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An SIRS epidemic model with a modified nonlinear incidence rate is studied, which describes that the infectivity is strong at first as the emergence of a new disease or the reemergence of an old disease, but then the psychological effect will weaken the infectivity. Lastly, the infectivity goes to a saturation state as a result of a crowding effect. The nonlinearity of the functional form of the incidence of infection is modified, which is more reasonable biologically. We analyze the stability of the associated equilibria, and the basic reproduction number and the critical value which determine the dynamics of the model are derived. The bifurcation analysis is presented, including backward bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension two and Hopf bifurcation. To study Hopf bifurcation of codimension three of the model when some assumptions hold, the focus values are calculated. Numerical simulations are shown to verify our results. [ABSTRACT FROM AUTHOR]

A natural biological system under human interventions may exhibit complex dynamical behaviors which could lead to either the collapse or stabilization of the system. The bifurcation theory plays an important role in understanding this evolution process by modeling and analyzing the biological system. In this paper, we examine two types of biological models that Fred Brauer made pioneer contributions: predator–prey models with stocking/harvesting and epidemic models with importation/isolation. First we consider the predator–prey model with Holling type II functional response whose dynamics and bifurcations are well-understood. By considering human interventions such as constant harvesting or stocking of predators, we show that the system under human interventions undergoes imperfect bifurcation and Bogdanov-Takens bifurcation, which induces much richer dynamical behaviors such as the existence of limit cycles or homoclinic loops. Then we consider an epidemic model with constant importation/isolation of infective individuals and observe similar imperfect and Bogdanov-Takens bifurcations when the constant importation/isolation rate varies. [ABSTRACT FROM AUTHOR]

In this article, we will investigate a retarded van der Pol-Duffing oscillator with multiple delays. At first, we will find conditions for which Bogdanov-Takens (B-T) bifurcation occurs around the trivial equilibrium of the proposed system. The center manifold theory has been used to extract second order normal form of the B-T bifurcation. After that, we derived third order normal form. We also provide a few bifurcation diagrams, including those for the Hopf, double limit cycle, homoclinic, saddle-node, and Bogdanov-Takens bifurcation. In order to meet the theoretical requirements, extensive numerical simulations have been presented in the conclusion.

This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities, specifically the number of hospital beds, are limited. The feasibility and stability of the obtained equilibria are analyzed, and the basic reproduction number ( R 0 ) is obtained. We show that the system exhibits transcritical bifurcation. To show the existence of Bogdanov–Takens bifurcation, we have derived the normal form. We have also discussed a generalized Hopf (or Bautin) bifurcation at which the first Lyapunov coefficient evanescences. To show the existence of saddle-node bifurcation, we used Sotomayor's theorem. Furthermore, we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure. An optimal control setting is studied analytically using optimal control theory, and numerical simulations of the optimal regimen are presented as well. [ABSTRACT FROM AUTHOR]

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell‐to‐cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so‐called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov–Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle‐node bifurcation, Bogdanov–Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell‐to‐cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so‐called new type bifurcation persist. Finally, some conclusions and discussions are also given. [ABSTRACT FROM AUTHOR]

In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: β i (1 + ν i I i) I i S i and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov–Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates β i (due to single contact) and ν i (due to double exposures). In a connected environment, we establish a threshold R 0 = 1 between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when ν i = 0 and patch 1 has a lower infection rate, our results indicate: (i) R 0 can be nonmonotonic in dispersal rates and R 0 ≤ max { R 01 , R 02 } ( R 0 i is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When ν i > 0 and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier. [ABSTRACT FROM AUTHOR]

