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A general time-delay reaction–diffusion–advection system with the Dirichlet boundary condition and spatial heterogeneity is investigated in this paper. By using the implicit function theorem, we obtain the existence and asymptotic expression of the spatially non-homogeneous positive steady-state solution. This is the steady-state bifurcation from zero equilibrium. Via analyzing the corresponding characteristic equation, the stability of the spatially non-homogeneous positive steady-state solution and the occurrence of Hopf bifurcation at the positive steady-state solution are obtained, and the spatially non-homogeneous periodic solution is derived from Hopf bifurcation, this is the secondary bifurcation behavior of the system. Utilizing the normal form method and center manifold theory, we prove that the direction of Hopf bifurcation is supercritical and the bifurcating spatially non-homogeneous periodic solution is stable. Furthermore, We show that there exist two sequences Hopf bifurcation values and the orders of two sequences Hopf bifurcation values are given. Moreover, theoretical and numerical results are applied to competition and cooperation systems, respectively. Finally, the effect of the advection rate and spatial heterogeneity are discussed. [ABSTRACT FROM AUTHOR]

This paper is devoted to the high codimension bifurcations of a classical predator–prey system with Allee effects and generalized Holling type III functional response p (x) = m x 2 a x 2 + b x + 1 where b > - 2 a . We show that the maximal orders of nilpotent saddle, cusp singularity and weak focus are all three. The unfoldings of a cusp singularity and of a nilpotent saddle of order 3 with a fixed invariant line are developed. The dependence of codimension of degenerate Hopf bifurcation on b is thoroughly investigated. It is proven that there exist a homoclinic loop of order 2 and a heteroclinic loop of order 2 for - 2 a < b < 0 , and three limit cycles for b > 0 . Together with existing work for Holling type I, II and IV functional responses, our results complement the analysis of the classical predator–prey systems with Allee effects and four types of Holling functional responses. Furthermore, simple formulas are derived to characterize the order of nilpotent saddle, through which the existence and order of the heteroclinic loop can be easily obtained for a general class of predator–prey systems with any smooth functional response. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence, multiplicity, stability, and Hopf bifurcation of steady-state solutions of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and nonlinear boundary condition. It is found that there is no Hopf bifurcation when the interior reaction term is weaker than the boundary reaction term, and that the interior reaction delay determines the existence of Hopf bifurcation only when the interior reaction term is stronger than the boundary reaction term. This observation helps us to understand the nonlinear balance between the interior reaction and boundary flux in nonlinear boundary problems. Moreover, the general results are illustrated by applications to a model with homogeneous kernels. [ABSTRACT FROM AUTHOR]

The tea green leafhopper Empoasca onukii is one kind of insect pest threatening the tea production, and the mite Anystis baccarum has been used as an agent for pest control. In this paper, we introduce a generalist predator-prey model to study the dynamics for informing biological control. There have been some bifurcation studies of the generalist predator-prey model in the last few years. Except for the bifurcations include saddle-node bifurcation of codimension 1 and 2, Hopf bifurcations, and Bogdanov-Takens bifurcation of codimension 2 and 3, we also present the bifurcations of nilpotent singularities of elliptic and focus type of codimension 3. We find that the nilpotent singularities are associated with a cubic Liénard system, and the nilpotent bifurcations are three-parameter bifurcations of a codimension 4 nilpotent focus. Furthermore, we show that the nilpotent focus serves as an organizing center to connect all the codimension 3 bifurcations in the system. We also present the bifurcation diagrams to unfold the nilpotent singularities of codimension 3. One interesting observation is that we show numerically the existence of three limit cycles in the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single species population model over patches with delay and nonlocal interactions, for which no symmetry for the dispersion (connection) matrix is assumed. We show that there exists a positive equilibrium when the dispersal rate is large. We also discuss the stability/instability of this positive equilibrium, establish the threshold dynamics and explore the associated Hopf bifurcation. Moreover, we demonstrate our theoretical results by a nonlocal logistic population model and by the Nicholson's blowflies model. [ABSTRACT FROM AUTHOR]

The dynamics of a general reaction–diffusion–advection two species model with nonlocal delay effect and Dirichlet boundary condition is investigated in this paper. The existence and stability of the positive spatially nonhomogeneous steady state solution are studied. Then by regarding the time delay τ as the bifurcation parameter, we show that Hopf bifurcation occurs near the steady state solution at the critical values τ n (n = 0 , 1 , 2 , ...) . Moreover, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to a Lotka–Volterra competition–diffusion–advection model with nonlocal delay. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of a Hopf bifurcation in a delayed single population model with patch structure. The effect of the dispersal rate on the Hopf bifurcation is considered. Especially, if each patch is favorable for the species, we show that when the dispersal rate tends to zero, the limit of the Hopf bifurcation value is the minimum of the "local" Hopf bifurcation values over all patches. On the other hand, when the dispersal rate tends to infinity, the Hopf bifurcation value tends to that of the "average" model. [ABSTRACT FROM AUTHOR]

