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Abstract This paper is concerned with the stationary patterns of a prey–predator model with a protection zone and fractional type cross-diffusion for the prey. It is shown that the fractional type cross-diffusion has negative effects on the survival of the prey when the intrinsic growth rate of the predator is positive. Moreover, our mathematical analysis shows that, compared with the results obtained in K. Oeda (2011) and K. Oeda (2012), the large cross-diffusion coefficient and large growth rate of the predator species have some essentially different effects on the profiles of the solutions. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the diffusive minimal sediment model with no-flux boundary conditions. The steady-state bifurcation with two-dimensional kernel is firstly obtained with respect to this model by the space decomposition and the implicit function theorem. The existence of the Hopf bifurcation is next attained, and the direction and the stability of the Hopf bifurcation are analyzed in detail. Numerical simulations are done to support and complement the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we develop a diffusive plant invasion model with delay under the homogeneous Neumann boundary condition. First, the existence and uniqueness of a non-negative solution, persistence property, and local asymptotic stability of the constant steady states are established. Then by analyzing the associated characteristic equation, the stability of steady states and the existence of Hopf bifurcation are demonstrated. Under special circumstance, the discontinuous Hopf bifurcation is also investigated. Furthermore, the existence and non-existence of nonconstant positive steady states of this model are studied through considering the effect of large diffusivity. Finally, in order to verify our theoretical results, some numerical simulations are also included. It shows that the numerically observed behaviors are in good agreement with the theoretically proposed results. On the basis of numerical simulations, we provide some useful comparisons for the readers between our results and the existing ones, and discuss the effects of the delay and the diffusion term on dynamic behaviors. Numerical results indicate that diffusion can make the system unstable and increasing delay may cause the plant to go extinct. [ABSTRACT FROM AUTHOR]

Abstract: This paper concerns with the number and distributions of limit cycles in a -equivariant quintic planar polynomial system. 24 limit cycles are found in this system and two different configurations of them are shown by combining the methods of double homoclinic loops bifurcation, Poincaré bifurcation and qualitative analysis. The two configurations of 24 limit cycles obtained in this paper are new. The results obtained are useful to the study of weakened 16th Hilbert Problem. [Copyright &y& Elsevier]

In this paper, we study the appearance of limit cycles from the equator in a class of cubic polynomial vector fields with no singular points at infinity and the stability of the equator of the systems. We start by deducing the recursion formula for quantities at infinity in these systems, then specialize to a particular case of these cubic systems for which we study the bifurcation of limit cycles from the equator. We compute the quantities at infinity with computer algebraic system Mathematica 2.2 and reach with relative ease an expression of the first six quantities at infinity of the system, and give a cubic system, which allows the appearance of six limit cycles in the neighborhood of the equator. As far as we know, this is the first time that an example of cubic system with six limit cycles bifurcating from the equator is given. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of quantities at infinity is linear and then avoids complex integrating operations. Therefore, the calculation can be readily done with using computer symbol operation system such as Mathematica. [Copyright &y& Elsevier]

Abstract This paper is concerned with the steady state problem of a predator–prey cross-diffusion system with herd effect, Allee effect and a protection zone. Some sufficient conditions for the existence of positive steady state solutions are given. Our proof is based on the local and global bifurcation theory and some a priori estimates. Some limiting behavior of positive steady states with respect to the Allee effect constant (α) or the diffusion coefficient (d) , are discussed. [ABSTRACT FROM AUTHOR]

This paper is concerned with a cross-diffusion predator–prey system with prey-taxis incorporating Holling type II functional response under homogeneous Neumann boundary condition. By employing global bifurcation theory, it is obtained that a branch of nonconstant solutions can bifurcate from the positive constant solution whenever the chemotactic is attractive or repulsive. Furthermore, by using perturbation of simple eigenvalues it is found that the bifurcating solutions are locally stable near the bifurcation point under suitable conditions. These results imply that cross-diffusion can create coexistence for the predators and preys under the above special case. [ABSTRACT FROM AUTHOR]