In this paper, a class of respiratory disease model is established by taking into account simultaneously two situations that susceptible individuals directly got sick by inhaling air pollutants and indirectly fallen ill from infection by patients with respiratory diseases. The sufficient conditions for equilibria existence of the system and locally asymptotically stability of endemic equilibria are obtained. The existence of saddle‐node bifurcation is derived by using the Sotomayor theorem. The stability of periodic solution of Hopf bifurcation is determined by calculating the first Lyapunov number. The direct conversion rate and indirect infection rate in the system are selected as bifurcation parameters; the existence of Bogdanov–Takens bifurcation of codimension 2 of the system is proved by calculating the universal unfolding near the cusp. This research shows that direct and indirect ways of getting sick have a significant influence on dynamical behaviors of the respiratory disease system. [ABSTRACT FROM AUTHOR]

The dynamic of a system of two mutualist preys which are preyed by the same predator is analyzed. As an effect of the mutualism, the carrying capacity of each prey population increases as the population size of the other increases. First, the dynamics of a prey and predator population are studied, and the conditions for coexistence are found. Then, the effect of the inclusion of a new prey on the previous predator–prey relationship is studied. In particular, dynamical properties such as Bogdanov–Takens bifurcation are preserved. The coexistence of the three species occurs by a stable equilibrium point or a stable limit cycle. Numerical simulations to exemplify the analytical results are given. [ABSTRACT FROM AUTHOR]

In the present article, we have investigated the impact of the fear effect and non-linear predator harvesting in a prey-predator interaction model with Holling type-II functional response. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. We have investigated all the biologically feasible equilibrium points and the positivity and boundedness of the system solutions. We have analyzed the local and global stability of the feasible equilibrium points in terms of the model parameters. Analytically we have established that the intrinsic growth rate of the prey population can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. The model system undergoes through Transcritical, Saddle-Node, Hopf bifurcation by considering the intrinsic growth rate of the prey population as a bifurcation parameter and Bogdanov–Takens bifurcation with respect to the prey intrinsic growth rate and rate of predator harvesting. Numerically we have identified two parametric planes, which are divided into sub-regions associated with different numbers and nature of the equilibrium points by various bifurcation curves. We also found that the system may exhibit bi-stability behavior by producing stable axial and interior equilibrium points. Lastly, the manuscript is concluded with some recommendations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the bifurcations of a susceptible-infectious-recovered-susceptible (SIRS) epidemic model with a general saturated incidence rate $ k I^p/(1+\alpha I^p) $. For general $ p > 1 $, it is shown that the model can undergo saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, and degenerate Hopf bifurcation of codimension two with the change of parameters. Combining with the results in [1] for $ 0 < p\leq 1 $, this type of SIRS model has Hopf cyclicity $ 2 $ for any $ p > 0 $. These results also improve some previous ones in [2] and [3], which are dealt with the special case of $ p = 2 $.

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 this paper, a diffusive predator–prey system with nonlocal effect and prey refuge subjects to the homogeneous Neumann boundary condition is studied. Firstly, the conditions for the occurrence of Turing, Hopf, Turing–Turing and Bogdanov–Takens bifurcations are established. Then, in order to meticulously describe the spatiotemporal dynamics resulting from the Bogdanov–Takens bifurcation, we derive an algorithm for calculating the third-order truncated normal form of Bogdanov–Takens bifurcation of this system by using the center manifold theory and normal form method, which can be applied to general diffusive predator–prey system with nonlocal effect. With the aid of the derived third-order truncated normal form, the complex spatiotemporal dynamics are theoretically predicted. At last, we carry out some numerical simulations to support the theory analysis, and the stable positive constant steady state and a pair of stable spatially inhomogeneous steady states are found. [ABSTRACT FROM AUTHOR]

