Takens Bifurcation Of Codimension Research Articles

The effect of human heterogeneity on the transmission of viral respiratory diseases under air pollution

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 SI1I2P 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.

Bifurcation Analysis of a Holling–Tanner Model with Generalist Predator and Constant-Yield Harvesting

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.

BIFURCATION ANALYSIS OF AN SIR MODEL WITH SATURATED INCIDENCE RATE AND STRONG ALLEE EFFECT

In this paper, rich dynamics and complex bifurcations of an SIR epidemic model with saturated incidence rate and strong Allee effect are investigated. First, the existence of disease-free and endemic equilibria is explored, and we prove that the system has at most three positive equilibria, which exhibit different types such as hyperbolic saddle and node, degenerate unstable saddle (node) of codimension 2, degenerate saddle-node of codimension 3 at disease-free equilibria, and cusp, focus, and elliptic types Bogdanov–Takens singularities of codimension 3 at endemic equilibria. Second, bifurcation analysis at these equilibria are investigated, and it is found that the system undergoes a series of bifurcations, including transcritical, saddle node, Hopf, degenerate Hopf, homoclinic, cusp type Bogdanov–Takens of codimensional 2, and focus and elliptic type Bogdanov–Takens bifurcation of codimension 3 which are composed of some bifurcations with lower codimension. The system shows very rich dynamics such as the coexistence of multiple periodic orbits and homoclinic loops. Finally, numerical simulations are conducted on the theoretical results.

Bifurcation Analysis of a Predator–Prey Model with Alternative Prey and Prey Refuges

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.

Bogdanov–Takens Bifurcation of Codimension 3 in the Gierer–Meinhardt Model

Bogdanov–Takens Bifurcation of Codimension 3 in the Gierer–Meinhardt Model

Degenerate codimension-2 cusp of limit cycles in a Holling–Tanner model with harvesting and anti-predator behavior

In this paper, a Holling–Tanner predator–prey model with constant-yield prey harvesting and anti-predator behavior is investigated. It is shown that the codimension of a nilpotent cusp is 3 and a degenerate Bogdanov–Takens bifurcation of codimension 3 acts as an organizing center for rich dynamical behaviors including saddle–node bifurcation, Hopf bifurcation, saddle–node bifurcation of limit cycles, and homoclinic bifurcation. Moreover, a Hopf bifurcation of codimension 2 is found, a codimension-2 cusp of limit cycles is firstly observed, and the corresponding normal form coefficients are also obtained, which indicate that there exists an acute angle region of three coexistent limit cycles. In particular, a reversed S-shaped saddle–node bifurcation curve of limit cycles is found. In addition, we observe that the anti-predator behavior may cause nearly extinction of the predator population while the prey population reaches its maximum which is less than the carrying capacity due to the effect of harvesting; however, the constant-yield prey harvesting may drive both species to extinction suddenly. Finally, numerical simulations including bifurcation diagrams and phase portraits are given to illustrate the theoretical results.

Dynamical behaviors of an epidemic model with partial immunity having nonlinear incidence and saturated treatment in deterministic and stochastic environments

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.

Bifurcation of an SIRS Model with a Modified Nonlinear Incidence Rate

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.

Bifurcation analysis of an SIR model considering hospital resources and vaccination

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.

Dynamics of a SIR epidemic model with variable recruitment and quadratic treatment

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].

Bifurcation analysis of a respiratory disease model about air pollution direct and indirect effects

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.

Multiple bifurcations in a predator–prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting

In this paper, a modified Leslie-type predator–prey model with simplified Holling type IV functional response is established, in which double Allee effect on prey and nonlinear prey harvesting are considered. The analysis of the model shows that there exists a Bogdanov–Takens singularity (focus case) of codimension 4, and also multiple other nonhyperbolic and degenerate equilibria. Bifurcations are explored and it is found that transcritical bifurcation, saddle–node bifurcation, Bogdanov–Takens bifurcation of codimension 2, degenerate cusp type Bogdanov–Takens bifurcation of codimension 3, and degenerate focus type Bogdanov–Takens bifurcation of codimension 4 occur as parameters vary. The bifurcations result in complex dynamic behaviors, such as double limit cycle, triple limit cycle, quadruple limit cycle, cuspidal loop, (multiple) homoclinic loop, saddle–node loop, and limit cycle(s) simultaneously with homoclinic loop. We run numerical simulations to verify the theoretical results, and it is found that the system admits bistability, tristability, or even tetrastability.

DYNAMIC INTERACTIONS BETWEEN PREY AND PREDATOR WITH COOPERATION AND ALLEE EFFECT: DETERMINISTIC AND STOCHASTIC APPROACH

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.

Complex Dynamics of Predator–Prey Systems with Allee Effect

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.

