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This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey's growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter (θ) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter? [ABSTRACT FROM AUTHOR]

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov–Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)- C ∞ -smooth and (non)-analytic periodic center manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a model called the k-nearest neighbor queue with the possibility of having delayed queue length feedback. We prove fluid limits for the stochastic queueing model and show that the fluid limit converges to a system of delay differential equations. Using the properties of circulant matrices, we derive a closed form expression for the value of the critical delay, which governs whether the delayed information will induce oscillations or a Hopf bifurcation in our queueing system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a quadratic chaotic system modeling self-excited and hidden attractors which is described by a system of three nonlinear ordinary differential equations with three real parameters. The primary goal is to establish the existence of two limit cycles that bifurcate based on the system's nature as an electronic circuits model, specifically via Hopf bifurcation. Notably, the application of the first and second Lyapunov coefficients is utilized to demonstrate the bifurcation of two limit cycles from an equilibrium point near a Hopf critical point. Furthermore, employing the first-order averaging theory enables us to confirm the presence of unstable periodic orbits originating from the zero-Hopf equilibrium. [ABSTRACT FROM AUTHOR]

A two-degree-of-freedom nonlinear high-speed railway wheelset model with two time delays in the lateral and yaw dampers is studied. The aim is to investigate the effect of time delays on stability and Hopf bifurcation characteristics of the wheelset model. The local stability of the trivial equilibrium under different time delay conditions is qualitatively analyzed. Analytical studies reveal that the wheelset model undergoes stability switches with the variation of the time delays. The stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Furthermore, properties of Hopf bifurcation including direction and stability of bifurcating limit cycles are studied by using the normal form theory and the center manifold theorem. Our findings indicate that time delays in both lateral dampers and yaw dampers influence the stability and direction of Hopf bifurcation. Additionally, the numerical results show that time delays in the lateral and yaw dampers not only affect the amplitude of the hunting motion of the wheelset but also the periodic and chaotic motions. If the time delays gradually increase, the wheelset will vibrate irregularly with large lateral displacements. The analytical results presented in this paper offer a theoretical reference for the stability design of wheelsets. [ABSTRACT FROM AUTHOR]

The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator's food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Leslie–Gower predator–prey model with strong Allee effect on prey and fear effect on predator. We discuss the existence and local stability of equilibria by making full use of qualitative analytical theory. It is shown that the above system exhibits at most two positive equilibria and it can undergo a series of bifurcation phenomena. We indicate that the dynamical behavior of the model is closely related to the fear effect on predator. In detail, when the fear effect parameter p = p ∗ , the system will undergo degenerate Hopf bifurcation. There exist two limit cycles (the inner is stable and the outer is unstable). However, when p = p ∗ , the system will undergo degenerate Bogdanov–Takens bifurcation. Also, by numerical simulation, we conclude that the stronger the fear effect, the bigger the density of prey species. The above shows that fear effect on predator is beneficial to the persistence of the prey species. Our results can be seen as a complement to previous works [González-Olivares et al., 2011; Pal & Mandal, 2014]. [ABSTRACT FROM AUTHOR]

Hidden chaotic attractors is a fascinating subject of study in the field of nonlinear dynamics. Jerk systems with a stable equilibrium may produce hidden chaotic attractors. This paper seeks to enhance our understanding of hidden chaotic dynamics in jerk systems of three variables (x , y , z) with nonlinear terms from a predefined set: { − x y + d z 2 , − x y + d z tanh (z) } , where d is a real parameter. The behavior of the systems is analyzed using rigorous Hopf bifurcation analysis and numerical simulations, including phase portraits, bifurcation diagrams, Lyapunov spectra, and basins of attraction. For certain jerk systems with a subcritical Hopf bifurcation, adjusting the coefficient of a linear term can lead to hidden chaotic behavior. The adjustment modifies the subcritical Hopf equilibrium, transforming it from an unstable state to a stable one. One such jerk system, while maintaining its equilibrium stability, experiences a sudden transition from a point attractor to a stable limit cycle. The latter undergoes a period-doubling route to chaos, which may be followed by a reverse route. Therefore, by perturbing certain jerk systems with a subcritical Hopf equilibrium, we can gain insights into the formation of hidden chaotic attractors. Furthermore, adjusting the coefficient of the nonlinear term z tanh (z) in certain systems with a stable equilibrium can also lead to period-doubling routes or reverse period-doubling routes to hidden chaotic dynamics. Both findings are significant for our understanding of the hidden chaotic dynamics that can emerge from nonlinear systems with a stable equilibrium. [ABSTRACT FROM AUTHOR]

