameter. Second, global exponential stability of the model is explored with the aid of linear matrix inequalities method. Finally, numerical simulations are carried out to illustrate that the obtained results and suggestions to ensure security of the cloud computing are given in the conclusion according to analyzing the dynamics of the proposed delayed malware propagation model for cloud computing security.

In this paper, we investigate the mathematical analysis of a mathematical model describing the virotherapy treatment of a cancer with logistic growth and the effect of viral cycle presented by a time delay. The cancer population size is divided into uninfected and infected compartments. Depending on time delay, we prove the positivity and boundedness and the stability of equilibria. We give conditions on which the viral cycle leads to “Jeff’s phenomenon” observed in laboratory and causes oscillations in cancer size via Hopf bifurcation theory. We establish an algorithm that determines the bifurcation elements via center manifold and normal form theories. We give conditions which lead to a supercritical or subcritical bifurcation. We end with numerical simulations illustrating our theoretical results.

Due to the random search of species and from the economic point of view, combined harvesting is more suitable than selective harvesting. Thus, we have developed and analyzed a prey–predator model with the combined effect of nonlinear harvesting in this research paper. Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system. We have examined the local stability analysis of all the equilibrium points. Besides these various types, rich and complex dynamical behaviors such as backward, saddle-node, Hopf and Bogdanov–Takens (BT) bifurcations, homo-clinic loop and limit cycles appear in this model. Furthermore, interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points. Also, we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species. We have observed that the extinction of predator species may not happen due to backward bifurcation, although a stable predator-free equilibrium (PFE) exists. Finally, numerical simulations are discussed using MATLAB to verify all the theoretical results.

Pine wilt disease is a destructive forest disease with strong infectivity, a wide spread range and high difficulty in prevention and control. Since controlling Monochamus alternatus, the vector of pine wood nematode (Bursaphelenchus xylophilus) can reduce the occurrence of pine wilt disease efficiently, the parasitic natural enemy of M. alternatus, Dastarcus helophoroides, is introduced in this paper. Considering the influence of parasitic time of D. helophoroides on the control effect, based on the mutualistic symbiosis and parasitic relationship among pine wood nematode, M. alternatus and D. heloporoides, this paper establishes a pine wood nematode prevention and control model with delay. Then, the stability of positive equilibrium and the existence of Hopf bifurcation are discussed. Besides, we obtain the normal form of Hopf bifurcation by applying the multiple time scales method. Finally, numerical simulations with two sets of meaningful parameters selected by means of statistical analysis are carried out to support the theoretical findings. Through the comparative analysis of numerical simulations, the factors affecting the control effect of pine wilt disease are obtained, and some suggestions are put forward for practical control in the forest.

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

In this work, we propose and investigate a human immunodeficiency virus (HIV) infection model that considers CD4${}^{+}$ T cell homeostasis and the cytotoxic T lymphocyte (CTL) response. The local stability of the disease-free and endemic equilibrium point is established. Further, the global stability of the disease-free and endemic equilibrium points is investigated. Under specific parametric conditions, it is shown that the model exhibits backward, forward (transcritical) bifurcation, Hopf and Hopf–Hopf bifurcation. We have further considered a time lag in the model to represent the time delay between CD4${}^{+}$ T cell infection and the viral particle production and performed the stability and bifurcation analysis for the delay model. We conduct comprehensive numerical experiments to visualize the dynamical behavior of the HIV model and validate our findings.

In this paper, a delayed fractional-order epidemic model with general incidence rate and incubation period is proposed for the Corona Virus Disease 2019 (COVID-19) pandemic. The corresponding sufficient conditions are established to analyze the existence and stability of disease-free equilibrium and endemic equilibrium of the proposed model. The conditions for the existence of Hopf bifurcation are obtained by selecting the time delay as the bifurcation parameter. The control strategies for the COVID-19 pandemic are designed, and the corresponding delay fractional order optimal control problem (DFOCP) is analyzed based on the generalized Euler–Lagrange equation. The parameters of the model are identified based on the data of multiple types of the COVID-19 pandemic. Further, the effectiveness of the model in describing the trend of the COVID-19 pandemic is verified. Based on the results of parameter identification, the influence of incubation period on the COVID-19 pandemic is discussed. The forward–backward sweep method (FBSM) is adopted to numerically solve DFOCP, and the control effects under different control measures are analyzed.

