Codimension one and two bifurcations of a discrete stage-structured population model with self-limitation

In this paper, the bifurcations of a discrete stage-structured population model with self-limitation between the two subgroups are investigated. We explore all possible codimension-one bifurcations associated with transcritical, flip (period doubling) and Neimark-Sacker bifurcations and discuss the stabilities of the fixed points in these non-hyperbolic cases. Meanwhile, we give the explicit approximate expression of the closed invariant curve which is caused by the Neimark-Sacker bifurcation. After that, through the theory of approximation by a flow, we explore the codimension two bifurcations associated with 1:3 strong resonance. We convert the nondegenerate condition of 1:3 resonance into a parametric polynomial, and determine its sign by the theory of complete discrimination system. We introduce new parameters and utilize some variable substitutions to obtain the bifurcation curves around 1:3 resonance, which are returned to the original variables and parameters to express for easy verification. By using a series of complicated approximate identity transformations and polar coordinate transformation, we explore 1:6 weak resonance. Moreover, we calculate the two boundaries of Arnold tongue which are caused by 1:6 weak resonance and defined as the resonance region. Numerical simulations and numerical bifurcation analyzes are made to demonstrate the effective of the theoretical analyzes and to present the relations between these bifurcations. Furthermore, our theoretical analyzes and numerical simulations are explained from the biological point of view.     

Role of transmissible disease in an infected prey-dependent predator – prey system

The role of disease in ecological systems is a very important issue from both mathematical and ecological points of view. This paper deals with the qualitative analysis of a prey-dependent predator – prey system in which a disease is spreading among the prey species only. We have analysed the behaviour of the system around each equilibrium and obtained conditions for global stability of the system around an equilibrium by using suitable Lypunov functions. We have also worked out the region of parametric space under which the system enters a Hopf bifurcation and a transcritical bifurcation but does not experience either saddle-node bifurcations or pitchfork bifurcations around the disease-free equilibrium E ₂. Finally, we have given an example of a real ecological situation with experimental data simulations.     

Dynamics of two-cell systems with discrete delays

We consider the system of delay differential equations (DDE) representing the models containing two cells with time-delayed connections. We investigate global, local stability and the bifurcations of the trivial solution under some generic conditions on the Taylor coefficients of the DDE. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including pitchfork, transcritical and Hopf bifurcation) and Takens-Bogdanov bifurcation as a codimension two bifurcation. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. Finally, we show that the analytical results agree with numerical simulations.     

Homoclinic Flip Bifurcations Accompanied by Transcritical Bifurcation

The bifurcations of orbit flip homoclinic loop with nonhyperbolic equilibria are investigated. By constructing local coordinate systems near the unperturbed homoclinic orbit, Poincaré maps for the new system are established. Then the existence of homoclinic orbit and the periodic orbit is studied for the system accompanied with transcritical bifurcation.     

Delay induced subcritical Hopf bifurcation in a diffusive predator-prey model with herd behavior and hyperbolic mortality

In this paper, we consider the dynamics of a delayed diffusive predator-prey model with herd behavior and hyperbolic mortality under Neumann boundary conditions. Firstly, by analyzing the characteristic equations in detail and taking the delay as a bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations induced by delay are investigated. Then, applying the normal form theory and the center manifold argument for partial functional differential equations, the formula determining the properties of the Hopf bifurcation are obtained. Finally, some numerical simulations are also carried out and we obtain the unstable spatial periodic solutions, which are induced by the subcritical Hopf bifurcation.     

Dynamics of a Discrete Two-species Competitive Model with Michaelies-Menten Type Harvesting in the First Species

In this paper, we use a semidiscretization method to derive a discrete two-species competitive model with Michaelis-Menten type harvesting in the first species. First, the existence and local stability of fixed points of the system are investigated by employing a key lemma. Subsequently, the transcritical bifurcation, period-doubling bifurcation and pitchfork bifurcation of the model are investigated by using the Center Manifold Theorem and bifurcation theory. Finally, numerical simulations are presented to illustrate corresponding theoretical results.     

Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

Bifurcations of solitary waves are classified for the generalized nonlinear Schrödinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely, saddle‐node, pitchfork, and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obtained. It is shown that for pitchfork and transcritical bifurcations, their power diagrams look differently from their familiar solution‐bifurcation diagrams. Numerical examples for these three types of bifurcations are given as well. Of these numerical examples, one shows a transcritical bifurcation, which is the first report of transcritical bifurcations in the generalized nonlinear Schrödinger equations. Another shows a power loop phenomenon which contains several saddle‐node bifurcations, and a third example shows double pitchfork bifurcations. These numerical examples are in good agreement with the analytical results.     

Bifurcation analysis of the flour beetle population growth equations

In this article, we study a discrete delayed flour beetle population equation. Firstly, we study the existence of period-doubling bifurcation and Neimark–Sacker bifurcations for the system by analysing its characteristic equations. Secondly, we investigate the direction of the two bifurcations and the stability of the bifurcation periodic solutions by using normal form theory. Finally, some numerical simulations are carried out to support the analytical results.     