The lumbar disc has complex structure and material properties, and if it develops disease, it cannot heal itself. Therefore, prevention of lumbar disc herniation is very important. In this paper, the model of lumbar disc is built, and the nonlinear dynamic response of the system is researched. A modified Van der Pol model is imported to describe the stress–strain curves of the lumbar disc. The system's dynamic model is set up, and harmonic balance method is applied to revise the natural frequency of the system. The product of numerical simulation reveals that the lumbar disc has complex dynamic characteristics, including balance point, limit cycle bifurcation and stochastic Hopf bifurcation. By changing the parameters, we can avoid large-scale periodic movement of the lumbar disc which causes lumbar disc herniation. These results contribute to the prevention of lumbar disc herniation. [ABSTRACT FROM AUTHOR]

In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter 0 < ≪ 1 , the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the codimensions of Hopf bifurcation and Bogdanov–Takens bifurcation of a predator–prey model with alternative prey and prey refuges, which was proposed by Chen et al. [2023]. The results show that the predator–prey model can undergo a supercritical Hopf bifurcation or a Bogdanov–Takens bifurcation of codimension two under certain parameter conditions. It means that there are some predator–prey models with an alternative prey and prey refuges which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Hopf bifurcation is at most one and codimension of Bogdanov–Takens bifurcation is at most two. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Holling type II predator–prey system with prey refuge, Allee effect, fear effect and time delay. The existence and stability of the equilibria of the system are investigated. Under the variation of the delay as a parameter, the system experiences a Hopf bifurcation at the positive equilibrium when the delay crosses some critical values. We also analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution by the center manifold theorem and normal form theory. We show that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. In particular, the birth rate plays an important role in the stability of the equilibria. Examples with associated numerical simulations are provided to prove our main results. [ABSTRACT FROM AUTHOR]

The Leslie–Gower model, a kind of predator–prey model with weak Allee effect, is studied in this paper. The existence and stability of non-negative equilibria are first discussed. Then, we investigate several bifurcation phenomena undergoing positive equilibria, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, etc. Some possible dynamical behaviors of this model are illustrated by numerical simulation. The bifurcation diagrams for the cases of codimensions 2 and 3 are given respectively. The coexistence of a periodic cycle and a homoclinic cycle, and two limit cycles enclosing an unstable equilibrium are also proved. This appears to be the first study of the Leslie–Gower model including the influence of weak Allee effect on prey. [ABSTRACT FROM AUTHOR]

In the course of reverse transcription, drug-sensitive strain can mutate to a drug-resistant one. Cell-to-cell transmission is proved to be more efficient than virus-to-cell infection. The population of Cytotoxic T Lymphocytes (CTLs) can be divided into CTL precursors and CTL effectors. Naive CTL precursors are those that never have contacted antigen, circulating around the body. Once the host is infected, CTL precursors (CTLp) begin to proliferate and differentiate into CTL effectors (CTLe), which have antiviral activity. In this paper, we study an HIV model with drug-sensitive and drug-resistant mixed strains, CTLp and CTLe, both viral and cellular infections. The basic reproduction ratios for each strain are derived and the stability of each feasible equilibrium is analyzed by using Routh–Hurwitz criteria. The periodic solutions may occur near the equilibrium with CTL immune response via Hopf bifurcation. By using normal form theory, we find more information about the limit cycles, such as the amplitude, phase and stability. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Allee effect and predator competition in the Bazykin's model with Holling I functional response. Theoretically, we analyze the existence and stability of equilibria, and derive the existence conditions of saddle-node bifurcation and Hopf bifurcation. In addition, in order to determine the stability of limit cycles, we explicitly calculate the first Lyapunov coefficient and prove that the positive equilibrium is not a center, but a weak focus with a multiplicity of at least two. Therefore, the system has Hopf bifurcation and Bautin bifurcation with two limit cycles. Our results indicate that the Allee effect and predator competition lead to a series of complex dynamic phenomena. Finally, numerical simulation verifies the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey model in which the prey has the additive Allee effect and the predator has artificially controlled migration is proposed. When the system introduces additive Allee effect and artificially controlled migration, more complicated dynamical behavior is obtained. The system can undergo saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Two limit cycles are found and discussed. The influence of the additive Allee effect and artificially controlled migration on the dynamics of the system is also presented. In detail, when the Allee effect is large, the prey will become extinct. When the artificially controlled migration rate is larger, the intensity of the prey (pest) will be smaller and the intensity of the predator will be larger. This indicates that artificially controlled migration can be effectively used to control the pest. [ABSTRACT FROM AUTHOR]

