Global Bifurcation Structure Research Articles

Global bifurcation in a diffusive Beddington-DeAngelis predator–prey model with population flux by attractive transition

The study involves examining the global bifurcation structure associated with the nonconstant steady states of a reaction-diffusion predator-prey system where both the species interact in accordance with the Beddington DeAngelis response and the movement flux of the predator incorporates attractive transition. We consider the magnitude of population flux by attractive transition as the bifurcation parameter and employ the Crandall-Rabinowitz bifurcation theorem to study the global bifurcation structure associated with the problem. We have also derived some a priori estimates associated with the problem and carried out numerical simulations to support our theoretical results. This work can be regarded as the first step towards inclusion of population flux by attractive transition in scenarios where interactions are governed by complex functional responses.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.

Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis

Abstract This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.

Dynamics of hydrodynamically unstable premixed flames in a gravitational field – local and global bifurcation structures

The dynamics of hydrodynamically unstable premixed flames are studied using the nonlinear Michelson–Sivashinsky (MS) equation, modified appropriately to incorporate effects due to gravity. The problem depends on two parameters: the Markstein number that characterises the combustible mixture and its diffusion properties, and the gravitational parameter that represents the ratio of buoyancy to inertial forces. A comprehensive portrait of all possible equilibrium solutions are obtained for a wide range of parameters, using a continuation methodology adopted from bifurcation theory. The results heighten the distinction between upward and downward propagation. In the absence of gravity, the nonlinear development always leads to stationary solutions, namely, cellular flames propagating at a constant speed without change in shape. When decreasing the Markstein number, a modest growth in amplitude is observed with the propagation speed reaching an upper bound. For upward propagation, the equilibrium states are also stationary solutions, but their spatial structure depends on the initial conditions leading to their development. The combined Darrieus–Landau and Rayleigh–Taylor instabilities create profiles of invariably larger amplitudes and sharper crests that propagate at an increasingly faster speed when reducing the Markstein number. For downward propagation, the equilibrium states consist in addition to stationary structures time-periodic solutions, namely, pulsating flames propagating at a constant average speed. The stabilising influence of gravity dampens the nonlinear growth and leads to spatiotemporal changes in flame morphology, such as the formation of multi-crest stationary profiles or pulsating cell splitting and merging patterns, and an overall reduction in propagation speed. The transition between these states occurs at bifurcation and exchange of stability points, which becomes more prominent when reducing the Markstein number and/or increasing the influence of gravity. In addition to the local bifurcation characterisation the global bifurcation structure of the equation, obtained by tracing the continuation of the bifurcation points themselves unravels qualitative features such as the manifestation of bi-stability and hysteresis, and/or the onset and sustenance of time-periodic solutions. Overall, the results exhibit the rich and complex dynamics that occur when gravity, however small, becomes physically meaningful.

Global bifurcation structure and some properties of steady periodic water waves with vorticity

This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work [5], we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using the modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Moreover, we show a connection between the concavity and convexity of wave profile and the monotonicity of the vertical velocity component v along the free surface. As an important application, we make up the missing major aspect on the behavior of v as mentioned in [4, Sect 4.4] for small amplitude waves. In addition, for favorable vorticity, we prove that the vertical displacement of water waves decreases with depth.

Representation formulas for stationary solutions of a cell polarization model

We are interested in the global bifurcation structure of solutions for a nonlinear boundary value problem with nonlocal constraint (SLP) that appears in a cell polarization model with mass conservation proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. We obtained primitive representation formulas of all solutions and investigated a surface \({\mathcal {S}}\) consisting of all bifurcation diagrams with heights. However, we could not find any parameterization of the surface \({\mathcal {S}}\). In this paper, we show parameterizations of the surface \({\mathcal {S}}\) and concrete representation formulas of all global bifurcation diagrams of (SLP). These results are also beneficial for numerical computations to get all bifurcation diagrams. We propose new methods in view of them.

Global bifurcation and pattern formation for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

Of concern in this paper is to propose an accurate description for the global bifurcation structure of the nonconstant steady states for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses systematically. By treating the coefficient of nonlinear prey-taxis as a bifurcation parameter and utilizing the user-friendly version of Crandall–Rabinowitz bifurcation theory, we study the global bifurcation theory of the system. Meanwhile, the existence of nonconstant steady states will be offered by the exported global bifurcation theorem under a rather natural condition. In the proof, a priori estimate of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given.