We investigate spontaneous critical dynamics of excitatory and inhibitory (EI) sparsely connected populations of spiking leaky integrate-and-fire neurons with conductance-based synapses. We use a bottom-up approach to derive a single neuron gain function and a linear Poisson neuron approximation which we use to study mean-field dynamics of the EI population and its bifurcations. In the low firing rate regime, the quiescent state loses stability due to saddle-node or Hopf bifurcations. In particular, at the Bogdanov-Takens (BT) bifurcation point which is the intersection of the Hopf bifurcation and the saddle-node bifurcation lines of the 2D dynamical system, the network shows avalanche dynamics with power-law avalanche size and duration distributions. This matches the characteristics of low firing spontaneous activity in the cortex. By linearizing gain functions and excitatory and inhibitory nullclines, we can approximate the location of the BT bifurcation point. This point in the control parameter phase space corresponds to the internal balance of excitation and inhibition and a slight excess of external excitatory input to the excitatory population. Due to the tight balance of average excitation and inhibition currents, the firing of the individual cells is fluctuation-driven. Around the BT point, the spiking of neurons is a Poisson process and the population average membrane potential of neurons is approximately at the middle of the operating interval [ V Rest , V th ] . Moreover, the EI network is close to both oscillatory and active-inactive phase transition regimes. [ABSTRACT FROM AUTHOR]

To investigate the impact of the number of hospital beds on the control of infectious diseases and help allocate the limited medical resources in a region, a SEIHR epidemic model including exposed and hospitalized classes is established. Different from available models, the hospitalization rate is expressed as a function of the number of empty beds. The existence and stability of the equilibria are analyzed, and it is proved that the system undergoes backward bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation of codimension 2 under certain conditions by using the center manifold theory and normal form theory. In particular, our results show that there is a threshold value for the capacity of hospital beds in a region. If the capacity of hospital beds is lower than this threshold value, there will be a backward bifurcation, which means that even if the basic reproduction number, ℝ0, is less than unity, it is not enough to prevent the outbreaks. Before taking disease control measures, one should compare the number of beds with the threshold value to avoid misjudgment and try to increase the capacity of hospital beds above this threshold value. The method to estimate the threshold value is also given. In addition, the impacts of the duration of the exposed period on the basic reproduction number and disease transmission are investigated. [ABSTRACT FROM AUTHOR]