In this paper, we apply bifurcation theory to consider four predator–prey systems which include the Allee effect, and show that the species having a strong Allee effct may affect their predation and hence extinction risk. It is shown that the models with the Allee effect exhibit more complex dynamical behaviors compared with that without the Allee effect. In particular, two models with no Allee effect do not have Hopf bifurcation, but can have Hopf bifurcation with the Allee effect; and one model, which does not have Bogdanov–Takens bifurcation if no Allee effect is involved, can have Bogdanov–Takens bifurcation of codimension two. Especially, for one model with Holling type II functional response of the predator to the prey, the Allee effect not only completely changes the stability of the equilibrium at the origin, but also changes the supercritical Hopf bifurcation arising from an interior equilibrium to subcritical Hopf bifurcation with very limited parameter values to yield unstable limit cycles, and further increases the system's stability. [ABSTRACT FROM AUTHOR]

In this paper, a dynamic model given by three-dimensional ordinary differential equations is studied to determine how the dynamics of tumor growth is controlled by some key parameters. By varying the competition coefficient between healthy host cells and tumor cells, a Hopf bifurcation occurs in this system, leading to the creation of a stable limit cycle. Through numerical analysis of the continuity of this limit cycle, we find that the cascade of period-doubling bifurcations leads to the generation of a chaotic attractor. Moreover, the region of attractors is shown in the parameter space. Numerical simulations, bifurcation diagrams, Lyapunov exponent graph and phase portraits permit to highlight the rich and complex phenomena presented by the model. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a lower bound for the maximum number of crossing limit cycles for cubic planar switching systems with two regions separated by a straight line. By pseudo Hopf bifurcation and -order Lyapunov constants, we prove that there exist cubic near-integrable switching systems with at least 26 small-amplitude limit cycles bifurcating from an elementary center after suitable perturbations. To the best of our knowledge, this is the best lower bound so far for the cubic class by using a first order analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator–prey system in which the prey population is infected from a parasite and the growth of susceptible prey is suppressed due to fear of predation. We consider that the predators have the ability to distinguish between the susceptible and infected prey items, and they avoid the infected ones to reduce fitness cost. The predators are assumed to die naturally and also due to intraspecific competition. The proposed model is analyzed mathematically for the feasibility and stability of the system's equilibria. We also discuss the existence of Hopf bifurcation by taking the feeding preference of predators as a bifurcation parameter. We perform global sensitivity analysis to identify model parameters having significant impact on the density of predator population in the ecosystem. Our simulation results show the stabilizing role of selective feeding of predators whereas fear factor and disease prevalence induce limit cycle oscillations. Feeding more the predators with additional foods bring stability in the system by evacuating the persistent oscillations. To model the situation more realistically, we consider that the parameters representing the cost of fear and the feeding preference of predators vary with time. For the seasonally forced system, conditions are obtained for which the system has at least one positive periodic solution; global attractivity of the positive periodic solution is also discussed. Our seasonally forced model demonstrates the appearance of a unique periodic solution, higher periodic solutions and complex bursting patterns. [ABSTRACT FROM AUTHOR]

