Steady-state Bifurcation Research Articles

Extended Fisher-Kolmogorov系统在Direchlet边界条件下的定态分歧

Extended Fisher-Kolmogorov系统在Direchlet边界条件下的定态分歧

Local bifurcation of a Ronsenzwing-MacArthur predator prey model with two prey-taxis.

The paper investigates the steady state bifurcation analysis in a general Ronsenzwing-MacArthur predator prey model with two prey-taxis under Neumann boundary conditions. The results show that the rich dynamics in predator prey systems with two prey taxis.

Dynamics and pattern formation of a diffusive predator–prey model in the presence of toxicity

In this paper, we develop a diffusivepredator–prey model with toxins under the homogeneous Neumann boundary condition. First, the persistence property and global asymptotic stability of the constant steady states are established. Then by analyzing the associated characteristic equation, we derive explicit conditions for the existence of nonconstant steady states that emerge through steady-state bifurcation from related constant steady states. Furthermore, the existence and nonexistence of nonconstant positive steady states of this model are studied by considering the effect of large diffusivity. Finally, in order to verify our theoretical results, some numerical simulations are also included. These explicit conditions are numerically verified in detail and further compared to those conditions ensuring Turing instability. It is shown that the numerically observed behaviors are in good agreement with the theoretically proposed results. All theoretical analyses and numerical simulations show that toxic substances have a perceptible effect on the system.

The steady-state lifting bifurcation problem associated with the valency on networks

In this paper we consider homogeneous coupled cell networks with asymmetric inputs — networks where each cell receives exactly one input of each edge type. The coupled cell systems associated with a network are the dynamical systems that respect the network structure. There are subspaces, determined solely by the network structure, that are flow-invariant under any such coupled cell system — the synchrony subspaces. For a homogeneous network with asymmetric inputs, one of the eigenvalues of the Jacobian matrix of any coupled cell system at an equilibrium in the full-synchrony subspace corresponds to the valency of the network. In this work, we study the codimension-one steady-state bifurcations of coupled cell systems with a bifurcation condition associated with the valency. We start by giving an adaptation of the Perron–Frobenius Theorem for the eigenspace associated with the valency showing that the dimension of that eigenspace equals the number of the network source components. A network source component is a strongly connected component of the network whose cells receive inputs only from cells in the component. Each synchrony subspace determines a smaller network called quotient network. The lifting bifurcation problem addresses the issue of understanding when the bifurcation branches of a network can be lifted from one of its quotient networks. We consider the lifting bifurcation problem when the bifurcation condition is associated with the valency. We give sufficient conditions on the number of source components for the answer to the lifting bifurcation problem to be positive and prove that those conditions are necessary and sufficient for a class of networks.

Rich spatial–temporal dynamics in a diffusive population model for pioneer–climax species

A general diffusive population model for interactions of pioneer and climax species subject to the no-flux boundary condition is considered. Local and global steady-state bifurcations as well as Hopf bifurcations are investigated. A condition for Turing instability not to happen is obtained, and the conditions for occurrences of Turing bifurcations and Hopf bifurcations are also obtained. Numerical simulations are carried out to demonstrate and extend the obtained analytic results which suggest that the spatial diffusion may make the climax species more dominant. The results indicate that the model, with spatial diffusion incorporated , can have very rich spatial–temporal dynamics.

Bifurcation solutions in the diffusive minimal sediment

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.

Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey

It is well known that the Leslie-Gower prey-predator model (without Allee effect) has a unique globally asymptotically stable positive equilibrium point, thus there is no Hopf bifurcation branching from positive equilibrium point. In this paper we study the Leslie-Gower prey-predator model with strong Allee effect in prey, and perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the steady-state bifurcation of PDE model. Moreover, by the center manifold theory and the normal form method, the direction and stability of Hopf bifurcation solutions are established. Finally, some numerical simulations are presented. Apparently, Allee effect changes the topology structure of the original Leslie-Gower model.

The lifting bifurcation problem on feed-forward networks

We consider feed-forward networks, that is, networks where cells can be divided into layers, such that every edge targeting a layer, excluding the first one, starts in the prior layer. A feed-forward system is a dynamical system that respects the structure of a feed-forward network. The synchrony subspaces for a network, are the subspaces defined by equalities of some cell coordinates, that are flow-invariant by all the network systems. The restriction of a network system to each of its synchrony subspaces is a system associated with a smaller network, which may be, or not, a feed-forward network. The original network is then said to be a lift of the smaller network. We show that a feed-forward lift of a feed-forward network is given by the composition of two types of lifts: lifts that create new layers and lifts inside a layer. Furthermore, we address the lifting bifurcation problem on feed-forward systems. More precisely, the comparison of the possible codimension-one local steady-state bifurcations of a feed-forward system and those of the corresponding lifts is considered. We show that for most of the feed-forward lifts, the increase of the center subspace is a sufficient condition for the existence of additional bifurcating branches of solutions, which are not lifted from the restricted system. However, when the bifurcation condition is associated with the internal dynamics and the lift occurs inside an intermediate layer, we prove that the existence of a bifurcation branch not lifted from the restricted system does depend generically on the chosen feed-forward system.

On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system

<p style='text-indent:20px;'>In this paper, the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system is investigated. By using the Lyapunov-Schmidt method, combining with the implicit function theorem, we prove that this system bifurcates from the trivial solution to the nontrivial solution branch as parameter crosses certain critical value. The expression of bifurcated solution is also obtained.

Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates

<p style='text-indent:20px;'>The main aim of this paper is to study the bifurcation solutions associated with the spinor Bose-Einstein condensates. Based on the Principle of Hamilton Dynamics and the Principle of Lagrangian Dynamics, a general pattern formation equation for the spinor Bose-Einstein condensates is established. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis and the different external confining potentials. With the change of different external potentials, the different topological structures of bifurcation solutions for the spinor Bose-Einstein condensates system are derived from steady state bifurcation theory.

Bifurcations and Pattern Formation in a Predator–Prey Model

In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

In this paper, the stability and bifurcation analysis of symmetrical and asymmetrical micro-rotating shafts are investigated when the rotational speed is in the vicinity of the critical speed. With the help of Hamilton’s principle, nonlinear equations of motion are derived based on non-classical theories such as the strain gradient theory. In the dynamic modeling, the geometric nonlinearities due to strains, and strain gradients are considered. The bifurcations and steady state solution are compared between the classical theory and the non-classical theories. It is observed that using a non-classical theory has considerable effect in the steady-state response and bifurcations of the system. As a result, under the classical theory, the symmetrical shaft becomes completely stable in the least damping coefficient, while the asymmetrical shaft becomes completely stable in the highest damping coefficient. Under the modified strain gradient theory, the symmetrical shaft becomes completely stable in the least total eccentricity, and under the classical theory the asymmetrical shaft becomes completely stable in the highest total eccentricity. Also, it is shown that by increasing the ratio of the radius of gyration per length scale parameter, the results of the non-classical theory approach those of the classical theory.

Bifurcation and spatio-temporal patterns in a diffusive predator–prey system

A diffusive predator–prey system with predator functional response subject to Neumann boundary conditions is considered. Existence, nonexistence, and boundedness of non-constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation. Hopf and steady-state bifurcation analyses are carried out in detail. These results provide theoretical evidences to the complex spatio-temporal dynamics found by numerical simulation.

Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems

We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in B. Rink and J. Sanders, "Coupled cell networks and their hidden symmetries", SIAM J. Math. Anal., 46 (2014). It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group - finite or compact Lie. Our generalization also includes non-compact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.

Numerical bifurcation analysis of mathematical models with time delays with the package DDE-BIFTOOL

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and making predictions in various areas of the life sciences, e.g., population dynamics, epidemiology, immunology, physiology, neural networks. The time delays in these models take into account a dependence of the present state of the modeled system on its past history. The delay can be related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period and so on. Due to an infinite-dimensional nature of DDEs, analytical studies of the corresponding mathematical models can only give limited results. Therefore, a numerical analysis is the major way to achieve both a qualitative and quantitative understanding of the model dynamics. A bifurcation analysis of a dynamical system is used to understand how solutions and their stability change as the parameters in the system vary. The package DDE-BIFTOOL is the first general-purpose package for bifurcation analysis of DDEs. This package can be used to compute and analyze the local stability of steady-state (equilibria) and periodic solutions of a given system as well as to study the dependence of these solutions on system parameters via continuation. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. In this paper we describe the structure of DDE-BIFTOOL, numerical methods implemented in the package and we illustrate the use of the package using a certain DDE system.

Dynamics in a diffusive predator–prey system with ratio-dependent predator influence

A diffusive predator–prey system with predator functional response subject to Neumann boundary conditions is considered. Existence, nonexistence, and boundedness of positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation. Hopf and steady-state bifurcation analyses are carried out in detail. These results provide theoretical evidences to the complex spatio-temporal dynamics found by numerical simulation.

Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior

• A predator-prey model with diffusion and herd behavior is investigated. • The local and global stability are determined. • The steady-state bifurcations from both simple and double eigenvalues are studied. • Numerical simulations illustrate and supplement our theoretical analysis. This paper is concerned with a diffusive predator-prey model with herd behavior. The local and global stability of the unique homogeneous positive steady state U * is obtained. Treating the conversion or consumption rate γ as the bifurcation parameter, the steady-state bifurcations both from simple and double eigenvalues are studied near U *. The techniques include the Lyapunov function, the spectrum analysis of operators, the bifurcation theory, space decompositions and the implicit function theorem.

Front propagation in weakly subcritical pattern-forming systems.

The speed and stability of fronts near a weakly subcritical steady-state bifurcation are studied, focusing on the transition between pushed and pulled fronts in the bistable Ginzburg-Landau equation. Exact nonlinear front solutions are constructed and their stability properties investigated. In some cases, the exact solutions are stable but are not selected from arbitrary small amplitude initial conditions. In other cases, the exact solution is unstable to modulational instabilities which select a distinct front. Chaotic front dynamics may result and is studied using numerical techniques.

Analysis on bifurcation solutions of an atherosclerosis model

This paper is concerned with the inflammatory process model resulting in the development of atherosclerosis subject to no-flux boundary conditions. The dissipation and persistence of the system are obtained. The steady-state bifurcations are also studied in two cases. The bifurcation from the simple eigenvalue can be extended to infinity by increasing d2 to infinity, and the bifurcation from the double eigenvalue is intensively investigated. The techniques include the spectrum analysis of operators, the bifurcation theory, space decompositions and the implicit function theorem.

The local bifurcation and stability of nontrivial steady states of a logistic type of chemotaxis

Chemotaxis is a type of oriented movement of cells in response to the concentration gradient of chemical substances in their environment. We consider local existence and stability of nontrivial steady states of a logistic type of chemotaxis. We carry out the bifurcation theory to obtain the local existence of the steady state and apply the expansion method on the chemotaxis to investigate the bifurcation direction. Moreover, by applying the bifurcation direction, we obtain the bifurcating steady state is stable when the bifurcation curve turns to right under certain conditions.