Refuge provides an important mechanism for preserving many ecosystems. Prey refuges directly benefit prey but also indirectly benefit predators in the long term. In this paper, we consider the complex dynamics and bifurcations in Holling-Tanner model with generalist predator and prey refuge. It is shown that the model admits a nilpotent cusp or focus of codimension 3, a nilpotent elliptic singularity of codimension at least 4, and a weak focus with order at least 3 for different parameter values. As the parameters vary, the model can undergo three types degenerate Bogdanov-Takens bifurcations of codimension 3 (cusp, focus and elliptic cases), and degenerate Hopf bifurcation of codimension 3. The system can exhibit complex dynamics, such as multiple coexistent periodic orbits and homoclinic loops. Moreover, our results indicate that the constant prey refuge prevents prey extinction and causes global coexistence. A preeminent finding is that refuge can induce a stable, large-amplitude limit cycle enclosing one or three positive steady states. Numerical simulations are provided to illustrate and complement our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a Beddington–DeAngelis prey–predator model with fear effect, refuge and harvesting is investigated. First, the positivity of solutions and boundedness of system are given. Then, the existence and local stability of equilibria of such system are obtained. Next, not only different codimension-one bifurcations, such as saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation take place, but also Bogdanov–Takens bifurcation of codimension-two occurs as predicted by the center manifold theorem and bifurcation theory. Finally, some numerical simulations are carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a Leslie–Gower model with weak Allee effect on prey and fear effect on predator is proposed. Compared with the case without fear effect on predator, the new model undergoes richer dynamic behaviors such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Also, different from the strong Allee effect on prey, the system with weak Allee effect has bistable attractors which are a largely stable limit cycle and a stable positive equilibrium, two stable equilibria, or a stable limit cycle and a stable trivial equilibrium. When the Allee effect coefficient is intermediate, fear effect on the predator can protect the prey and the predator from being extinguished. The results in this paper can be seen as a complement to those in the literatures about the Leslie–Gower model with Allee effect and fear effect. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Predator–prey interactions are the ubiquitous and natural phenomenon in an ecological system. Predators reduce the prey population's density by direct killing, which is an essential part of any ecological system. Based on the experimental works, for overcoming predation pressure, prey uses a variety of mechanisms. With Holling type-II functional response, we examined a prey–predator system incorporating anti-predator behavior and the cost of fear into prey. Prey anti-predator activity is a counterattacking strategy in which adult prey targets adolescent predators in order to counteract the potential predation pressure. Fear of predation may disrupt the physiological state of prey species and lead to long loss of prey species. In this study, we investigated this aspect to use a dynamical modeling approach. This research finds a plethora of fascinating phenomena. The studied system exhibits a wide range of dynamics and bifurcations, including saddle-node, Hopf, homoclinic, and a Bogdanov–Takens bifurcation in co-dimension two are among the dynamics and bifurcations observed in the analyzed system. We performed some numerical simulations to investigate the effects of anti-predator behavior and fear on prey and found both affect the prey–predator dynamics significantly. Our numerical examples clearly show that as prey carrying capacity increases, so does the prey's ability to perceive the risk of predation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of a predator–prey model with cooperation and Allee effect considering both deterministic and stochastic approaches. The main aim of this paper is to investigate the effect of environmental fluctuation in a deterministic predator–prey model. During the analysis of the deterministic model, it is shown that the system has saddle-node point of co-dimension 1, Hopf bifurcation and Bogdanov–Takens bifurcation of co-dimension 2. To study the effect of environmental fluctuation, we use perturbation to the birth rate of prey and death rate of predator density by Gaussian white noise. The persistence of the model and the stationary distribution is shown by forming a suitable Lyapunov function. Finally, numerical simulations are performed to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a modified May–Holling–Tanner predator–prey model by considering the Allee effect in the prey and alternative food sources for the predator. The model is analyzed theoretically as well as numerically to determine all local codimension one and two bifurcations. Using the local parametrization method and Hopf bifurcation theorem, we analyze the Hopf bifurcation and compute its corresponding normal form coefficient to reveal its criticality. We specially derive normal form of the system near Bogdanov–Takens and generalized Hopf bifurcations, to determine possible bifurcation scenarios near each bifurcation. For Bogdanov–Takens, we further determine a set of parameters for which curves of saddle-node, Hopf and Homoclinic bifurcations can be observed. By numerical continuation technique, we compute several curves of equilibria and bifurcations, and detect different bifurcation points on these curves. For the computed bifurcation points, we numerically compute their corresponding normal form coefficients. We specially compute a family of limit cycles and limit point cycle bifurcation on this family. [ABSTRACT FROM AUTHOR]

A Leslie–Gower predator–prey model with square root response function and generalist predator is considered, and the existence and stability of equilibria of the system are discussed. It is shown that the system undergoes a degenerate Hopf bifurcation of codimension exactly two, where there exist two limit cycles. In addition, we find that the system has a cusp of codimension two and exhibits a Bogdanov–Takens bifurcation of codimension two. Our results reveal richer dynamics than the system with no generalist predator. [ABSTRACT FROM AUTHOR]