In this paper, a single reaction-diffusion population model with memory effect and the heterogeneity of the environment, equipped with the Neumann boundary, is considered. The global existence of a spatial nonhomogeneous steady state is proved by the method of super and subsolutions, which is linearly stable for relatively small memory-induced diffusion. However, after the memory-induced diffusion rate exceeding a critical value, spatial inhomogeneous periodic solution can be generated through Hopf bifurcation, if the integral of intrinsic growth rate over the domain is negative. Such phenomenon will never happen, if only memory-induced diffusion or spatially heterogeneity is presented, and therefore must be induced by their joint effects. This indicates that the memory-induced diffusion will bring about spatial-temporal patterns in the overall hostile environment. When the integral of intrinsic growth rate over the domain is positive, it turns out that the steady state is still linearly stable. Finally, the possible dynamics of the model is also discussed, if the boundary condition is replaced by Dirichlet condition. [ABSTRACT FROM AUTHOR]

West Nile virus (WNv) transmission shows both seasonal pattern in every single year and cyclic pattern over years. In this paper we formulate a compartmental model with bird demographics and maturation time of mosquitoes during metamorphosis to study the impact of ambient temperature on the transmission and recurrence of disease. We show that avian birds serve as a reservoir of viruses, whilst maturation time affects disease transmission in sophisticated ways. It turns out that large maturation delay will lead to the extinction of mosquitoes and the disease; small maturation delay will stabilize the epidemic level of the disease; and intermediate maturation delay will cause sustainable oscillations of mosquito population, recurrence of diseases, and even mixed-mode oscillation of diseases with an alternation between oscillations of distinct large and small amplitudes. With bifurcation theory, we prove that temperature can drive the oscillation of mosquito population, which leads recurrence of WNv through the incidence interaction between mosquitoes and hosts, while the biting and transmission process itself will not generate sustained oscillations. Our results provide a sound explanation for understanding interactions between vectors and hosts, and driving mechanisms of periodic phenomena in the transmission of WNv and other mosquito-borne diseases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a delayed reaction–diffusion–advection equation, which models the population dynamics in the advective heterogeneous environment. The existence of the nonconstant positive steady state and associated Hopf bifurcation are obtained. A weighted inner product associated with the advection rate is introduced to compute the normal forms, which is the main difference between Hopf bifurcation for delayed reaction–diffusion–advection model and that for delayed reaction–diffusion model. Moreover, we find that the spatial scale and advection can affect Hopf bifurcation in the heterogenous environment. [ABSTRACT FROM AUTHOR]

In this paper we prove that the following delay differential equation d dt x (t) = r x (t) 1 - ∫ 0 1 x (t - s) d s , has a periodic solution of period two for r > π 2 2 (when the steady state, x = 1 , is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke (J Math Anal Appl 48:317–324, 1974). The periodic solution is expressed in terms of the Jacobi elliptic functions. [ABSTRACT FROM AUTHOR]

In the existing literatures, there exist at least four limit cycles for 3D Lotka-Volterra competitive systems in Zeeman's classes 26 and 27 with no explicit critical parameter values, and three limit cycles in Zeeman's classes 27 and 29 with explicit critical parameter values. Giving explicit critical parameter values in Zeeman's classes 26 - 31 and accurate focus values at them except class 30, we rigorously prove that Zeeman's classes 27 - 30 have at least three limit cycles, and Zeeman's classes 26 and 31 have at least two small amplitude limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, we apply the recently developed normal form theory for abstract Cauchy problems with non-dense domain in Liu et al. (J Diff Equ 257:921-1011, 2014) to study normal forms for an age structured model. We provide detailed computations for the Taylor's expansion of the reduced system on the center manifold, from which explicit formulae are given to determine the direction of the Hopf bifurcation and the stability and amplitude of the bifurcating periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, the discrete-time mathematical model for tissue inflammation obtained by Euler is investigated in detail. Conditions of the existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, and chaos in the sense of Marotto is proved by analytical method. Numerical simulations, including bifurcation diagrams, Lyapunov exponents, fractal dimensions, and phase portraits are plotted to perfectly show the consistence with the theoretical analysis. [ABSTRACT FROM AUTHOR]

We indicate how a Hopf bifurcation theorem for retarded functional differential equations can be proved correctly, following an approach that is described, but not carried out properly in the literature. Then we state and prove a Hopf bifurcation theorem for general semiflows in Banach spaces, using different methods. The general statement includes the case of retarded functional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the universal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are characterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov-Takens bifurcation with $$Z_2$$ symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a 'figure-eight' homoclinic orbit. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a diffusive predator–prey model with memory-based diffusion, which is described by a delayed predator-taxis term. We first obtain the existence and uniform boundedness of solutions. Then we study the stability of positive constant equilibria. We find that the memory delay may induce Hopf bifurcation, generating spatially inhomogeneous periodic solutions. This result implies that the memory-based diffusion is beneficial to pattern formations and is contrary to the conclusion on predator–prey model only with predator-taxis. The normal form of Hopf bifurcation is also derived. The theoretical results are demonstrated through the case with the Holling II functional response. Numerical simulations also suggest the phenomenon of stability switches and the occurrence of double Hopf bifurcation. [ABSTRACT FROM AUTHOR]

