Stability and Hopf bifurcation for a three-component reaction–diffusion population model with delay effect

A delayed three-component reaction–diffusion population model with Dirichlet boundary condition is investigated. The existence and stability of the positive spatially nonhomogeneous steady state solution are obtained via the implicit function theorem. Moreover, taking delay τ$\tau$ as the bifurcation parameter, Hopf bifurcation near the steady state solution is proved to occur at the critical value _τ0${\tau }_0$. The direction of Hopf bifurcation is forward. In particular, by using the normal form theory and the center manifold reduction for partial functional differential equations, the stability of bifurcating periodic solutions occurring through Hopf bifurcations is investigated. It is demonstrated that the bifurcating periodic solution occurring at _τ0${\tau }_0$ is orbitally asymptotically stable. Finally, the general results are applied to four types of three species population models. Numerical simulations are presented to illustrate our theoretical results.     

Stability of steady-state solutions to a prey-predator system with cross-diffusion

This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response

We consider a delayed predator-prey system with Beddington-DeAngelis functional response. The stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are performed to illustrate the obtained results.     

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.     

Stability and Hopf bifurcation analysis in a three-level food chain system with delay

A class of three level food chain system is studied. With the theory of delay equations and Hopf bifurcation, the conditions of the positive equilibrium undergoing Hopf bifurcation is given regarding τ as the parameter. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument, and numerical simulations are performed to illustrate the analytical results.     

Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the 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.     

Bifurcations for a predator-prey system with two delays

In this paper, a predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838], we may show the global existence of periodic solutions.     

Stability and bifurcation in a harmonic oscillator with delays

We consider a harmonic oscillator with delays. Linear stability is investigated by analyzing the associated characteristic transcendental equation. The bifurcation analysis of the equation shows that Hopf bifurcation can occur as the delay τ (taken as a parameter) crosses some critical values. The direction and stability of the Hopf bifurcation are considered by using the normal form theory due to Faria and Magalhães. An example is given to explain the results. Numerical simulations support our results.     

Hopf bifurcation in numerical approximation of the sunflower equation

In this paper we consider the numerical solution of the sunflower equation. We prove that if the sunflower equation has a Hopf bifurcation point ata =a ₀, then the numerical solution with the Euler-method of the equation has a Hopf bifurcation point ata _h =a ₀ +O(h).     

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.     

Hopf bifurcation and topological horseshoe of a novel finance chaotic system

This paper deals with the existence of both Hopf bifurcation and topological horseshoe for a novel finance chaotic system. First, through rigorous mathematical analysis, we show that a Hopf bifurcation occurs at systems’ three equilibriums S_0,1,2 and Hopf bifurcation at equilibrium S₀ is non-degenerate and supercritical. Second, the computer-assisted verifications for horseshoe chaos in the system are given. Simulation results are presented to support the analysis.     

Bifurcation of infinite Prandtl number rotating convection

We consider infinite Prandtl number convection with rotation which is the basic model in geophysical fluid dynamics. For the rotation free case, the rigorous analysis has been provided by Park (2005, 2007, revised for publication) [5], [6] and [25] under various boundary conditions. By thoroughly investigating we prove in this paper that the solutions bifurcate from the trivial solution u=0 to an attractor Σ_R which consists of only one cycle of steady state solutions and is homeomorphic to S¹. We also see how intensively the rotation inhibits the onset of convective motion. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation which was developed by Ma and Wang (2005); see [15].     

Stability and Bifurcation in a Delayed Reaction–Diffusion Equation with Dirichlet Boundary Condition

In this paper, we study the dynamics of a diffusive equation with time delay subject to Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady-state solution is investigated by applying Lyapunov–Schmidt reduction. The existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution is derived by analyzing the distribution of the eigenvalues. The direction of Hopf bifurcation and stability of the bifurcating periodic solution are also investigated by means of normal form theory and center manifold reduction. Moreover, we illustrate our general results by applications to the Nicholson’s blowflies models with one- dimensional spatial domain.     

Bifurcation analysis for a regulated logistic growth model

In this paper, we consider a regulated logistic growth model. We first consider the linear stability and the existence of a Hopf bifurcation. We show that Hopf bifurcations occur as the delay τ passes through critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit algorithm determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, numerical simulation results are given to support the theoretical predictions.     

Bifurcations in a Diffusive Predator–Prey Model with Beddington–DeAngelis Functional Response and Nonselective Harvesting

In this paper, we discuss the dynamics of a predator–prey model with Beddington–DeAngelis functional response and nonselective harvesting. By using the Lyapunov–Schmidt reduction, we obtain the existence of spatially nonhomogeneous steady-state solution. The stability and existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the change of a specific parameter are investigated by analyzing the distribution of the eigenvalues. We also get an algorithm for determining the bifurcation direction of the Hopf bifurcating periodic solutions near the nonhomogeneous steady-state solution. Finally, we show some numerical simulations to verify our analytical results.

     

The delay effect on reaction–diffusion equations

In this work we study a reaction–diffusion problem with delay and we make an analysis of the stability of solutions by means of bifurcation theory. We take the delay constant as a parameter. Special conditions on the vector field assure existence of a spatially nonconstant positive equilibrium U_k , which is stable for small values of the delay. An increase of the delay destabilizes the equilibrium of U_k and leads to super or subcritical Hopf bifurcation.     

Numerical Hopf bifurcation analysis in nonlinear ordinary and partial differential systems from chemical reactor theory

A code has been developed which will automatically locate and analyze points of Hopf bifurcation in autonomous ordinary differential systems. The code first locates critical value(s) v_c of a user-specified parameter v (the bifurcation parameter) such that a stationary (equilibrium) solution x_*(v) loses linear stability by virtue of a complex conjugate pair of eigenvalues. The code computes x_*(v) during the location of v_c. Then the code computes the various coefficients in a local approximation to the family of periodic solutions which arise, a process which involves computation of second and third partial derivatives by numerical differencing of the user-supplied Jacobian matrix. The current version of the code, called BIFOR2, is fully described in Hassard, Kazarinoff, and Wan, Theory and Applications of Hopf Bifurcation, Cambridge U.P., 1981. In this paper we demonstrate the code in applications to two systems drawn from chemical reactor theory. The first application is to a 4th order ordinary differential system modeling a coupled tank reactor. The second application is to a partial differential system modeling a catalyst particle system. These represent the first applications of the code to chemically reacting systems other than the Brusselator. The second application demonstrates how collocation methods may be used in conjunction with BIFOR2 to perform Hopf bifurcation analysis of partial differential systems.     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

Bifurcation formulae derived from center manifold theory

Applicable formulae for the parameters μ₂, β₂, τ₂, μ₄, β₄ and τ₄ of N-dimensional Hopf bifurcation theory are presented. The center manifold theorem is used to reduce the system from N dimensions to 2 dimensions. Approximate solution of the system in Poincaré normal form provides the formulae. The formulae are explicit so that the parameters may be computed directly from partial derivatives of the system in real canonical form. The formula for μ₂ is shown to be identical to that of I. D. Hsu and N. D. Kazarinoff. Formulae for the parameters μ₁, μ₂, μ₃, τ₁, τ₂, τ₃, β₂, β₃ and β₄ in the “tangency” case Re λ₁′(v_c) = 0, Re λ₁′'(v_c) ≠ 0 are also presented.