In this paper, we concern about the dynamics of a diffusive enzyme-catalyzed system arising from glycolysis, describing a biochemical reaction in which a substrate is converted into a product with positive feedback and into a branched sink. The temporal and spatiotemporal dynamics of the system under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equation is presented. For the reaction–diffusion model, firstly the parameter regions for the stability or instability of the unique constant steady state are discussed. Finally, bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied. Numerical simulations are presented to verify and illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction–diffusion system describing a three-species Lotka–Volterra food chain model with homogeneous Dirichlet boundary conditions. By regarding the birth rate of prey r 1 as a bifurcation parameter, the global bifurcation of positive steady-state solutions from the semi-trivial solution set is obtained via the bifurcation theory. The results show that if the birth rate of mid-level predator and top predator are located in the regions 0 < r 2 < λ 1 a 23 u 3 r 3 and r 3 > λ 1 , respectively. Then the three species can co-exist provided the birth rate of prey exceeds a critical value. Moreover, an explicit expression of coexistence steady-state solutions is constructed by applying the implicit function theorem. It is demonstrated that the explicit coexistence steady-state solutions is locally asymptotically stable. [ABSTRACT FROM AUTHOR]

This paper is purported to investigate a ratio-dependent prey–predator system with cross-diffusion in a bounded domain under no flux boundary condition. The asymptotical stabilities of nonnegative constant solutions are investigated to this system. Moreover, without estimating the lower bounds of positive solutions, the existence, multiplicity of positive steady states are considered by using fixed points index theory, bifurcation theory, energy estimates and the differential method of implicit function and inverse function. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stage structure model with spatiotemporal delay and homogeneous Dirichlet boundary condition. The existence of steady state solution bifurcating from the trivial equilibrium is obtained by using Lyapunov–Schmidt reduction. The stability analysis of the positive spatially nonhomogeneous steady state solution is investigated by a detailed analysis of the characteristic equation. Using the properties of the omega limit set, we obtain the global convergence of the solution with finite delay. [ABSTRACT FROM AUTHOR]

In this paper, we study a diffusive predator–prey model with a transmissible disease in the prey population. We offer a complete discussion of the dynamical properties under the homogeneous Neumann boundary condition. We analyze local and global stabilities of nonnegative constant steady states and long time behaviors of the positive solutions. Moreover, we study the existence, nonexistence and bifurcation of nonconstant positive stationary solutions. Finally, some numerical simulations are presented to support and strengthen our analytical analysis. [ABSTRACT FROM AUTHOR]

In this paper we analytically and numerically investigate the emerging process of pattern formation for a reaction–diffusion–chemotaxis model with volume-filling effect. We first apply globally asymptotic stability analysis to show that the chemotactic flux is the key mechanism for pattern formation. Then, by weakly nonlinear analysis with multiple scales and the adjoint system theory, we derive the cubic and the quintic Stuart–Landau equations to describe the evolution of the amplitude of the most unstable mode, and thus the analytical approximate solutions of the patterns are obtained. Next, we present the selection law of principal wave mode of the emerging pattern by considering the competition of the growing modes, and for this we deduce the change rule of the most unstable mode and the coupled ordinary differential equations that indicates the significant nonlinear interaction of two competing modes. Finally, in the subcritical case we clarify that there exists the phenomenon of hysteresis, which implies the existence of large amplitude pattern for the bifurcation parameter values smaller than the first bifurcation point. Therefore, we answer the open problems proposed in the known references and improve some of results obtained there. All the theoretical results are tested against the numerical results showing excellent qualitative and good quantitative agreement. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a class of diffusive Lotka–Volterra equation with time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady state solution is investigated by applying Lyapunov–Schmidt reduction. The stability and nonexistence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the changes of a specific parameter are obtained by analyzing the distribution of the eigenvalues. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the positive solution set of the following quasilinear elliptic system { − Δ [ ( 1 + α v ) u ] = u ( a − u − c v 1 + m u ) , x ∈ Ω , − Δ [ ( 1 + γ 1 + β u ) v ] = v ( b − v + d u 1 + m u ) , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω , where Ω is a bounded smooth domain in R N , α , β , γ are nonnegative constants, a , c , d , m are positive constants, b is a real constant. This elliptic system is the stationary problem of a predator–prey model, in which u and v denote the population densities of the prey and predator, respectively. Regarding b as the bifurcation parameter, the global bifurcation structure of the positive solution set is shown. Then the nonlinear effect of either large α or β on the bifurcation point is deduced. Moreover, as α is large, we show that the positive solution set is only of one type, whereas, as β is large, the positive solution set is of two types. Additionally, in certain circumstances, we also show that which one of the two types can characterize the positive solution set. [ABSTRACT FROM AUTHOR]