This paper investigates the Andronov–Hopf and Bogdanov–Takens bifurcations in a three-dimensional Hindmarsh–Rose system with switching dynamics. A switching policy is applied to the slow recovery variable, dividing the system into two smooth subsystems by setting the switching threshold to zero. The presence of visible or invisible fold and cusp singularities in the sliding vector field is examined. By examining conditions satisfied by the trace and determinant, this study establishes the existence of Andronov–Hopf and Bogdanov–Takens bifurcations. The necessary and sufficient conditions for the occurrence of Andronov–Hopf and Bogdanov–Takens bifurcations are derived. Through numerical simulations, we have obtained the irregular limit cycle phase portrait with sawtooth-shaped and the corresponding time series of spike peaks gradually attenuating within the firing period, these new firing phenomena attributed to the switching policy. We have gathered statistics on the spike counts of membrane potential during firing events within a specific range of parameter d. Thus, we can achieve a wide variety of bursting patterns through this switching system. Furthermore, single-parameter and two-parameter bifurcation diagrams are generated by varying the key parameters b and d , we found that the chaotic and periodic regions intertwine with each other, with chaotic regions existing within the periodic regions, and periodic states found within the chaotic regions. The number of spikes during a firing period in neurons can provide important clues about the neuronal activity state, information processing, and potential diseases. In summary, this study demonstrates that the Filippov Hindmarsh–Rose system, characterized by its slow variable switching threshold, significantly influences the firing behavior of the system. • The study derived the necessary and sufficient conditions for Hopf and Bogdanov-Takens bifurcations. • The irregular limit cycle phase portrait with sawtooth-shaped attributed to the switching policy. • Choosing the slow recovery variable as the switching threshold enriched various bursting states. [ABSTRACT FROM AUTHOR]

Dynamics of an interconnected population of excitatory and inhibitory spiking neurons wandering around a Bogdanov-Takens (BT) bifurcation point can generate the observed scale-free avalanches at the population level and the highly variable spike patterns of individual neurons. These characteristics match experimental findings for spontaneous intrinsic activity in the brain. In this paper, we address the mechanisms causing the system to get and remain near this BT point. We propose an effective stochastic neural field model which captures the dynamics of the mean-field model. We show how the network tunes itself through local long-term synaptic plasticity by STDP and short-term synaptic depression to be close to this bifurcation point. The mesoscopic model that we derive matches the directed percolation model at the absorbing state phase transition. [ABSTRACT FROM AUTHOR]

A tritrophic food chain model of Kolmogorov type is analyzed. The mesopredator and superpredator populations are specialist, the functional responses are general and the prey has general growth rate function. Independently of the functional responses and of the growth rate of the prey, conditions to have the existence of stable limit sets obtained by a Bogdanov–Takens bifurcation are proved. The general results, fixing Holling type functional responses and logistic growth rate for the prey are illustrated. Moreover, derived from our Bogdanov–Takens bifurcation results, the coexistence of the three species by means of stable limit cycles or quasiperiodic orbits with chaotic motion is numerically shown. Finally, general formulae for the second-order homoclinic predictor are computed and numerically illustrated. [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]

Nonmonotone incidence and saturated treatment are incorporated into an SIRS model under constant and changing environments. The nonmonotone incidence rate describes the psychological or inhibitory effect: when the number of the infected individuals exceeds a certain level, the infection function decreases. The saturated treatment function describes the effect of infected individuals being delayed for treatment due to the limitation of medical resources. In a constant environment, the model undergoes a sequence of bifurcations including backward bifurcation, degenerate Bogdanov-Takens bifurcation of codimension 3, degenerate Hopf bifurcation as the parameters vary, and the model exhibits rich dynamics such as bistability, tristability, multiple periodic orbits, and homoclinic orbits. Moreover, we provide some sufficient conditions to guarantee the global asymptotical stability of the disease-free equilibrium or the unique positive equilibrium. Our results indicate that there exist three critical values r 1 , r 2 and r 3 for the treatment rate r: (i) when r ≥ max { r 1 , r 2 } , the disease will disappear; (ii) when r < min { r 1 , r 3 } , the disease will persist. In a changing environment, the infective population starts along the stable disease-free state (or an endemic state) and surprisingly continues tracking the unstable disease-free state (or a limit cycle) when the system crosses a bifurcation point, and eventually tends to the stable endemic state (or the stable disease-free state). This transient tracking of the unstable disease-free state when R 0 > 1 predicts regime shifts that cause the delayed disease outbreak in a changing environment. Furthermore, the disease can disappear in advance (or belatedly) if the rate of environmental change is negative and large (or small). The transient dynamics of an infectious disease heavily depend on the initial infection number and rate or the speed of environmental change. [ABSTRACT FROM AUTHOR]

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov-Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions.