This paper considers a Leslie-Gower predator–prey system with Allee effect and prey refuge. By considering the prey refuge constant as a parameter, we analyze the stability of the equilibria in the system, and find that there are abundant dynamic behaviors. It is shown that the model can undergo a sequence of bifurcations including saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension two or three as the parameters vary. Moreover, the model underdoes a degenerate Hopf bifurcation of codimension two and has two limit cycles, where the inner one is stable and the outer one is unstable. Through some numerical simulations, the occurrence of Bogdanov–Takens bifurcation and Hopf bifurcation of codimension two are confirmed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nongeneric quadratic reversible system with piecewise polynomial perturbations. We use the expansion of the first order Melnikov function to obtain the maximal number of small-amplitude limit cycles produced by Hopf bifurcation in the perturbed systems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a system in which both prey and predator are harvested with constant rate. Our main objective is to find the effects of harvesting on equilibria, stability, and bifurcations in the system, which may be useful for biological management. The existence and stability of equilibrium points of the model are further investigated. A thorough qualitative analysis has been carried out based on bifurcation theory in dynamical systems and to validate our analytical findings, a large scale numerical simulation has been performed by using plausible values of parameters involved. It is shown that the model can exhibit Hopf bifurcation. The first Lyapunov coefficient is calculated to determine the direction of limit cycle of Hopf bifurcation. Also, it has been proven analytically that the system exhibits Bogdanov–Takens bifurcation of codimension 2. Moreover, discrete-time delay effect has been included due to gestation of the predator species on the same system and observed Hopf bifurcation with respect to the delay parameter. This study renders important tools for investigations of the dynamics of biotic organisms for the management and control of over harvesting. Some phase plane analysis has been carried out to support our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an upper bound for the number of small-amplitude limit cycles produced by Hopf bifurcation in one particular type of rational Liénard systems in the form of ẋ = y − q n (x) / p m (x) , ẏ = − g (x) , where q n (x) and p m (x) are polynomials in x with degrees n and m , respectively. Furthermore, we show that the upper bound presented here is sharp in the case of m = n. [ABSTRACT FROM AUTHOR]

By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space. [ABSTRACT FROM AUTHOR]

In this paper, we present a numeric bifurcation analysis of the normal form of degenerate Hopf bifurcation truncated up to seventh order with an equilibrium point located at the origin. By applying the genericity nondegenerate conditions and normal form theory, we study the bifurcation analysis of the codimension-3 Takens–Hopf bifurcation for the difficult case, where a rich bifurcation scenario is displayed. The third Lyapunov coefficient is used to distinguish the different cases of a codimension-3 Takens–Hopf bifurcation point, which can be efficiently computed with the aid of a software program based on the symbolic package Maple, presented in Appendix A. The normal form analysis results can be used to depict the complete bifurcation diagrams and phase portraits. In order to investigate the mechanism of the transitions between equilibrium and limit cycles, the methods of two scales in frequency domain are employed to study the evolutions. [ABSTRACT FROM AUTHOR]

In this work, we study a predator–prey model of Gause type, in which the prey growth rate is subject to an Allee effect and the action of the predator over the prey is determined by a generalized hyperbolic-type functional response, which is neither differentiable nor locally Lipschitz at the predator axis. This kind of functional response is an extension of the so-called square root functional response, used to model systems in which the prey have a strong herd structure. We study the behavior of the solutions in the first quadrant and the existence of limit cycles. We prove that, for a wide choice of parameters, the solutions arrive at the predator axis in finite time. We also characterize the existence of an equilibrium point and, when it exists, we provide necessary and sufficient conditions for it to be a center-type equilibrium. In fact, we show that the set of parameters that yield a center-type equilibrium, is the graph of a function with an open domain. We also prove that any center-type equilibrium is stable and it always possesses a supercritical Hopf bifurcation. In particular, we guarantee the existence of a unique limit cycle, for small perturbations of the system. [ABSTRACT FROM AUTHOR]

The dynamical behavior of a variable recruitment SIR model has been investigated with the nonlinear incidence rate and the quadratic treatment function for a horizontally transmitted infectious disease that sustains for a long period (more than one year). For a long duration, we have incorporated human fertility in variable recruitment. The societal effort, i.e. all types of medical infrastructures, have a vital role in controlling such a disease. For this reason, we have considered the quadratic treatment function, which divides the system into two subsystems. We have established the existence and stability of different equilibrium points that depend mainly on the societal effort parameter in both subsystems and also global stability. Different rich dynamics such as forward bifurcation, Hopf bifurcation, limit cycle, and Bogdanov–Takens bifurcation of co-dimension 2 have been established by using bifurcation theory and the biological significance of these dynamics has been explained. Different numerical examples have been considered to illustrate the theoretical results. Finally, we have discussed the advantage of our model with the model by Eckalbar and Eckalbar [Nonlinear Anal.: Real World Appl. 12 (2011) 320–332]. [ABSTRACT FROM AUTHOR]