In this paper, we study global bifurcation phenomena for the following Kirchhoff type problem { − M ( ∫ Ω | ∇ u ( x ) | 2 d x ) Δ u = λ f ( x , u ) in Ω , u = 0 on ∂ Ω , where M is a continuous function. Under some natural hypotheses, we show that ( λ 1 ( a ) M ( 0 ) , 0 ) is a bifurcation point and there is a global continuum C emanating from ( λ 1 ( a ) M ( 0 ) , 0 ) , where λ 1 ( a ) denotes the first eigenvalue of the above problem with f ( x , s ) = a ( x ) s . As an application of the above result, we study the existence of positive solution for this problem with asymptotically linear nonlinearity. [ABSTRACT FROM AUTHOR]

This paper presents a two stage SIS epidemiological model in animal population with bovine tuberculosis (BTB) in African buffalo as a guiding example. The proposed model is rigorously analyzed. The analysis reveals that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium (DFE) coexists with a stable endemic equilibrium (EE) when the associated reproduction number ( R v ) is less than unity. It is shown under two special cases of the presented model, that this phenomenon of backward bifurcation does not arise depending on vaccination coverage and efficacy of vaccine. Numerical simulations of the model show that, the use of an imperfect vaccine can lead to effective control of the disease if the vaccination coverage and the efficacy of vaccine are high enough. [ABSTRACT FROM AUTHOR]

The nonlinear thermal bifurcation buckling behavior of carbon nanotube reinforced composite (CNTRC) beams with surface-bonded piezoelectric layers is studied in this paper. The governing equations of piezoelectric CNTRC beam are obtained based on the Euler–Bernoulli beam theory and von Kármán geometric nonlinearity. Two kinds of carbon nanotube-reinforced composite (CNTRC) beams, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRC beam are assumed to be graded in the thickness direction. The SWCNTs are assumed aligned, straight and with a uniform layout. Exact solutions are presented to study the thermal buckling behavior of beams made of a symmetric single-walled carbon nanotube reinforced composite with surface-bonded piezoelectric layers. The critical temperature load is obtained for the nonlinear problem. The effects of the applied actuator voltage, temperature, beam geometry, boundary conditions, and volume fractions of carbon nanotubes on the buckling of piezoelectric CNTRC beams are investigated. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with the existence of positive radial solutions of the elliptic system where , is a constant, is a parameter and , is continuous and for all . Under some appropriate conditions on the nonlinearity , we show that the above system possesses at least one positive radial solution for any . The proof of our main results is based upon bifurcation techniques. [Copyright &y& Elsevier]

Abstract: The bifurcations in a three-variable ODE model describing the oxygen reduction reaction on a platinum surface is studied. The investigation is motivated by the fact that this reaction plays an important role in fuel cells. The goal of this paper is to determine the dynamical behaviour of the ODE system, with emphasis on the number and type of the stationary points, and to find the possible bifurcations. It is shown that a non-trivial steady state can appear through a transcritical bifurcation, or a stable and an unstable steady state can arise as a result of saddle-node bifurcation. The saddle-node bifurcation curve is determined by using the parametric representation method, and this enables us to determine numerically the parameter domain where bistability occurs, which is important from the chemical point of view. [Copyright &y& Elsevier]

Abstract: This paper studies the bifurcation and nonlinear behaviors of a united gas-lubricated bearing (UGB) system by a hybrid numerical method combining the differential transformation method and the finite difference method. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, quasi-periodic and chaotic responses of the rotor center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those of other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of UGB systems. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the effect of delayed feedbacks on the 3-D chaotic system only with two stable node-foci by Yang et al. The stability of equilibria and the existence of Hopf bifurcations are considered. The explicit formulas determining the direction, stability and period of the bifurcating periodic solutions are obtained by employing the normal form theory and the center manifold theorem. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodic orbit in the chaotic system with direct time delay feedback. We also find that the control law can be applied to the chaotic system only with two stable node-foci for the purpose of control and anti-control of chaos. Finally, some numerical simulations are given to illustrate the effectiveness of the results found. [Copyright &y& Elsevier]

