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The impact of air pollutants on the human respiratory system has received more and more attention. However, due to the differences in human characteristics and the strength of protective measures, different people have different symptoms in air pollution and viral infection environments. In this paper, an S I 1 I 2 P dynamic model of respiratory diseases infected by virus under air pollution is established by introducing air pollutant concentration compartment. Qualitative research shows that there are seven equilibria in the system, the daily average emission and clearance rate of pollutants are the key parameters affecting the existence and stability of equilibria. In this paper, by using Sotomayor's theorem and center manifold theory, it is proved that the system undergoes saddle–node bifurcation or Bogdanov–Takens bifurcation of co-dimension 3 at the boundary equilibrium, and exhibits saddle–node bifurcation at the endemic equilibrium. Numerical simulation results show that wearing masks in haze weather can not only reduce the number of allergic patients, but also reduce the number of viral patients. In addition, reducing the daily average emission of air pollution and increasing its clearance rate can also reduce the number of allergic patients. It is worth mentioning that this measure is more effective in controlling viral respiratory diseases. • The heterogeneity of patients is taken into consideration. • The average daily emission of air pollution, the clearance rate and the proportion of illness are the key parameters affecting the existence, local asymptotic stability and bifurcation. • Respiratory diseases can be controlled by reducing the daily average emission of air pollution and its clearance rate, and this measure has a better control effect on viral respiratory diseases. [ABSTRACT FROM AUTHOR]

In this paper, an SIR epidemic model considering hospital resources and vaccination is established and the rich dynamics and complex bifurcations are investigated. Firstly, the existence of disease-free equilibrium and endemic equilibria is explored. It is founded that when the vaccination rate is not high, the number of endemic equilibrium points changed easily with the number of hospital resources and vaccination, resulting in transcritical bifurcation and saddle–node bifurcations. Secondly, different types singularities such as degenerate saddle–node of codimension 1 at the disease-free equilibrium, and cusp or focus type Bogdanov–Takens singularities of codimension 3 at endemic equilibria are presented. Thirdly, bifurcation analysis at these equilibria is investigated, and it is found that the system undergoes a sequence of bifurcations, including Hopf, degenerate Hopf bifurcation, homoclinic bifurcation, the cusp type Bogdanov–Takens bifurcation of codimension 2, and the focus type Bogdanov–Takens bifurcation of codimension 3 which are the organizing centers for a series of bifurcations with lower codimension. And the system shows very rich dynamics such as the existence of multiple coexistent periodic orbits, homoclinic loops. Finally, numerical simulations are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a delayed virotherapy model including infected tumor cells, uninfected tumor cells and free virus. In this model, both infected and uninfected tumor cells have special growth patterns, and there are at most two positive equilibria. We mainly analyze the stability and Hopf bifurcation of the model under different time delays. For the model without delay, we study the Hopf and Bogdanov–Takens bifurcations. For the delayed model, by center manifold theorem and normal form theory of functional differential equation, we study the direction of Hopf bifurcation and stability of the bifurcated periodic solution. Moreover, we prove the existence of Zero-Hopf bifurcation. Finally, some numerical simulations show the results of our theoretical calculations, and the dynamic behaviors near Zero-Hopf and Bogdanov–Takens point of the system are also observed in the simulations, such as bistability, periodic coexistence and chaotic behavior. • A delayed virotherapy model including infected and uninfected tumor cells is established. • We mainly analyze the stability and Hopf bifurcation of the model under different time delays. The system also undergoes Bogdanov–Takens and Zero-Hopf bifurcation. • In this paper, some interesting dynamic phenomena are simulated, such as bistability, periodic coexistence and chaotic behavior. [ABSTRACT FROM AUTHOR]

Study of a food chain model under uncertainty is quite difficult. Because, in an uncertain food chain model, the biological parameters can't be determined accurately. The aim of this work is to study the stability and local bifurcations (Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation) of an imprecise prey–predator system in an uncertain environment. The proposed imprecise model is formulated by considering two more realistic factors: the effect of a fear factor on the growth rate of prey population and non-linear harvesting of predator population. To study the proposed imprecise system mathematically, the dynamical interactions between the imprecise species are presented by the system of governing interval differential equations. And to study the dynamics of the proposed imprecise system theoretically, it is modelled in a precise way by the linear parametric representation of the interval. Then all the theoretical analyses, including Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens (BT) bifurcation of the interior equilibrium point of the proposed imprecise model are discussed in parametric form. To verify all the theoretical analyses of the proposed imprecise model, numerical simulations with interval-valued hypothetical data of the imprecise parameters are performed graphically. Finally, the work is concluded with some biological consequences. [ABSTRACT FROM AUTHOR]