In this paper, the dynamical behaviors of a new cytokine-enhanced HIV infection model with intracellular delay ${\tau }_1$, virus replication delay ${\tau }_2$ and immune response delay ${\tau }_3$ are investigated. The positivity and boundedness of all solutions for the model with non-negative initial values have been proved. Moreover, two important biological parameters, called the virus reproductive number $R_0$ and the antibody immune reproductive number $R_1$ are established. By constructing suitable Lyapunov functionals and using LaSalle’s invariance principle, the global dynamics of the equilibria is completely determined by $R_0$ and $R_1$. On the one hand, the results show that intracellular delay ${\tau }_1$, viral replication delay ${\tau }_2$ and immune response delay ${\tau }_3$ have no effect on the stability of $E_0$, $E_1$, and if ${\tau }_1\ge 0$, ${\tau }_2\ge 0$, ${\tau }_3=0$, the endemic equilibrium with the presence of antibody response $E_2$ is globally asymptotically stable. On the other hand, when ${\tau }_1\ge 0$, ${\tau }_2\ge 0$, ${\tau }_3>0$, numerical analysis confirms the theorems and suggests that time delay play a positive role in virus infection, with the increase of ${\tau }_3$, the dynamic behavior of the equilibrium $E_2$ will change as follows: locally asymptotically stable $\to$ unstable; Hopf bifurcation appears.

This work focuses on studying high-codimensional bifurcations in the glucose model with obesity’s effect. We examine the related dynamical behaviors, taking into account the risk and adverse health effects associated with obesity’s impact on the glucose model. Through the application of the normal form method, we demonstrate that the model exhibits codimension-1 and codimension-2 bifurcations such as saddle node bifurcation and cusp bifurcation. Additionally, we introduce and prove a theorem that establishes the presence of Hopf bifurcation in the model. The computation of the first Lyapunov coefficient is achieved using center manifold theory. To support our theoretical analysis and showcase the model’s complex dynamical behaviors, including periodic curve families, we provide numerical simulations. These simulations contribute to understanding clinical observations regarding the effects of obesity on glucose levels, oscillations and disorders. Ultimately, this research can aid in the early control of blood glucose levels.

In this study, we delve into the dynamics of a generalist predator–prey system with prevalence of a disease in the prey population and the presence of predator-induced fear. In this intricate ecological web, the fear response triggered by predators influences the birth rate of susceptible prey population and also escalates intra-specific competition among them. In an effort to encapsulate a more realistic scenario, we extend our deterministic model to its stochastic counterpart by introducing environmental white noise that impacts the mortality rates of both prey and predator species. Comprehensive mathematical analyses are performed on both the deterministic and stochastic systems to elucidate their qualitative behaviors. Specifically, we derive conditions under which the stochastic system exhibits a unique global positive solution and explore the potential extinction of species within the ecosystem. To unravel the intricate dynamics numerically, we perform sensitivity of parameters, and construct one- and two-parameter bifurcation diagrams. Our investigations reveal that when the costs of fear surpass specific thresholds and disease incidence is high, both the susceptible and infected prey populations face extinction. Notably, the introduction of supplementary foods can lead to system’s destabilization, potentially pushing it towards a state where only predators can persist. Moreover, we observe that lower intensities of white noise have minimal impact on the system’s dynamics, while higher noise intensities introduce substantial fluctuations and may precipitate the extinction of species within the ecosystem. Furthermore, we elucidate the abundances of prey and predator species within the ecosystem through histogram plots.

The menstrual cycle of fertile female is regulated by multiple organs and hormones, controlling the reproductive health and fertility of females. This paper is dedicated to exploring how two negative feedback loops regulate and influence the dynamic behavior of the menstrual cycle. We simplified the regulation circuit of the menstrual cycle into a nonlinear differential equation with two time delays. This simplified mathematical model integrates the hypothalamus–pituitary–ovary primary negative feedback loop and the negative feedback auxiliary loop between the ovary and the pituitary. We investigated the delay-dependent conditions for the stability of the model’s equilibrium points and also demonstrated the presence of Hopf bifurcations. Theoretical analysis suggests that the primary feedback loop is the major factor driving oscillation in the system. Further numerical simulations indicate that the negative feedback auxiliary loop allows the system to adjust its amplitude while maintaining the robustness of the cycle length. Consequently, the results obtained from this model provide new insights into the regulation of oscillations in menstrual cycle irregularities.

This paper presents a brucellosis disease model with reaction–diffusion and time delay. The model takes into account both the direct and indirect transmission of infected animals and pathogens in the environment. By analyzing the associated characteristic equation, the local stability of the unique positive equilibrium point is established. The existence of Hopf bifurcations at the positive equilibrium point is also examined by considering the discrete time delay as a bifurcation parameter. Additionally, an optimal control analysis is conducted to minimize disease outbreaks and control costs. This includes reducing the exposure of susceptible animals to infected animals, removing infected animals from herds, and reducing emissions of brucella into the environment. By constructing Hamiltonian function and applying Pontryagin’s maximum principle, the necessary conditions for the existence of optimal control are given. Finally, the existence of bifurcation periodic solutions and the effectiveness of control strategies are illustrated through numerical simulations.