Abstract: In this paper, we show the combined use of analytical and numerical techniques in the study of bifurcations of equilibria of low-dimensional chaotic problems. We study in detail different aspects of the paradigmatic Rössler model. We provide analytical formulas for the stability of the equilibria as well as some of their codimension one, two, and three bifurcations. In particular, we carry out a complete study of the Andronov–Hopf bifurcation, establishing explicit formulas for its location and studying its character numerically, determining a curve of generalized-Hopf bifurcation, where the Hopf bifurcation changes from subcritical to supercritical. We also briefly study some routes among the different Andronov–Hopf bifurcation curves and how these routes are influenced by the local and global bifurcations of limit cycles. Finally, we show the U-shape of the homoclinic bifurcation curve at the studied parameter values. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a HIV infection model with CTL-response delay and analyze the effect of time delay on stability of equilibria. We obtain the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the CTL-absent equilibrium and CTL-present equilibrium. By choosing the CTL-response delay as a bifurcation parameter, we prove that the CTL-present equilibrium is locally asymptotically stable in a range of delays and a Hopf bifurcation occurs as crosses a critical value. Numerical simulations are given to support the theoretical results. [Copyright &y& Elsevier]

Abstract: The AFM (atomic force microscope) has become a popular and useful instrument for measuring intermolecular forces with atomic resolution, that can be applied in electronics, biological analysis, and studying materials, semiconductors etc. This paper conducts a systematic investigation into the bifurcation and chaotic behavior of the probe tip of an AFM using the differential transformation (DT) method. The validity of the analytical method is confirmed by comparing the DT solutions for the displacement and velocity of the probe tip at various values of the vibrational amplitude with those obtained using the Runge–Kutta (RK) method. The behavior of the probe tip is then characterized utilizing bifurcation diagrams, phase portraits, power spectra, Poincaré maps, and maximum Lyapunov exponent plots. The results indicate that the probe tip behavior is significantly dependent on the magnitude of the vibrational amplitude. Specifically, the tip motion changes first from subharmonic to chaotic motion, then from chaotic to multi-periodic motion, and finally from multi-periodic motion to subharmonic motion with windows of chaotic behavior as the non-dimensional vibrational amplitude is increased from 1.0 to 5.0. [Copyright &y& Elsevier]

Abstract: The paper presents the dynamical features of a vector–host epidemic model with direct transmission. First, we extended the model by taking into account the exposed individuals in both human and vector population with the impact of disease related deaths and total time dependent population size. Using Lyapunov function theory some sufficient conditions for global stability of both the disease-free equilibrium and the endemic equilibrium are obtained. For the basic reproductive number , a unique endemic equilibrium exists and is globally asymptotically stable. Furthermore, it is found that the model exhibits the phenomenon of backward bifurcation, where the stable disease-free equilibria coexists with a stable endemic equilibrium. Finally, numerical simulations are carried out to investigate the influence of the key parameters on the spread of the vector-borne disease, to support the analytical conclusion and illustrate possible behavioral scenarios of the model. [Copyright &y& Elsevier]

Abstract: The mathematical modelling of the keloid disease triggered by a virus has been recently investigated by one of the authors, Bianca (2011) , where it was shown that the model is able to depict the emerging behaviours which occur during the keloid formation. This paper deals with further numerical investigations of that model related to the bifurcation analysis of the measurable macroscopic variables associated to each functional subsystem. It is shown that there exists a critical value of a bifurcation parameter separating situations where the immune system controls the keloid formation from those where malignant effects are not contrasted. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with an unstirred chemostat model with the Beddington–DeAngelis functional response. First, a sufficient condition to the existence of positive steady state solutions is established. Second, the effect of the parameter in the Beddington–DeAngelis functional response which models mutual interference between species is considered. The result shows that if is sufficiently large, the solution of this model is determined by a limiting equation. The main tool used here includes the fixed point index theory, the perturbation technique and the bifurcation theory. [Copyright &y& Elsevier]

Abstract: Like for smooth quadratic systems, it is important to determine the maximum order of a fine focus and the cyclicity of discontinuous quadratic systems. Previously, examples of discontinuous quadratic systems with five limit cycles bifurcated from a fine focus of order 5 have been constructed. In this paper we construct a class of discontinuous quadratic systems with a fine focus of order 9. In addition, by using a method similar to that developed by C. Christopher for smooth systems, which allows one to estimate the cyclicity just from the lower order terms of Lyapunov constants, we show that the cyclicity of discontinuous quadratic systems is at least 9, thus improving on previous results. [Copyright &y& Elsevier]

Abstract: In this paper, a delayed SIRS epidemic model with saturation incidence and temporary immunity is investigated. The immunity gained by experiencing a disease is temporary, whenever infected the diseased individuals will return to the susceptible class after a fixed period. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a novel delayed chaotic neural model, in which a non-monotonously increasing transfer function is employed as activation function. Local stability and existence of Hopf bifurcation are analyzed in details. Chaos behavior of the neuron model is observed in computer simulations. An electronic implementation of the neuron is also considered. The dynamical behavior of the designed circuits is closely similar to the results simulated by numerical experiments. [Copyright &y& Elsevier]

