Turing-Hopf bifurcation in the reaction-diffusion system with delay and application to a diffusive predator-prey model

The interactions of diffusion-driven Turing instability and delay-induced Hopf bifurcation always give rise to rich spatiotemporal dynamics. In this paper, we first derive the algorithm for the normal forms associated with the Turing-Hopf bifurcation in the reaction-diffusion system with delay, which can be used to investigate the spatiotemporal dynamical classification near the Turing-Hopf bifurcation point in the parameter plane. Then, we consider a diffusive predator-prey model with weak Allee effect and delay. Through investigating the dynamics of the corresponding normal form of Turing-Hopf bifurcation induced by diffusion and delay, the spatiotemporal dynamics near this bifurcation point can be divided into six categories. Especially, stable spatially homogeneous/inhomogeneous periodic solutions and steady states, coexistence of two stable spatially inhomogeneous periodic solutions, coexistence of two stable spaially inhomogeneous steady states and the transition from one kind of spatiotemporal patterns to another are found.     

Deformations of functionals and bifurcations of extremals

We study homology characteristics of critical values and extremals of Lipschitz functionals defined on bounded closed convex subsets of a reflexive space that are invariant under deformations. Sufficient conditions for the existence of a bifurcation point of a multivalued potential operator (the switch principle for the typical number of an extremal) are established.     

Scaling Methods and Approximative Equations for Homogeneous Reaction—Diffusion Systems and Applications to Epidemics

Summary. We consider a reaction-diffusion equation that is homogeneous of degree one. This homogeneity is a symmetry. The dynamics is factorized into trivial evolution due to symmetry and nontrivial behavior by a projection to an appropriate hypermanifold. The resulting evolution equations are rather complex. We examine the bifurcation behavior of a stationary point of the projected system. For these purposes we develop techniques for dimension reduction similar to the Ginzburg-Landau (GL) approximation, the modulation equations. Since we are not in the classical GL situation, the remaining approximative equations have a quadratic nonlinearity and the amplitude does not scale with ε but with ε ² where ε ² denotes the bifurcation parameter. Moreover, the symmetry requires that not only one but two equations are necessary to describe the behavior of the system. We investigate traveling fronts for the modulation equations. This result is used to analyze an epidemic model. Received April 9, 1996; second revision received January 3, 1997; final revision received October 7, 1997; accepted January 19, 1998     

Global bifurcation of positive equilibria in nonlinear population models

Existence of nontrivial nonnegative equilibrium solutions for age-structured population models with nonlinear diffusion is investigated. Introducing a parameter measuring the intensity of the fertility, global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium. Moreover, for the parameter-independent model we establish existence of positive equilibria by means of a fixed point theorem for conical shells.     

一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

研究了一类不可压的广义neo-Hookean材料组成的球体的空穴分岔问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo-Hookean材料,得到了球体内部空穴生成的条件．与均匀各向同性的neo-Hookean球体的情况相比,证明了当摄动参数属于某些区域时,从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点,空穴生成时的临界载荷会比无摄动的材料的临界载荷小．用奇点理论证明了,空穴分岔方程在临界点附近等价于具有单边约束条件的正规形．用最小势能原理分别讨论了空穴分岔解的稳定性和实际稳定的平衡状态．     

Periodically forced nonlinear systems of difference equations

The existance of nontrivial (x=0( periodic solutions of a general class of periodic nonlinear difference equations is proved using bifurcation theory methods. Specifically, the existance of a global continuum of nontrivial periodicsolutions that bifurcates from the trivial solution (x=0) is proved. Conditions are given under which the nontrivial solutions are positive. A prerrequisite Fredholm and adjoint operator theory for linear periodic systems is developed. An application to application dynamics is made.     

Stability and Hopf bifurcation of a delayed Cohen–Grossberg neural network with diffusion

In this paper, a delayed Cohen–Grossberg neural network with diffusion under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equation, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation at the trivial steady state are established, respectively. By using the normal form theory and the center manifold reduction of partial function differential equations, formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.     

Bifurcation direction and exchange of stability for variational inequalities on nonconvex sets

This paper concerns Crandall–Rabinowitz type bifurcation for abstract variational inequalities on nonconvex sets and with multidimensional bifurcation parameter. We derive formulae which determine the bifurcation direction and, in the case of potential operators, the stability of all solutions close to the bifurcation point. In particular, it follows that in some cases an exchange of stability appears similar to the case of equations, but in some other cases stable nontrivial solutions bifurcate at points where there is no loss of stability of the trivial solution. As an application we consider a system of two second order ODEs with nonconvex unilateral boundary conditions.     

A nonintersection property for extremals of variational problems with vector-valued functions

In this work we study the structure of extremals of variational problems with vector-valued functions on [0,∞). We show that if an extremal is not periodic, then the corresponding curve in the phase space does not intersect itself.     

Bifurcation and stability of a Mimura–Tsujikawa model with nonlocal delay effect

In this paper, we investigate a Mimura–Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady‐state solutions bifurcating from the nonzero steady‐state solution. Moreover, we illustrate our general results by applications to models with a one‐dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd.     