On a numerical bifurcation analysis of a particle reaction-diffusion model for a motion of two self-propelled disks

AbstractTheoretical analysis using mathematical models is often used to understand a mechanism of collective motion in a self-propelled system. In the experimental system using camphor disks, several kinds of characteristic motions have been observed due to the interaction of two camphor disks. In this paper, we understand the emergence mechanism of the motions caused by the interaction of two self-propelled bodies by analyzing the global bifurcation structure using the numerical bifurcation method for a mathematical model. Finally, it is also shown that the irregular motion, which is one of the characteristic motions, is chaotic motion and that it arises from periodic bifurcation phenomena and quasi-periodic motions due to torus bifurcation.

Bifurcation analysis of the predator\u2013prey model with the Allee effect in the predator

The use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey ( λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey ( a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.

Optimal stock–enhancement of a spatially distributed renewable resource

We study the economic management of a renewable resource, the stock of which is spatially distributed and moves over a discrete or continuous spatial domain. In contrast to standard harvesting models where the agent can control the take-out from the stock, we consider the case of optimal stock enhancement. In other words, we model an agent who is, either because of ecological concerns or because of economic incentives, interested in the conservation and enhancement of the abundance of the resource, and who may foster its growth by some costly stock–enhancement activity (e.g., cultivation, breeding, fertilizing, or nourishment). By investigating the optimal control problem with infinite time horizon in both spatially discrete and spatially continuous (1D and 2D) domains, we show that the optimal stock–enhancement policy may feature spatially heterogeneous (or patterned) steady states. We numerically compute the global bifurcation structure and optimal time-dependent paths to govern the system from some initial state to a patterned optimal steady state. Our findings extend the previous results on patterned optimal control to a class of ecological systems with important ecological applications, such as the optimal design of restoration areas.

Stationary solutions of advective Lotka–Volterra models with a weak Allee effect and large diffusion

This paper is devoted to the Neumann problem of a stationary Lotka–Volterra model with diffusion and advection. In the model it is assumed that one population growth rate is described by weak Allee effect. We first obtain some sufficient conditions ensuring the existence of nonconstant solutions by using the Leray–Schauder degree theory. And then we study a limiting system (with nonlocal constraint) which stems from the original model as diffusion and advection of one of the species tend to infinity. Finally, we classify the global bifurcation structure of nonconstant solutions of the simplified 1D case.

Local and global bifurcations in magnetic resonance force microscopy

The focus of this paper is on the investigation of local and global bifurcations in a continuum mechanics-based resonator model proposed for measurement of electron spin via magnetic resonance force microscopy (MRFM). The resonator model, derived using the extended Hamilton’s principle incorporating the Bloch equations for magnetization, is investigated analytically and numerically. Analysis of both adiabatic and non-adiabatic equilibrium configurations enables formulation of the dynamical system bifurcation structure and identification of the parameter space required for stable MRFM operation. A multiple-scales analysis of the limiting adiabatic model enables estimation of the local bifurcation thresholds for bistable solutions and prediction of the frequency shift that enables spin detection. Orbital instabilities of the adiabatic model reveal a global bifurcation structure where lengthy chaotic transients occur below a homoclinic jump-to-contact threshold which is determined via a Melnikov–Holmes analysis. Both local and global bifurcations are verified numerically in the non-adiabatic model and reveal a dense power spectra for the magnetic moments. The computation of the parameter space governing the model orbital instabilities enables a consistent estimation of robust MRFM operation conditions.

Patterns and dynamics in the diffusive model of a nutrient–microorganism system in the sediment

This paper is devoted to study patterns and dynamics in the diffusive minimal sediment model. Firstly, we analyze nonnegative constant equilibrium solutions and investigate patterns induced by diffusions. Then we study the properties and existence of nonconstant steady states. Moreover, we describe the local bifurcation structure and global bifurcation structure from two positive constant solutions, respectively. The main tools used here include the stability theory, degree theory, bifurcation theory. From extensive numerical simulations, the theoretical results are confirmed and complemented, and the influence of parameters δ1,δ2 on these patterns is depicted.

Reduction approach to the dynamics of interacting front solutions in a bistable reaction–diffusion system and its application to heterogeneous media

The dynamics of pulse solutions in a bistable reaction-diffusion system are studied analytically by reducing partial differential equations (PDEs) to finite-dimensional ordinary differential equations (ODEs). For the reduction, we apply the multiple-scales method to the mixed ODE-PDE system obtained by taking a singular limit of the PDEs. The reduced equations describe the interface motion of a pulse solution formed by two interacting front solutions. This motion is in qualitatively good agreement with that observed for the original PDE system. Furthermore, it is found that the reduction not only facilitates the analytical study of the pulse solution, especially the specification of the onset of local bifurcations, but also allows us to elucidate the global bifurcation structure behind the pulse behavior. As an application, the pulse dynamics in a heterogeneous bump-type medium are explored numerically and analytically. The reduced ODEs clarify the transition mechanisms between four pulse behaviors that occur at different parameter values.

Bifurcation structure of nonconstant positive steady states for a diffusive predator-prey model.