In the paper, stability and bifurcation behaviors of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response are studied theoretically and numerically. Mathematical theory works mainly give some critical threshold conditions to guarantee the existence and stability of all possible equilibrium points, and the occurrence of Hopf bifurcation and Bogdanov-Takens bifurcation. Numerical simulation works mainly display that the Bazykin's predator-prey ecosystem has complex dynamic behaviors, which also directly proves that the theoretical results are effective and feasible. Furthermore, it is easy to see from numerical simulation results that some key parameters can seriously affect the dynamic behavior evolution process of the Bazykin's predator-prey ecosystem. Moreover, limit cycle is proposed in view of the supercritical Hopf bifurcation. Finally, it is expected that these results will contribute to the dynamical behaviors of predator-prey ecosystem.

Dynamics of an interconnected population of excitatory and inhibitory spiking neurons wandering around a Bogdanov-Takens (BT) bifurcation point can generate the observed scale-free avalanches at the population level and the highly variable spike patterns of individual neurons. These characteristics match experimental findings for spontaneous intrinsic activity in the brain. In this paper, we address the mechanisms causing the system to get and remain near this BT point. We propose an effective stochastic neural field model which captures the dynamics of the mean-field model. We show how the network tunes itself through local long-term synaptic plasticity by STDP and short-term synaptic depression to be close to this bifurcation point. The mesoscopic model that we derive matches the directed percolation model at the absorbing state phase transition.

In this paper, as a complement to the works by Monterio and Notargiacomo, we analyze the dynamical behavior of a learning-process model in a case where the system admits a unique interior degenerate equilibrium. Meanwhile, we acquire the sufficient condition for the cusp of codimension 2 and verify that the system undergoes Bogdanov–Takens bifurcation around the cusp. Finally, we give a numerical simulation to support the theoretical results.

An SIRS epidemic model with a modified nonlinear incidence rate is studied, which describes that the infectivity is strong at first as the emergence of a new disease or the reemergence of an old disease, but then the psychological effect will weaken the infectivity. Lastly, the infectivity goes to a saturation state as a result of a crowding effect. The nonlinearity of the functional form of the incidence of infection is modified, which is more reasonable biologically. We analyze the stability of the associated equilibria, and the basic reproduction number and the critical value which determine the dynamics of the model are derived. The bifurcation analysis is presented, including backward bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension two and Hopf bifurcation. To study Hopf bifurcation of codimension three of the model when some assumptions hold, the focus values are calculated. Numerical simulations are shown to verify our results.

In this paper, we improve a new mathematical model associated with glue applying of particleboard. First, we study the existence and stability of the equilibria and the existence of fold, Hopf, and Bogdanov–Takens bifurcations in above system. Second, the normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation are derived, and the classifications of local dynamics near above bifurcation critical values are analyzed. Then, numerical simulation results show that the glue flow system associated with glue applying of particleboard exists stable equilibrium, stable periodic‐1, periodic‐2, and periodic‐4 solutions, and chaotic attractor phenomenon from a sequence of period‐doubling bifurcations. Finally, we compare the dynamical phenomena of glue flow system with frequency water supply system, showing that cubic terms which the latter lacks, could influence the dynamical behaviors of glue flow system. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by [Math Processing Error] I) exceeds a certain level, the incidence rate is a decreasing function with respect to [Math Processing Error] I. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with [Math Processing Error] I until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value [Math Processing Error] I 0 ~ [Math Processing Error] (= b d) for the infective level [Math Processing Error] I 0 at which the health care system reaches its capacity such that: (i) When [Math Processing Error] I 0 ≥ I 0 ~ , the transmission dynamics of the model is determined by the basic reproduction number [Math Processing Error] R 0 : [Math Processing Error] R 0 = 1 separates disease persistence from disease eradication. (ii) When [Math Processing Error] I 0 < I 0 ~ , the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation. [ABSTRACT FROM AUTHOR]