The differential equations of motion are established for a three-degree-freedom wing dynamic model subjected to unsteady aerodynamic loads and random perturbations. The system is dimensionally reduced by the improved average method to obtain the standard equations. Flutter problems of the deterministic wing system with high-order structural nonlinearity are studied using Hopf bifurcation theory and numerical simulation, the critical flutter speed is obtained and the effectiveness of the improved average method in the process of dimensionality reduction is verified. The stochastic P-bifurcation behaviors of the system are analyzed considering the effects of random perturbations of the longitudinal airflow by examining the qualitative variations of the probability density function curves. The results show that the deterministic wing system has a secondary bifurcation, a bistable phenomenon in which the equilibrium and the limit cycle oscillations coexist. The random disturbances significantly increases the critical flutter speed of the wing system, and the amplitude of limit cycle oscillations increases after including random perturbations for the same incoming flow speed. [ABSTRACT FROM AUTHOR]

Compared with most known singularly degenerate heteroclinic cycles consisting of two different equilibria of a line or a curve, or two parallel lines of semi-hyperbolic equilibria, little seems to be noticed about the one that connects two perpendicular lines of semi-hyperbolic equilibria, i.e. E z = (0 , 0 , z) and E x = (x , e x , k + e f) , z , x ∈ ℝ , which is found in the mathematical chaos model: ẋ = a (y − x) + d x z , ẏ = k x + f y − x z , ż = − e x 2 + x y + c z when c = 0 and e (a + f d) = (a − k d). Surprisingly, apple-shape attractors are also created nearby that kind of singularly degenerate heteroclinic cycles in the case of small c > 0. Further, some other rich dynamics are uncovered, i.e. global boundedness, Hopf bifurcation, limit cycles coexisting with one chaotic attractor, etc. We not only prove that the ultimate bound sets and globally exponentially attracting sets perfectly coincide under the same parameters, but also illustrate four limit cycles coexisting with one chaotic attractor, the saddle in the origin, and other two stable nontrivial node-foci, which are also trapped in the obtained globally exponentially attracting set, extending the recently reported results of the Lü-type subsystem. In addition, combining theoretical analysis and numerical simulation, the bidirectional forming mechanism of that kind of singularly degenerate heteroclinic cycles is illustrated, and their collapses may create three-scroll/apple-shape attractors, or limit cycles, etc. Finally, conservative chaotic flows are numerically found in the new system. We expect that the outcome of the study may provide a reference for subsequent research. [ABSTRACT FROM AUTHOR]

In this work, a prey–predator system with Holling type II response function including a Michaelis–Menten type capture and fear effect is put forward to be studied. Firstly, the existence and stability of equilibria of the system are discussed. Then, by considering the harvesting coefficient as bifurcation parameter, the occurrence of Hopf bifurcation at the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation are proved. Furthermore, through the analysis of fear effect and capture item, we find that: (i) the fear effect can either stabilize the system by excluding periodic solutions or destroy the stability of the system and produce periodic oscillation behavior; (ii) increasing the level of fear can reduce the final number of predators, but not lead to extinction; (iii) the harvesting coefficient also has significant influence on the persistence of the predator. Finally, numerical simulations are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

Modeling the interplay between immune system components and cancer cells via immunotherapy is the purpose of this work. We present a simple mathematical model of interaction between tumor cells and the immune system's effector cells. With rigorous mathematical analysis and numerical continuation, we study the generalized Hopf bifurcations (GH), known as Bautin bifurcation, including the Allee effect, presenting also Hopf bifurcation. The Hopf bifurcation indicates the existence of limit cycles. Bautin suggests two limit cycles; consequently, the model presents the equilibrium phase in immunoediting theory. Finally, the numerical continuation is performed to support the analytical results. [ABSTRACT FROM AUTHOR]