In this paper, we make a detailed descriptions for the local and global bifurcation structure of nonconstant positive steady states of a modified Holling-Tanner predator-prey system under homogeneous Neumann boundary condition. We first give the stability of constant steady state solution to the model, and show that the system exhibits Turing instability. Second, we establish the local structure of the steady states bifurcating from double eigenvalues by the techniques of space decomposition and implicit function theorem. It is shown that under certain conditions, the local bifurcation can be extended to the global bifurcation.

Dynamical transitions between equilibria in a dissipative Klein–Gordon lattice

We consider the energy landscape of a dissipative Klein–Gordon lattice with a ϕ 4 on-site potential. Our analysis is based on suitable energy arguments, combined with a discrete version of the Łojasiewicz inequality, in order to justify the convergence to a single, nontrivial equilibrium for all initial configurations of the lattice. Then, global bifurcation theory is explored, to illustrate that in the discrete regime all linear states lead to nonlinear generalizations of equilibrium states. Direct numerical simulations reveal the rich structure of the equilibrium set, consisting of non-trivial topological (kink-shaped) interpolations between the adjacent minima of the on-site potential, and the wealth of dynamical convergence possibilities. These dynamical evolution results also provide insight on the potential stability of the equilibrium branches, and glimpses of the emerging global bifurcation structure, elucidating the role of the interplay between discreteness, nonlinearity and dissipation. • Analysis of the dynamical transitions between equilibria in a dissipative Klein–Gordon lattice. • Application of a discrete variant of Łojasiewicz inequality. • Applications of the global bifurcation theorem in discrete set-ups. • Numerical simulations reveal the rich structure of the equilibrium set and relevant bifurcations.

Steady-states of a Leslie–Gower model with diffusion and advection

This paper focuses on a stationary Leslie–Gower model with diffusion and advection. Firstly, some existence conditions of nonconstant positive solutions are obtained by means of the Leray–Schauder degree theory. As diffusion and advection of one of the species both tend to infinity, we obtain a limiting system, which is a semi-linear elliptic equation with nonlocal constraint. In the simplified 1D case, the global bifurcation structure of nonconstant solutions of the limiting system is classified.

On the non-linear dynamics of a forced plate with boundary conditions correction in subsonic flow

• A complete modeling of a periodically driven plate with boundary conditions correction in subsonic ow is developed. • A transformation upon the Legendre polynomials is applied for converting the boundary conditions with correction. • The systems with or without correction show the similar global bifurcation structure and scaling property. • The correction can cause reversals which increase the diversity and complexity of the local bifurcation structure. When a plate is subjected to long-term vibration the original boundary conditions are no longer ideal and necessary correction should be considered during its dynamical modeling. This paper aims at the modeling of the boundary conditions correction and its influence on the nonlinear dynamics of a harmonic excited plate in subsonic flow . The plate boundary conditions with correction are non-autonomous and converted into autonomous by using a coordinate transformation upon the orthogonal Legendre polynomials. The fluid force is separated into the reactive and resistive parts. The reactive fluid force, due to plate motion, is derived from the bound and wake vorticity in Glauert’s expansion; the resistive force , which is independent on plate motion, is evaluated in drag coefficient . The governing nonlinear partial differential equation is discretized in space and time domains by using the Galerkin method. Results show that the present fluid model is in good agreement with other theories archived and is reliable. The unforced system loses its stability by divergence following a pitchfork-like bifurcation featured by the appearance of new bifurcated stable equilibrium points due to nonlinearity after instability. The boundary conditions correction can stabilize the system and does not change the bifurcation structure and scaling property. The symmetry-breaking/restoring bifurcations play an important role in the change of different period-1 motions, and the nature of such bifurcations is associated with the collision or division of attractors . Subharmonic motions and chaos appear alternatively, and the transition between chaos and periodic motions is generally in accompany with period-doubling bifurcations, explosion and condensation of attractors.

Turing patterns in a reaction–diffusion epidemic model

In this paper, we investigate the spatiotemporal dynamics of a reaction–diffusion epidemic model with zero-flux boundary conditions. The value of our study lies in two aspects: mathematically, by using maximum principle and the linearized stability theory, a priori estimates of the steady state system and the local asymptotic stability of positive constant solution are given. By using the implicit function theorem, the existence and nonexistence of nonconstant positive steady states are shown. Applying the bifurcation theory, the global bifurcation structure of nonconstant positive steady states is established. Epidemiologically, through numerical simulations, under the conditions of the existence of nonconstant positive steady states, we find that the smaller the space, the easier the pattern formation; the bigger the diffusion, the easier the pattern formation. These results are beneficial to disease control, that is, we must do our best to control the diffusion of the infectious to avoid disease outbreak.