The main purpose of the article is to classify all the possible bursting oscillations in a vector field with Bogdanov–Takens bifurcation at the origin. Based on the universal unfolding of the normal form of the vector field, the topological structure in the neighborhood of the bifurcation point on the unfolding parameters is presented. Replacing one of the unfolding parameters by a slow-varying periodic exciting term, the coupling of two scales in frequency domain involves the vector field, which may lead to the bursting oscillations. According to the bifurcation analysis, we focus on three typical cases to investigate the dynamical evolution with the increase of the exciting amplitude. By introducing the transformed phase portrait, the mechanism of bursting oscillations can be presented. Three types of bifurcations, that is, fold, Hopf, and saddle on the limit cycle bifurcations may cause the alternations of the trajectory between the quiescent states and the spiking states, different combinations of which may result in different bursting attractors. Furthermore, the inertia of the movement may result in the delay effect of the bifurcation, which may lead to the disappearance of the bifurcation influence and the corresponding spiking oscillations. [ABSTRACT FROM AUTHOR]

The stability and two kinds of bifurcations of a genetic regulatory network are considered in this paper. We have given a complete stability analysis involved in mentioned model. The Hopf bifurcation of codimension 1 and Bogdanov–Takens bifurcation of codimension 2 for the nonhyperbolic equilibria of the model is characterized analytically. In order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and a numerical example is given to illustrate graphically. Three bifurcation curves related to Bogdanov–Takens bifurcation, namely saddle-node, Hopf and homoclinic bifurcation curves, are given explicitly by calculating a universal unfolding near the cusp. Moreover, the numerical continuation results show that the model has other bifurcation types, including saddle-node and cusp bifurcations. The bifurcation diagram and phase portraits are also given to verify the validity of the theoretical results. These results show that there exists rich bifurcation behavior in the genetic regulatory network. [ABSTRACT FROM AUTHOR]

In this paper, we propose a predator-prey system with fear effect and predator death due to toxins based on the phenomenon that African predators death due to eating toxin-carrying carcasses. First, the existence and stability of equilibria are proved. Second, the conditions for the occurrence of different types of bifurcations are established by varying parameters. Our results indicate that (1) when the cost of minimum fear is taken as the bifurcation parameter, it can determine the stability and direction of periodic solution of Hopf bifurcation; (2) when the predation rate and the cost of minimum fear are taken as the bifurcation parameters, the system undergoes Bogdanov-Takens (BT) bifurcation of codimension two, and the BT bifurcation will disappear when the system is not affected by toxins. At last, some numerical simulations verify the validity of theoretical results, and we obtain that toxins not only inhibit the survival of predator populations but also affect the density of prey populations. Moreover, only an appropriate cost of minimum fear maintains long-term coexistence of prey and predator populations. [ABSTRACT FROM AUTHOR]

Abstract A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

Abstract A diffusive phytoplankton–zooplankton model with nonlinear harvesting is considered in this paper. Firstly, using the harvesting as the parameter, we get the existence and stability of the positive steady state, and also investigate the existence of spatially homogeneous and inhomogeneous periodic solutions. Then, by applying the normal form theory and center manifold theorem, we give the stability and direction of Hopf bifurcation from the positive steady state. In addition, we also prove the existence of the Bogdanov–Takens bifurcation. These results reveal that the harvesting and diffusion really affect the spatiotemporal complexity of the system. Finally, numerical simulations are also given to support our theoretical analysis.