Bifurcation analysis on a discrete model of Nicholson's blowflies

In this paper, the discrete Nicholson's blowflies model with delay is considered. First, the stability of the equilibria of the system is investigated by analyzing the characteristic equation and then the existence of fold and Neimark–Sacker bifurcations are verified. Subsequent to that, the direction and stability of the bifurcation are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out in order to support the results of mathematical analysis.     

Numerical investigation of bifurcations of equilibria and Hopf bifurcations in disease transmission models

One of the general SIRS disease transmission model is considered under the assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. A combination of analytical and numerical techniques is used to show that (for some parameters) the bifurcations of equilibria can occur and also asymptotically orbitally stable periodic solutions with asymptotic phase can arise through Hopf bifurcations. The investigation is based on computer simulation of bifurcation manifolds in the parameter space. Hopf bifurcations are investigated on the base of center manifold theory by the computation of bifurcation parameters and the approximation of Hopf-bifurcating cycles by bifurcation formulas. This method finds the limit cycle to a good approximation and also its stability. For computer simulations the necessary computer oriented algorithms were developed and encoded by C++. Some results of computer simulations are presented and numerical evidence of existence of bifurcations of equilibria and Hopf bifurcations for the considered model is provided.     

关于Guzowska-Luís-Elaydi模型的分岔

本文考虑了一个离散的Logistic竞争模型.为了讨论分岔,给出了不动点的拓扑类型及非双曲的情况.应用中心流行约化定理,证明了跨临界分岔会在三个不动点上发生.本文还证明了在两个不动点处,跳跃分岔会发生,同时稳定的周期2轨会出现.     

Imperfect transcritical and pitchfork bifurcations

Imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on Banach spaces. In particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established. Applications include semilinear elliptic equations, imperfect Euler buckling beam problem and perturbed diffusive logistic equation.     

具阶段结构的多时滞SIR模型的稳定性分析

考虑了一类具有两个阶段结构的SIR模型,得到了解的正性和有界性,通过分析特征方程根的分布,以(?)_1和(?)_2为参数分析了平衡点的稳定性和局部Hopf分支存在性.进一步地,利用规范型和中心流型理论,给出了决定Hopf分支方向和分支周期解的稳定性的隐式算法.最后利用一些数值模拟来支持所得到的理论分析结果.     

Global Dynamics of a Tuberculosis Epidemic Model and the Influence of Backward Bifurcation

In this paper, we propose and analyze a tuberculosis (TB) model with exogenous re-infection. We assume that treated individual may be again infected by infectious individual. The model exhibits two bifurcations viz. transcritical bifurcation when the basic reproductive number R ₀?=?1 and backward bifurcation where the disease transmission rate β plays as control parameter. The persistent of the model and, the local and global stability criteria of disease-free and endemic equilibria are discussed. By carrying out bifurcation analysis, it is shown that the model exhibits the bistability and undergoes the Hopf bifurcation when immunological memory is everlasting i.e. when σ?=?0. Lastly, some simulations are given to verify our analytical results.     

LOCALIZED PATTERNS OF THE SWIFT-HOHENBERG EQUATION WITH A DISSIPATIVE TERM

In this paper, the normal form analysis of quadratic-cubic Swift-Hohenberg equation with a dissipative term is investigated by using the multiple-scale method. In addition, we obtain Hamiltonian-Hopf bifurcations of two equilibria and homoclinic snaking bifurcations of one-peak and two-peak homoclinic solutions by numerical simulations.     

Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure

In this paper, an eco-epidemiological model with a stage structure is considered. The asymptotical stability of the five equilibria, the existence of stability switches about positive equilibrium, is investigated. It is found that Hopf bifurcation occurs when the delay τ passes though a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.     

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.     

一类具有非线性发生率和治疗函数的传染病模型研究

传染病动力学系统的数学建模中,合理的使用非线性发生率往往更能使模型与实际相吻合.并且在实际的疾病防治过程中,由于受到空间人力物力资源的影响一般存在最大治疗容量的限制.结合这两种情况建立了一类含非线性发生率和最大治疗容量限制的传染病模型.通过分析这个模型,得到无病平衡点和正平衡点的存在性、稳定性.进一步取发生率和治疗系统达到最大容量时的感染者人数作为分支参数,得到了Hopf分支和Bogdanov-Takens分支的存在条件,并进行了数值模拟.     

Bifurcation analysis in an age-structured model of a single species living in two identical patches

In this paper, we consider the age-structured model of a single species living in two identical patches derived in So et al. [J.W.-H. So, J. Wu, X. Zou, Structured population on two patches: modeling dispersal and delay, J. Math. Biol. 43 (2001) 37–51]. We chose a birth function that is frequently used but different from the one used in So et al. which leads to a different structure of the homogeneous equilibria. We investigate the stability of these equilibria and Hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation. By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out for supporting the analytic results.