Considering the food diversity of natural enemy species and the habitat complexity of prey populations, a pest-natural enemy model with non-monotonic functional response is proposed for biological management of Bemisia tabaci. The dynamic characteristics of the proposed model are analyzed. In addition, considering that the conversion from prey to predator has a certain time lag rather than instantaneous, a time delay is introduced into this model, and it is shown that the Hopf bifurcation occurs at the interior equilibrium when the time delay is used as the bifurcation parameter. Furthermore, the values of the parameters that determine the direction of the Hopf bifurcation as well as the stability of the periodic solution are calculated. In order to illustrate the theoretical analysis results, numerical simulations and validation are carried out to demonstrate the effects of non-monotonic functional response, additional food supply and habitat complexity.

In this paper, a delayed cytokine-enhanced viral infection model incorporating nonlinear incidence and immune response is proposed and studied. The global stabilities of the equilibria of the model are characterized by constructing suitable Lyapunov functionals. The existence and properties of local Hopf bifurcation are discussed by regarding the immunity delay as the bifurcation parameter. Moreover, the global existence of Hopf bifurcation has been proved by regarding the immune delay as the bifurcation parameter. Numerical simulations are carried out to validate the obtained results. The results show that ignoring the cytokine-enhanced effect makes the infection risk underevaluated. The simulations show that increasing the immune delay destabilizes the model and generates a Hopf bifurcation and stability switches occurs. Moreover, it shows that immune delay may dominate the intracellular delays in such a viral infection model which means that the immune system of the host itself is complicated during virus infection.

In this paper, the dynamical behavior in a delayed Aron–May model for malaria transmission is investigated. The basic reproduction number is defined. The global stability of the malaria-free equilibrium (MFE) is established. By using the Bendixson theorem, a sufficient condition for the global stability of the delay-free equilibrium (DFE) is also established. Furthermore, to deal with the local stability of endemic equilibrium (EE), by means of the stability switches analysis method proposed in [E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal.33(5) (2002) 1144–1165.] the related characteristic equation at EE is investigated, and the occurrence of Hopf bifurcation is discussed by using the incubation period in mosquito as a bifurcation parameter. Last, the simulation analysis is also performed to verify the dynamical behavior of the model.

In this paper, we investigate the spatiotemporal patterns of solutions to diffusive nonlocal Nicholson’s blowflies equations, wherein a natural death rate of the immature population is included in the distribution function. We first prove the positivity and boundedness of positive solutions in the model by using the minimum principle and the method of lower and upper solutions. Subsequently, we conduct a detailed bifurcation and stability analysis to obtain conditions on all the diffusion coefficients and the death rate coefficient of the immature population required for the emergence of spatiotemporal patterns, including spatially nonhomogeneous time periodic orbits. Our results indicate that the model can undergo Hopf bifurcation when the diffusion rate of the mature population passes through a sequence of critical values. Additionally, we examine the dependence of Hopf bifurcation points and bifurcated oscillations on model parameters, including the diffusion rate and death rate of the immature population. Finally, we report numerical simulations based on the bifurcation analysis to demonstrate the theoretical results, and it will help us better understand the ecological characteristics and behavioral patterns of the blowfly population.

In this paper, we study a prey–predator–top predator food chain model with nonlinear harvesting of top predator. We have derived two important thresholds: the top predator extinction threshold and the coexistence threshold. We found that the top predator will die out if the nonlinear harvesting from predator to top predator is larger than the top predator extinction threshold. On the other hand, the prey, predator and top predator coexist if the nonlinear harvesting from predator to top predator is less than the coexistence threshold. While the parameter value of nonlinear harvesting from predator to top predator is between two critical thresholds, the system displays bistability phenomena, implying that the top predator species either die out or exist with the prey and predator species, which largely depend on the initial condition. Thus, a bistable interval exists between two critical thresholds, which is a significant phenomenon for the model. Meanwhile, we performed bifurcation analysis for the model, showing that the system would arise backward/forward bifurcation and saddle-node bifurcation and Hopf bifurcation. Finally, we performed numerical simulations to verify the theoretical analysis.

In this paper, the effect of time delay is investigated on the system dynamics of a glucose-insulin model incorporating obesity. Treating the time delay as a bifurcation parameter, the stability switching on the positive equilibrium with global bifurcation is obtained. With the method of normal forms and central manifold theory, the direction and stability of limit cycles arising from Hopf bifurcation are analyzed. Using the method of multiple time scales, the normal form associated with non-resonant double Hopf bifurcation is derived. Moreover, the bifurcations are classified in the two-dimensional parameter plane near the critical point, and numerical simulations are presented to demonstrate the applicability of the theoretical results. Our results indicate that time delay in the glucose-insulin model can not only induce Hopf bifurcation and double Hopf bifurcation but also generate multiple stable periodic solutions. These results may help to understand the dynamical mechanism of glucose-insulin metabolic regulation systems, and to design control strategies for regulating and mitigating the occurrence of related diseases.