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.     

Interactions of Turing and Hopf bifurcations in an additional food provided diffusive predator-prey model

Complex spatiotemporal dynamics of a diffusive predator-prey system involving additional food supply to predator and intra-specific competition among predator, are investigated. We establish critical conditions of the occurrence of Turing instability, which are necessary and sufficient. Furthermore, we also establish conditions of the occurrence of codimension-2 Turing-Hopf bifurcation and Turing-Turing bifurcation, by exploring interactions of Turing bifurcations and Hopf bifurcation. For Turing-Hopf bifurcation, by analyzing normal form truncated to order 3 which are derived by applying normal form method, it is shown that under proper conditions, diffusive predator-prey system generates interesting spatial, temporal and spatiotemporal patterns, including a pair of spatially inhomogeneous steady states, a spatially homogeneous periodic solution and a pair of spatially inhomogeneous periodic solutions. And numerical simulations are also shown to support theory analysis. Moreover, it is found that proper intra-specific competition among predator helps generate complex spatiotemporal dynamics. And, proper additional food supply to predator helps control the population fluctuations of predator and prey, while large quantity and high quality of additional food supply will lead to the extinction of prey and make predator change the source of food, which finally destroy the ecosystem. These investigations might help understand complex spatiotemporal dynamics of predator-prey system involving additional food supply to predator and intra-specific competition among predator, and help conserve species in an ecosystem via supplying suitable additional food.     

Zero‐zero‐Hopf bifurcation and ultimate bound estimation of a generalized Lorenz–Stenflo hyperchaotic system

This paper is devoted to the analysis of complex dynamics of a generalized Lorenz–Stenflo hyperchaotic system. First, on the local dynamics, the bifurcation of periodic solutions at the zero‐zero‐Hopf equilibrium (that is, an isolated equilibrium with double zero eigenvalues and a pair of purely imaginary eigenvalues) of this hyperchaotic system is investigated, and the sufficient conditions, which insure that two periodic solutions will bifurcate from the bifurcation point, are obtained. Furthermore, on the global dynamics, the explicit ultimate bound sets of this hyperchaotic system are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Hopf‐pitchfork bifurcation in a two‐neuron system with discrete and distributed delays

Both discrete and distributed delays are considered in a two‐neuron system. We analyze the influence of interaction coefficient and time delay on the Hopf‐pitchfork bifurcation. First, we obtain the codimension‐2 unfolding with original parameters for Hopf‐pitchfork bifurcation by using the center manifold reduction and the normal form method. Next, through analyzing the unfolding structure, we give complete bifurcation diagrams and phase portraits, in which multistability and other dynamical behaviors of the original system are found, such as a stable periodic orbit, the coexistence of two stable nontrivial equilibria, and the coexistence of a stable periodic orbit and two stable equilibria. In addition, the obtained theoretical results are verified by numerical simulations. Finally, we perform the comparisons of the obtained results of Hopf‐pitchfork bifurcation with other Hopf‐fold bifurcation results in some biological neural systems and give the obtained mathematical results corresponding to the physical states of neurons. Copyright © 2015 JohnWiley & Sons, Ltd.     

Spatial resonance and Turing–Hopf bifurcations in the Gierer–Meinhardt model

Gierer–Meinhardt system as a molecularly plausible model has been proposed to formalize the observation for pattern formation. In this paper, the Gierer–Meinhardt model without the saturating term is considered. By the linear stability analysis, we not only give out the conditions ensuring the stability and Turing instability of the positive equilibrium but also find the parameter values where possible Turing–Hopf and spatial resonance bifurcation can occur. Then we develop the general algorithm for the calculations of normal form associated with codimension-2 spatial resonance bifurcation to better understand the dynamics neighboring of the bifurcating point. The spatial resonance bifurcation reveals the interaction of two steady state solutions with different modes. Numerical simulations are employed to illustrate the theoretical results for both the Turing–Hopf bifurcation and spatial resonance bifurcation. Some expected solutions including stable spatially inhomogeneous periodic solutions and coexisting stable spatially steady state solutions evolve from Turing–Hopf bifurcation and spatial resonance bifurcation respectively.     

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

In this paper, a reaction‐diffusion predator–prey system that incorporates the Holling‐type II and a modified Leslie‐Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation in a class of nonlocal delay differential equation with the zero‐flux boundary condition

The aim of this paper is to study the stability and Hopf bifurcation in a general class of differential equation with nonlocal delayed feedback that models the population dynamics of a two age structured spices. The existence of Hopf bifurcation is firstly established after delicately analyzing the eigenvalue problem of the linearized nonlocal equation. The direction of the Hopf bifurcation and stability of the bifurcated periodic solutions are then investigated by means of center manifold reduction. Subsequently, we apply our main results to explore the spatial‐temporal patterns of the nonlocal Mackey‐Glass equation. We obtain both spatially homogeneous and inhomogeneous periodic solutions and numerically show that the former is stable while the latter is unstable. We also show that the inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations and increasing of the immature mobility constant will shorten the transient oscillation time.     

Three‐dimensional internal gravity‐capillary waves in finite depth

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ₀) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.     

Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator–prey model with predator harvesting

The ratio-dependent predator–prey model exhibits rich dynamics due to the singularity of the origin. Harvesting in a ratio-dependent predator–prey model is relatively an important research project from both ecological and mathematical points of view. In this paper, we study the temporal, spatial and spatiotemporal dynamics of a ratio-dependent predator–prey diffusive model where the predator population harvest at catch-per-unit-effort hypothesis. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction–diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibit Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the existence and non-existence of positive non-constant steady-state solutions are established. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.     