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]

In the present paper we have investigated the impact of predator harvesting in a two-dimensional prey–predator model with Holling type III functional response. The main objective of this paper is to study the change of dynamical behaviour of the prey–predator model in the presence of non-linear predator harvesting. The model system shows complex dynamics with the change of different system parameters. We have established the positivity and boundedness of the solutions under a certain parametric condition with non-negative initial conditions. The existence and stability criterion of different equilibrium points are investigated in terms of system parameters. We have shown that the system undergoes through saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation under different parametric conditions. The structural changes or the system bifurcations predict the global dynamics of the model system. We have computed the first Lyapunov number to find the direction of the Hopf-bifurcating periodic solution of the system. Using numerical simulation we have studied here the effect of conversion efficiency, protection of the environment to the prey population and non-linear predator harvesting on the model dynamics. Finally the paper is ended with some conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Leslie–Gower predator–prey model with Allee effect on the prey and a linear functional response. Here the Allee effect impacts the birth rate of the prey, which is different from the common multiplicative and additive Allee effects. The model is well-posed, that is, all solutions are bounded. Applying the blow-up method indirectly, we prove that the origin which is not an equilibrium of the system is an attractor. Then we study the existence and stability of equilibria, which indicate that the system undergoes bifurcations. With the help of Sotomayor's theorem, we show the occurrence of saddle-node bifurcation. Moreover, there is degenerate Hopf bifurcation of codimension at least three. By choosing two (three) parameters of the system as bifurcation parameters and calculating a versal unfolding near the cusp, we demonstrate that the system undergoes Bogdanov–Takens bifurcation of codimension two (three). These theoretical results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated global dynamics of a two-species food chain model with the Holling type III functional response that includes linear harvesting for the prey and nonlinear harvesting for the predator. The long-time continued existence of both species is discussed using uniform persistence theory. Stability of various equilibrium points is described in terms of model parameters. The local asymptotic stability of non-hyperbolic equilibrium points is determined with the help of center manifold theorem. Global behavior of solutions of the model system when both species are present is determined by considering the global properties of the coexistence equilibrium. Here, we have taken a comprehensive view by considering different bifurcations of co-dimension one and two and have discussed the importance of various model parameters on the system dynamics. The model system shows much more complex and realistic behavior compared to a model system without any harvesting, with constant harvesting or linear-yield harvesting of either or both of the species. Numerical simulations have been conducted to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we systematical study the rich dynamics and complex bifurcations of a non-monotonic predator–prey system with a constant releasing rate for the predator. We prove that the system can have at most three positive equilibria, and can undergo a sequence of bifurcations, including transcritical, saddle-node, Hopf, degenerate Hopf, double limit cycle, saddle-node homoclinic bifurcation (or homoclinic loop with a saddle-node), cusp bifurcation of codimension 2, and Bogdanov–Takens bifurcation of codimension 2 and 3. And the system can generate very rich dynamics, such as the existence of a semi-stable limit cycle, multiple coexistent periodic orbits, homoclinic loops, etc. Moreover, our results show that the dynamical behaviors highly rely on the constant releasing rate of predators and the initial conditions. That is, there exists a critical value of the constant releasing rate of predators such that (i) when the constant releasing rate is greater than the critical value, the prey goes to extinction for all admissible initial populations of both species; (ii) when the constant releasing rate is less than the critical value, the prey can always coexist with the predator. Numerical simulations are presented to verify the main results. [ABSTRACT FROM AUTHOR]

In this paper, strong Allee effects on the bifurcation of the predator–prey model with ratio-dependent Holling type III response are considered, where the prey in the model is subject to a strong Allee effect. The existence and stability of equilibria and the detailed behavior of possible bifurcations are discussed. Specifically, the existence of saddle-node bifurcation is analyzed by using Sotomayor's theorem, the direction of Hopf bifurcation is determined, with two bifurcation parameters, the occurrence of Bogdanov–Takens of codimension 2 is showed through calculation of the universal unfolding near the cusp. Comparing with the cases with a weak Allee effect and no Allee effect, the results show that the Allee effect plays a significant role in determining the stability and bifurcation phenomena of the model. It favors the coexistence of the predator and prey, can lead to more complex dynamical behaviors, not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation. Numerical simulations and phase portraits are also given to verify our theoretical analysis. [ABSTRACT FROM AUTHOR]