In this paper, a Gause type predator–prey system with constant-yield prey harvesting and monotone ascending functional response is proposed and investigated. We focus on the influence of the harvesting rate on the predator–prey system. First, equilibria corresponding to different situations are investigated, as well as the stability analysis. Then bifurcations are explored at nonhyperbolic equilibria, and we give the conditions for the occurrence of two saddle–node bifurcations by analyzing the emergence, coincidence and annihilation of equilibria. We calculate the Lyapunov number and focal values to determine the stability and the quantity of limit cycles generated by supercritical, subcritical and degenerate Hopf bifurcations. Furthermore, the system is unfolded to explore the repelling and attracting Bogdanov–Takens bifurcations by perturbing two bifurcation parameters near the cusp. It is shown that there exists one limit cycle, or one homoclinic loop, or two limit cycles for different parameter values. Therefore, the system is susceptible to both the constant-yield prey harvesting and initial values of the species. Finally, we run numerical simulations to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Highlights • The nonlinear system has multiple internal equilibria. • Saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation may occur. • Heteroclinic and homoclinic orbit, bistability may appear. • Some numerical simulations are given. Abstract We propose and investigate a predator-prey model with selective nonlinear harvesting for the prey and predator, such harvesting increases smoothly to a limit value when the density of harvested population is large enough. The existence of nonlinear harvesting makes the dynamics of the model more complicated, including multiple equilibria, limit cycle, heteroclinic and homoclinic orbit, bistability, saddle-node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation and Bogdanov–Takens bifurcation. Furthermore, some numerical simulations are given to illustrate these results. [ABSTRACT FROM AUTHOR]

This manuscript deals with an epidemic model with partial immunity having nonlinear incidence and saturated treatment. Positivity and boundedness of the solutions have been established here. We discuss local stability of all equilibria. The proposed system experiences various types of bifurcations, namely Transcritical, Saddle–node, Hopf, and Bogdanov–Takens bifurcation of co-dimension 2. The system is reduced to a two-dimensional system using center manifold theorem to deduce normal form for Bogdanov–Takens bifurcation of co-dimension 2 when two eigenvalues at the endemic equilibrium point becomes zero. Whether deterministic model overestimates the condition for disease propagation, to observe this we also analysis stochastic model. We derive the condition for extinction, persistence in mean and stationary distribution. All theoretical findings are justified by numerical simulations. Finally, to check validity of the model, we fit it with real reported influenza data of Canada. • SIRS model with logistic growth, saturated type incidence and treatment. • Bogdanov–Takens bifurcation normal form using Center manifold theorem. • Introduction of stochasticity in the model and it's analysis. • Model is validated fitting with real influenza outbreak data of Canada. [ABSTRACT FROM AUTHOR]