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component. To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances. In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.     

On the explosive instability in a three‐species food chain model with modified Holling type IV functional response

In earlier literature, a version of a classical three‐species food chain model, with modified Holling type IV functional response, is proposed. Results on the global boundedness of solutions to the model system under certain parametric restrictions are derived, and chaotic dynamics is shown. We prove that in fact the model possesses explosive instability, and solutions can explode/blow up in finite time, for certain initial conditions, even under the parametric restrictions of the literature. Furthermore, we derive the Hopf bifurcation criterion, route to chaos, and Turing bifurcation in case of the spatially explicit model. Lastly, we propose, analyze, and simulate a version of the model, incorporating gestation effect, via an appropriate time delay. The delayed model is shown to possess globally bounded solutions, for any initial condition. Copyright © 2017 John Wiley & Sons, Ltd.     

Turing and Hopf Bifurcation in a Diffusive Tumor-immune Model

In order to understand the effect of the diffusion reaction on the interaction between tumor cells and immune cells, we establish a tumor-immune reaction diffusion model with homogeneous Neumann boundary conditions. Firstly, we investigate the existence condition and the stability condition of the coexistence equilibrium solution. Secondly, we obtain the sufficient and necessary conditions for the occurrence of Turing bifurcation and Hopf bifurcation. Thirdly, we perform some numerical simulations to illustrate the complex spatiotemporal patterns near the bifurcation curves. Finally, we explain spatiotemporal patterns in the diffusion action of tumor cells and immune cells.     

Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay

We consider a reaction-diffusion system with general time-delayed growth rate and kernel functions. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained. Moreover, taking minimal time delay τ as the bifurcation parameter, Hopf bifurcation near the steady-state solution is proved to occur at a critical value τ=τ₀. Especially, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to competitive and cooperative systems with weak or strong kernel function respectively.     

Dynamics in a diffusive predator–prey system with a constant prey refuge and delay

In this paper, a diffusive predator–prey system with a constant prey refuge and time delay subject to Neumann boundary condition is considered. Local stability and Turing instability of the positive equilibrium are studied. The effect of time delay on the model is also obtained, including locally asymptotical stability and existence of Hopf bifurcation at the positive equilibrium. And the properties of Hopf bifurcation are determined by center manifold theorem and normal form theorem of partial functional differential equations. Some numerical simulations are carried out.     

Turing Instabilities at Hopf Bifurcation

Turing–Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this paper we suggest a way for deriving asymptotic expansions to the limit cycle solutions due to a Hopf bifurcation in two-dimensional reaction systems and we use them to build convenient normal modes for the analysis of Turing instabilities of the limit cycle. They extend the Fourier modes for the steady state in the classical Turing approach, as they include time-periodic fluctuations induced by the limit cycle. Diffusive instabilities can be properly considered because of the non-catastrophic loss of stability that the steady state shows while the limit cycle appears. Moreover, we shall see that instabilities may appear even though the diffusion coefficients are equal. The obtained normal modes suggest that there are two possible ways, one weak and the other strong, in which the limit cycle generates oscillatory Turing instabilities near a Turing–Hopf bifurcation point. In the first case slight oscillations superpose over a dominant steady inhomogeneous pattern. In the second, the unstable modes show an intermittent switching between complementary spatial patterns, producing the effect known as twinkling patterns.     

Dynamics of a competitive Lotka-Volterra system with three delays

In this paper, a competitive Lotka-Volterra system with three delays is investigated. By choosing the sum τ of three delays as a bifurcation parameter, we show that in the above system, Hopf bifurcation at the positive equilibrium can occur as τ crosses some critical values. And we obtain the formulae determining direction of Hopf bifurcation and stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Focus quantities with applications to some finite‐dimensional systems

In studying small limit cycles of finite‐dimensional systems, one of the central problem is the computation of focus quantities. In practice, the computation is a challenging problem even for some simple low‐dimensional systems. This paper is devoted to the computation of focus quantities of all orders and to the study of Hopf bifurcations in some chaotic systems. A recursive formula for computing focus quantities is presented for a K + 2‐dimensional system. The formula is a generalization of previous results on low‐dimensional systems with K = 0 and K = 1. For a four‐dimensional hyper‐chaotic system, according to the sign of the first focus quantity, we prove that the simple Hopf bifurcation of the system is supercritical. For a five‐dimensional chaotic system with four equilibria of Hopf type, according to the signs of the first focus quantities, we prove that the simple Hopf bifurcations of the system are subcritical.     

Numerical bifurcation analysis of a ‘car‐following’ model

We study a system of ordinary differential equations describing a car‐following model for the motion of N car around a circular highway. All cars behave in the same way. The acceleration of each car is determined as a function of the headway (optimal velocity function). This model is known to have a solution with constant velocities and headways which, in a certain parameter regime, is stable and, varying the density of the cars, the loss of stability is generally due to a super‐ or subcritical Hopf bifurcation. Guided by analytical results, we numerically investigate the global bifurcation diagram for periodic solutions and obtain a complete picture of the dynamics of the model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation analysis of HIV infection model with antibody and cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV‐1 infection with antibody, cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response is investigated. The stability of the infection‐free and infected steady states is investigated. The basic reproduction number R₀ is identified for the proposed system. If R₀ < 1, then there is an infection‐free steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R₀ > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.