Abstract: Bifurcation phenomena of equilibrium states occur in both standard and complex materials. In this paper we study the equilibrium configurations close to a bifurcation point. In particular the attention is focused on bifurcations of pitchfork type [S.H. Strogatz, Non Linear Dynamics and Chaos, Addison-Wesley Publishing Company, 1994]. This problem is usually solved by using the Signorini’s compatibility of the solution expansion in a neighborhood of the critical point. We show how the same results can be reached in another way which involves just the linear term of the solution expansion. As a test, we analyze two bifurcation phenomena: the buckling of an elastic beam under an axial load and the magnetic field-induced optical switch in nematic liquid crystals. [Copyright &y& Elsevier]

In this paper, we study nonlinear discrete boundary value problems of the form where is an eigenvalue of the associated linear problem, is a parameter, and satisfies the sublinear growth condition for some and . We study the existence and multiplicity of solutions for the above-mentioned problems by establishing some a priori bounds, together with Leray–Schauder continuation and bifurcation arguments. [Copyright &y& Elsevier]

Abstract: In this paper, we give the global dynamical behaviors of a reduced SIRS epidemic model with a nonlinear incidence rate . We first discuss the qualitative properties of the equilibria in the interior of the first quadrant, and study the bifurcations including saddle–node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Then we consider equilibria at infinity, determining the number of orbits in exceptional directions for the global tendency. In this discussion, the unspecified degree of polynomials and their high degeneracy prevent us from using the methods of blowing-up or normal sectors in some cases. We lastly discuss the existence and uniqueness of limit cycles. [Copyright &y& Elsevier]

Abstract: This paper is about the number of limit cycles for a quintic near-Hamiltonian system. It is proved that the system can have 20, 22, 24 limit cycles with different distributions of limit cycles for each case. The limit cycles are obtained by using the methods of bifurcation theory and qualitative analysis. [Copyright &y& Elsevier]

Abstract: Of concern in the paper is the stability of populations which change their size according to the Hassell growth function, not only in isolation, but also under external effects such as migration. The mathematical model developed is applied to a host-parasite system and the dynamics of the resulting system are studied analytically. The numerical simulations demonstrate the inherent instabilities in the single-species system, which manifest itself in the form of period-doubling bifurcations and chaos. It is found that the effect of migration is to suppress such an unstable behaviour. [Copyright &y& Elsevier]

Abstract: Based on an inverted pendulum impacting on rigid walls under external periodic excitation, a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation. The Melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one. For nonlinear impact systems, closed form solutions between impacts are generally unavailable. The absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold. In this paper, we give a method to compute the Melnikov functions up to the n th-order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule. [Copyright &y& Elsevier]

Abstract: In this paper, we show that the solution flows generated by a one-parameter family of ordinary differential equations are stable under their numerical approximations in a vicinity of a saddle node. Our result sharpens the one in [] and the proof is adapted from the method of Sotomayor in []. [Copyright &y& Elsevier]

Abstract: In this paper, the number of limit cycles in a family of polynomial systems was studied by the bifurcation methods. With the help of a computer algebra system (e.g., Maple 7.0), we obtain that the least upper bound for the number of limit cycles appearing in a global bifurcation of systems (2.1) and (2.2) is 5n + 5 + (1 − (−1) n )/2 for c ≠ 0 and n for c ≡ 0. [Copyright &y& Elsevier]

Abstract: Consider a polynomial differential system of degree p + q, which was given from a general multimolecular reaction in biochemistry as a theoretical problem of concentration kinetics. Although its local bifurcations are investigated in [1], a bifurcation of codimension 2 at a cusp remains to be considered. In this paper, such a bifurcation, called Bogdanov-Takens bifurcation, is discussed and the corresponding universal unfolding is given so as to complete the analysis of local bifurcations for the system. [Copyright &y& Elsevier]

In this paper, we construct a conjugacy between the time-1-map of the solution flow generated by an ordinary differential equation and its numerical approximation in a neighborhood of a fold bifurcation point. Our main result is that the conjugacy is O(hp)-close to the identity on the center manifold where h is the step size and p is the order of the numerical method. [Copyright &y& Elsevier]