Bifurcation Analysis of Multiscale Malaria Model with Serratia AS1 Bacteria and Saturated Treatment

Recent experimental results have implied that Serratia AS1 bacteria efficiently suppress the development of P. falciparum, and rapidly disseminate via vertical transmission and sexual transmission throughout mosquito population. In this paper, we propose a new malaria model to investigate the roles of saturated treatment and AS1 bacteria in malaria transmission. One feature of the model is that human and mosquito populations dynamics run on disparate temporal scales. Singular perturbation techniques are used for the separation of fast and slow dynamics. It also enables us to overcome the difficulties of exploring bifurcations in higher dimensional system. We demonstrate that the reduced systems may undergo backward bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension 2, which indicates that the delayed effect for treatment has important implications for malaria control. Numerical simulations substantiate these analytical results and show epidemiological insights. Finally, the model without AS1 bacteria transmission is applied to simulate the accumulative malaria cases in Burundi. We observe that malaria may not be eliminated by a single control in Burundi. A variety of available control measures should be adopted to eradicate malaria, such as improving treatment rate, reducing the delayed effect for treatment, and possibly releasing mosquitoes that carry AS1 bacteria.

Dynamics of a ratio-dependent Leslie–Gower predator–prey model with Allee effect and fear effect

We propose a ratio-dependent Leslie–Gower predator–prey model with the Allee effect and fear effect on prey and study its dynamic behaviors. On the basis of Poincaré transformation and blow-up method, we find that the solutions of the system are bounded and the origin is attractive. We consider the existence of equilibria and analyze their stability. The bifurcation of the system was analyzed, including the occurrence of saddle–node bifurcation, degenerate Hopf bifurcation, and Bogdanov–Takens bifurcation. The results show that the system has a cusp of codimension two and undergoes a Bogdanov–Takens bifurcation of codimension two. Numerical simulation results show that there exist two limit cycles (the inner one is stable and the outer one is unstable) and a Bogdanov–Takens bifurcation of codimension two in the system.

Bifurcation analysis of a predator–prey model with Beddington–DeAngelis functional response and predator competition

In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response and predator competition, which is a five‐parameter family of planar vector field. It is shown that the model can undergo a sequence of bifurcations including focus type degenerate Bogdanov–Takens bifurcation of codimension 3 and Hopf bifurcation of codimension at least 2 as the parameters vary. Our theoretical results indicate that predator competition can cause richer dynamics such as two limit cycles enclosing one or three hyperbolic positive equilibria and three kinds of homoclinic orbits (homoclinic to hyperbolic saddle, saddle‐node, or neutral saddle). Moreover, there exists a threshold value for predator capturing rate , below or equal to which the predators always tend to extinction, above which the predators and preys will coexist in the form of multiple steady states or periodic oscillations for all positive initial populations. Numerical simulations are presented to illustrate the theoretical results.

Modeling Allee Effect in the Leslie-Gower Predator–Prey System Incorporating a Prey Refuge

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.

Stability and Bifurcation in a Leslie–Gower Predator–Prey Model with Allee Effect

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.

Linking bifurcation analysis of Holling–Tanner model with generalist predator to a changing environment

AbstractBifurcation theory has been highly popular in the analysis of mathematical models. However, stability and bifurcation analyses are only for asymptotic dynamics while applied scientists care more about transient dynamics. In this paper, we first rigorously analyze Holling–Tanner model with generalist predators who have alternative food sources, and then discuss transient dynamics via a changing environment. For a constant environment, we provide a complete bifurcation analysis with high codimension. It is shown that the highest codimension of a nilpotent cusp is 3, and the model can undergo degenerate Bogdanov–Takens bifurcation of codimension 3. Moreover, by using resultant elimination to solve the semialgebraic varieties of Lyapunov coefficients, we show that a center‐type equilibrium is a weak focus with order at most 2, and the model can exhibit Hopf bifurcation of codimension 2. Our results indicate that generalist predators can cause not only richer dynamics and bifurcations, but also the extinction of prey for some positive initial densities. Numerical simulations, including the coexistence of a limit cycle and a homoclinic cycle, tristability, two limit cycles, are presented to illustrate the theoretical results. In a changing environment, the populations start along one stable state but can track unstable states or oscillations when the system crosses a bifurcation point, and then tend to another stable state or oscillations. This tracking on transient dynamics predicts regime shifts under environmental changes. When environmental conditions vary, the populations can track unstable states in the constant environment. The rate of environmental change determines how long the system tracks an unstable state although finally the solution under environmental change is attracted to a stable steady state or limit cycle. Finally, we focus on a periodic environment and find that the populations converge to a periodic solution or an invariant torus depending on both the initial environmental capacity and the amplitude of periodic fluctuation.