In this work, an analytical framework to understand nonlinear dynamics of plasma perturbations model is introduced. In particular, we analyze the model presented by Constantinescu et al. [20] which consists of three coupled ODEs and contains three parameters. The basic dynamical properties of the system are first investigated by the ways of bifurcation diagrams, phase portraits and Lyapunov exponents. Then, the normal form technique and perturbation methods are applied so as to the different types of bifurcations that exist in the model are investigated. It is proved that pitcfork, Bogdanov–Takens, Andronov–Hopf bifurcations, degenerate Hopf and homoclinic bifurcation can occur in phase space of the model. Also, the model can exhibit quasiperiodicity and chaotic behavior. Numerical simulations confirm our theoretical analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation and stability of a ratio-dependent predator-prey model with nonconstant predator harvesting rate. The analysis is carried out both analytically and numerically. We determine stability and dynamical behaviours of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Fold, Hopf, Cusp, and Bogdanov–Takens bifurcations. We use the numerical software MATCONT, to compute curves of equilibria and to compute several bifurcation curves. We especially approximate a family of limit cycles emanating from a Hopf point. Our results generalize and improve some known results and show that the model has more rich dynamics than the ratio-dependent predator-prey model without harvesting rate. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model describing the interaction of malignant glioma cells, macrophages and glioma specific CD8+T cells is discussed. The biologically feasible equilibria and corresponding local stability are deduced. The bifurcations and related dynamical behaviors of this model are further studied thoroughly. The existence of transcritical bifurcation and saddle–node bifurcation is derived based on Sotomayor's theorem and Hopf bifurcation is well discussed. The codimension 2 bifurcation such as Bogdanov–Takens bifurcation is investigated using the normal form theory and center manifold theorem in more detail. Finally, numerical simulations are obtained to validate our analytical findings by varying the parameters. • The dynamics of a mathematical model describing the interaction of malignantglioma cells, macrophages and glioma specific CD8+T cells are discussed. • The co-existence of equilibria of the model are obtained by numerical simulation. • The existence of codimension 1 and codimension 2 bifurcations is investigated. • Extensive numerical simulations are obtained to validate our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the fear effect and Leslie–Gower function on the dynamic behavior of the predator–prey model is considered. First, the well-posedness analysis of this model is demonstrated, and the existence and local stability of the equilibrium point are given. Then by using the bifurcation theory and selecting the appropriate bifurcation parameters, many types of bifurcation phenomena in the system are discovered, including transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Meanwhile, the numerical simulations on the basis of theoretical analysis are further carried out to intuitively show the influence of fear effect on population. As the degree of fear increases, the system will undergo multiple dynamic behaviors switching until the final extinction of prey population, while the predator population will survive due to the presence of substitute prey. In addition, the initial density of predator and prey can determine whether the solution of system tends to a coexisting steady state or a periodic oscillation. • A novel prey–predator model with the impact of fear is introduced. • The system can experience many types of bifurcation phenomena. • Numerical simulations and sensitivity analysis are performed. • Our results show that fear effect plays an important role on deriving the complex dynamics. [ABSTRACT FROM AUTHOR]

Highlights: [•] Comparative analysis of Bogdanov–Takens and Andronov–Poincare–Hopf bifurcations. [•] Tuning a PID controller using Bogdanov–Takens bifurcation and describing function. [•] Tuning a PID controller by using the Andronov–Poincaré–Hopf bifurcation. [•] Analysis of self-oscillating and chaotic behavior for the servomechanism. [•] Use of chaotic behavior to improve the PID controller performance. [ABSTRACT FROM AUTHOR]

Highlights: [•] The effects of the hunting delay on the dynamics of the predator–prey model are investigated. [•] The procedure for the computation of normal forms of type-II Bogdanov–Takens bifurcation is addressed. [•] The dynamical classification near the Bogdanov–Takens bifurcation point is determined. [•] The delay-induced homoclinic orbit, heteroclinic orbit and subcritical Hopf bifurcation are obtained. [ABSTRACT FROM AUTHOR]

• Local dynamics analysis of ecological model, for host-generalist parasitoid system. • Parametric and geometric conditions to obtain Coexistence Equilibrium Points (CEPs). • Hopf and non-degenerate Bogdanov-Takens bifurcation. • Biological control of hosts can be controlled via an unstable limit cycle. • Generalist parasitoids are of fundamental importance in the stability of the host-generalist parasitoid system. Parasitoids are ubiquitous in nature and they play a fundamental role in the equilibrium of ecosystems. Their interactions with other species are particularly crucial for agricultural activities, since they act as a biological control of invasive hosts. In this paper we study Hopf, generalized Hopf and nondegenerate Bogdanov-Takens bifurcations in a simple model which describes the biological control of invasive species through predation by generalist parasitoids. We assume logistic growth for the generalist parasitoid in the absence of host species, and their interaction is described by a Holling type II functional response. We determine a geometric condition which characterizes local bifurcations associated with the appearance of periodic orbits. Further, we find conditions on the parameters which yield to the extinction of the host species, independently of the initial conditions and the charge capacity of the generalist parasitoid. Our results are supported by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a classical van der Pol’s equation with delayed feedback. Triple-zero bifurcation is investigated by using center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at the triple-zero bifurcation and show that the model can exhibit transcritical bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and zero-Hopf bifurcation. Some numerical simulations are given to support the analytic results. [Copyright &y& Elsevier]

This paper presents a purely dynamic economic-environmental growth model with a S -shaped production function and a pollution externality. We first outline the role of the saving rate in inducing/suppressing a dual steady-state, one with lower capital and pollution (the "green" steady state) than the other (the "dirty" steady state). Assuming the saving rate is such that the economy possesses a dual steady state, the second part of the paper exploits global analysis tecniques to study conditions such that the economy is able to escape the dirty steady state and approach the green steady state. [ABSTRACT FROM AUTHOR]