The propagation of travelling waves in an open cubic autocatalytic chemical system

The reaction-diffusion travelling waves that can be initiatedin an open isothermal chemical system governed by cubic autocatalytickinetics are discussed. The system is shown to be capable ofsustaining up to three spatially uniform steady states, the(trivial) unreacted state, which is always stable (a node),and two nontrivial states, one of which is always unstable (asaddle point). The third state can change its stability throughHopf bifurcation (both subcritical and supercritical). Thisallows the possibility of two sorts of travelling wave beingestablished; there are wave profiles which connect the unreactedstate ahead to the nontrivial state at the rear, and wave profiles(pulse waves) which have the unreacted state at both the frontand rear. The conditions under which a particular wave is initiatedare considered by both a discussion of the (ordinary) differentialequations governing the travelling waves and by numerical integrationsof an initial-value problem. This treatment also reveals thepossibility of a stable travelling wave propagating throughthe system, leaving behind a temporally unstable stationarystate. Under these conditions, spatiotemporal chaotic behaviouris seen to develop after the passage of the wave.     

Direction and stability of bifurcation branches for variational inequalities

We consider a class of variational inequalities with a multidimensional bifurcation parameter under assumptions guaranteeing the existence of smooth families of nontrivial solutions bifurcating from the set of trivial solutions. The direction of bifurcation is shown in a neighborhood of bifurcation points of a certain type. In the case of potential operators, also the stability and instability of bifurcating solutions and of the trivial solution is described in the sense of minima of the potential. In particular, an exchange of stability is observed.     

参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞,可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．     

Stability,Steady‐State Bifurcations,and Turing Patterns in a Predator–Prey Model with Herd Behavior and Prey‐taxis

In this paper, we consider a predator–prey model with herd behavior and prey‐taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey‐tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey‐taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots–strip pattern, strip pattern, stable nonconstant steady‐state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.     

Existence and stability of nontrivial periodic solutions of periodically forced discrete dynamical Systems

The local existence and local asymptotic stability of nontrivial p-periodic solutions of p-periodically forced discrete systems are proven using Liapunov-Schmidt methods. The periodic solutions bifurcate transcritically from the trivial solution at the critical value n=n^cr of the bifurcation parameter with a typical exchange of stability. If the trivial solution loses (gains) stability as n is increased through n^cr , then the periodic solutions on the nontrivial bifurcating branch are locally asymptotically stable if and only if they correspond to n>n^cr (n n^cr ).     

A periodic predator–prey-chain system with impulsive perturbation

A periodic predator–prey-chain system with impulsive effects is considered. By using the global results of Rabinowitz and standard techniques of bifurcation theory, the existence of its trivial, semi-trivial and nontrivial positive periodic solutions is obtained. It is shown that the nontrivial positive periodic solution for such a system may be bifurcated from an unstable semi-trivial periodic solution. Furthermore, the stability of these periodic solutions is studied.     

The onset of positive periodic solutions for a biochemical pest management model

In this paper, the bifurcation of nontrivial periodic solutions for an impulsively perturbed system of ordinary differential equations which models an integrated pest management strategy is studied by means of a fixed point approach. A biological control, consisting in the periodic release of infective pests, and a chemical control, consisting in pesticide spraying, are employed to maintain susceptible pests below an acceptable level. It is assumed that the biological and chemical control act with the same periodicity, but not in the same time. It is then shown that if the constant amount of infective pests released each time reaches a certain threshold value, then the trivial susceptible pest-eradication periodic solution loses its stability, which is transferred to a newly emerging nontrivial periodic solution.     

Transformation of Quadratic Forms to Perfect Squares for Broken Extremals

In this paper we study a quadratic form which corresponds to an extremal with piecewise continuous control in variational problems. This form, compared with the classical one, has some new terms connected with the set of all points of discontinuity of the control. Its positive definiteness is a sufficient optimality condition for broken extremals. We show that if there exists a solution to corresponding Riccati equation satisfying some jump condition at each point of the set , then the quadratic form can be transformed to a perfect square, just as in the classical case. As a result we obtain sufficient conditions for positive definiteness of the quadratic form in terms of the Riccati equation and hence, sufficient optimality conditions for broken extremals.     

Orthogonal polynomials and quadratic extremal problems

The purpose of this paper is to analyse a class of quadratic extremal problems defined on various Hilbert spaces of analytic functions, thereby generalizing an extremal problem on the Dirichlet space which was solved by S.D. Fisher. Each extremal problem considered here is shown to be connected with a system of orthogonal polynomials. The orthogonal polynomials then determine properties of the extremal function, and provide information about the existence of extremals.

     

On delay and asymmetry points of one-dimensional diffusion processes

A homogeneous second order linear differential equation is considered. On an open interval where the equation has sense, it generates a family of operators of the Dirichlet problem on the set of all subintervals; this family is a generalized semi-group. Let the equation be defined on two disjoint intervals with a common boundary point z. It is shown that an extension of the corresponding two semi-groups of operators to the semi-group of operators corresponding to the union of the intervals and the point z is not unique and depends on two abritrary constants. To give an interpretation of these arbitrary constants, we use a one-dimensional locally Markov diffusion process with special properties of passage of the point z. One of these arbitrary constants determines the delay of the process at the point z, and the second one induces an asymmetry of the process with respect to z. The two extremal values of the latter constant, 0 and ∞, determine the reflection of the process from the point z when the process approaches the point from the left and right, respectively. Bibliography: 4 titles.