The Hopf bifurcations for the classical Gray–Scott system and the Schnakenberg system in an one-dimensional interval are considered. For each system, the existence of time-periodic solutions near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of limit cycles are determined and it is shown that the Hopf bifurcation is supercritical for the Schnakenberg system, and hence the bifurcating periodic solutions are linearly stable. For the Gray–Scott system, there is a critical feeding rate, when the feeding rate of the system is greater than this critical feeding rate, the Hopf bifurcation is supercritical which implies the bifurcating periodic solutions are linearly stable, while when the feeding rate is smaller than this critical feeding rate, the Hopf bifurcation is subcritical, implying the bifurcating periodic solutions are linearly unstable. [ABSTRACT FROM AUTHOR]

It is shown that time delay induces self-excited vibrations in one dimensional damped wave equations, via Hopf bifurcation. In order to obtain classical solutions and C ∞ smoothness with respect to the amplitude parameter, we avoid the use of any abstract Hopf bifurcation theorem, and instead work directly in Sobolev spaces on the 2-torus. As a result, a "derivative loss" problem arises, which is overcome using the regularizing effects of the time delay and damping. Only the classical implicit function theorem is used. The direction of bifurcation is also obtained. [ABSTRACT FROM AUTHOR]

We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the S 1 -equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with threshold type state-dependent delay vanishing at the stationary state, for a description of the global continuation of the periodic oscillations. [ABSTRACT FROM AUTHOR]

The dynamics of a diffusive predator-prey system with Holling type-III functional response subject to Neumann boundary conditions is investigated. The parameter region for the stability and instability of the unique constant steady state solution is derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method and Leray-Schauder degree theory. The effect of various parameters on the existence and nonexistence of spatiotemporal patterns is analyzed. These results show that the impact of Holling type-III response essentially increases the system spatiotemporal complexity. [ABSTRACT FROM AUTHOR]

The local asymptotic stability and stability switches of the positive equilibrium in a logistic population model with mixed instantaneous and delayed density dependence is analyzed. It is shown that when the delayed dependence is more dominant, either the positive equilibrium becomes unstable for all large delay values, or the stability of equilibrium switches back and force several times as the delay value increases. Compared with the logistic model with the instantaneous term and a delayed term, our finding here is that the incorporation of another delayed term can lead to the occurrence of multiple stability switches. [ABSTRACT FROM AUTHOR]

We investigate local dynamics of a 4-dimensional system with small and slowly varying time periodic forcing. By assuming the unperturbed system is autonomous and has a fixed point with eigenvalues $$(0,0,i,-i)$$ , we study homoclinic, subharmonic solutions and Hopf bifurcation in a $$O(\epsilon )$$ neighborhood of the fixed point, where $$\epsilon $$ is the perturbation parameter. [ABSTRACT FROM AUTHOR]

We apply our recently developed global Hopf bifurcation theory to examine global continuation with respect to the parameter for periodic solutions of functional differential equations with state-dependent delay. We give sufficient geometric conditions to ensure the uniform boundedness of periodic solutions, obtain an upper bound of the period of non-constant periodic solutions in a connected component of Hopf bifurcation, and establish the existence of rapidly oscillating periodic solutions. [ABSTRACT FROM AUTHOR]

This work is concerned with the rigorous analysis of the effects of small periodic forcing (perturbations) on the dynamical systems which present some interesting phenomena known as delayed bifurcations. We study the dynamical behavior of the system where u0( I) is the solution of f(u0( I), I)=0 and I(t)= Ii+ εt is a slowly varying parameter that moves past a critical point I_ of the system so that the linear stability around u0( I) changes from stable to unstable at I_. General results are given with respect to the effects of the perturbation εg( u, I(t), ε, t) to several important types of dynamical systems which present dynamical patterns that there exist persistent unstable solutions in the dynamical systems (delayed bifurcations) in contrast to bifurcations in the classical sense. It is shown that (1) the delayed bifurcations persist if the frequency of g(...,...,..., t) on t is a constant Ω which is not a resonant frequency; (2) in case the frequency of g(...,...,..., t) on t is Ω=Ω( Ii+ εt) that is slowly varying, the resonance frequencies where the delayed bifurcations might be destructed are shifted downward or upward depending on Ω′(I_)>0 or Ω′(I_)<0; and (3) delayed pitchfork (simple eigenvalue) bifurcations occur in a codimension one parameter family of periodic perturbations. (1) is a rigorous analysis of the results in [3], (2) is a new and interesting phenomenon, and (3) is a generalization of the results of Diener [8] and Schecter [19]. [ABSTRACT FROM AUTHOR]