Turing-Hopf Bifurcation Point Research Articles

Spatiotemporal Dynamics of a General Two-Species System with Taxis Term

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one.

A high accuracy compact difference scheme and numerical simulation for a type of diffusive plant-water model in an arid flat environment

<abstract><p>In this paper, we investigate the numerical computation method for a one-dimensional self diffusion plant water model with homogeneous Neumann boundary conditions. First, a high accuracy compact difference scheme for the diffusive plant water model in an arid flat environment is constructed using the finite difference method. The fourth order compact difference scheme is used for the spatial derivative term, and the Taylor series expansion and residual correction function are used to discretize the time term. We obtain a difference scheme with second-order accuracy in time and fourth-order accuracy in space. Second, the Fourier analysis method is used to prove that the above format is unconditionally stable. Then, the numerical examples provided the convergence and accuracy of the difference scheme. Finally, numerical simulations are conducted near the Turing Hopf bifurcation point of the model to obtain the spatial distribution maps of vegetation and water under small disturbances of different parameters. In this paper, the evolution law of vegetation quantity and water density at any time is observed.Revealing the impact of small changes in parameters on the spatiotemporal dynamics of plant water models will provide a basis for understanding whether ecosystems are fragile.</p></abstract>

Turing instabilities in a glycolysis reaction-diffusion system

Pattern dynamics in reaction-diffusion modeling glycolysis are investigated. This paper includes linear stability analysis to deduce the threshold condition for Turing pattern formation and weakly nonlinear analysis to describe the time evolution of the pattern amplitude close to the instability threshold. We will assume that the emerging patterns do not have any spatial structure. We derive a criterion to find the region of parameters for which super/sub-critical Turing instability is possible. Furthermore, we focus on the Turing-Hopf (TH) bifurcation and obtain the reduction of the model to the TH normal forms in order to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the degenerate Turing-Hopf bifurcation point. Our theoretical results are confirmed by numerical simulations.

Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator.

It has been shown that Allee effect can change predator-prey dynamics and impact species persistence. Allee effect in the prey population has been widely investigated. However, the study on the Allee effect in the predator population is rare. In this paper, we investigate the spatiotemporal dynamics of a diffusive predator-prey model with digestion delay and Allee effect in the predator population. The conditions of stability and instability induced by diffusion for the positive equilibrium are obtained. The effect of delay on the dynamics of system has three different cases: (a) the delay doesn't change the stability of the positive equilibrium, (b) destabilizes and stabilizes the positive equilibrium and induces stability switches, or (c) destabilizes the positive equilibrium and induces Hopf bifurcation, which is revealed (numerically) to be corresponding to high, intermediate or low level of Allee effect, respectively. To figure out the joint effect of delay and diffusion, we carry out Turing-Hopf bifurcation analysis and derive its normal form, from which we can obtain the classification of dynamics near Turing-Hopf bifurcation point. Complex spatiotemporal dynamical behaviors are found, including the coexistence of two stable spatially homogeneous or inhomogeneous periodic solutions and two stable spatially inhomogeneous quasi-periodic solutions. It deepens our understanding of the effects of Allee effect in the predator population and presents new phenomena induced be delay with spatial diffusion.

BIFURCATIONS AND HYDRA EFFECTS IN A REACTION-DIFFUSION PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE

In this paper, through bifurcation analysis and numerical simulations, we consider a reaction-diffusion predator-prey model with Holling Ⅱ functional response to analyze the existence of hydra effect and the relationship between mortality independent of predator density and different steady-state solutions of the system. The hydra effect, which is a paradoxical result in both theoretical and applied ecology, refers to the phenomenon in which an increase in population mortality enhances its own population size. We investigate the existence of the hydra effect when the positive equilibrium point is locally asymptotically stable and Turing unstable. Meanwhile, numerical simulations verify the existence of the hydra effect when the one-dimensional reaction-diffusion system has a spatially inhomogeneous steady-state solution. In addition, we introduce the existence of the Turing bifurcation, the Hopf bifurcation, and the Turing-Hopf bifurcation with the parameters $ d_{2} $ and $ m_{C} $, respectively, as well as the normal form for the Turing-Hopf bifurcation. Based on the obtained normal form, we analyze the complex spatio-temporal dynamics near the Turing-Hopf bifurcation point. Finally, the numerical simulations are carried out to corroborate the obtained theoretical results.

Complex pattern formations induced by the presence of cross-diffusion in a generalized predator–prey model incorporating the Holling type functional response and generalization of habitat complexity effect

The interaction between predator and their prey is one of the most complex processes in ecosystems due to its structure and nature. Many extrinsic and intrinsic factors can affect the population dynamics. In this manuscript, we have focused our study on the process of Turing-pattern formation in a generalized predator–prey system with cross-diffusion and habitat complexity effects. By applying the linear stability theory, sufficient conditions for the occurrence of Hopf bifurcation and Turing-driven instability are successfully established. Moreover, the amplitude equations are derived near the critical value of Turing instability according to the standard multiple-scale analysis. From the mathematical view, our investigation reveals that the whole patterns generated by the system are governed by two control parameters, which are the cross-diffusion parameter δ21 and the predator mortality rate m. Complex but interesting dynamical pattern formations are specified theoretically and numerically such as spot pattern, stripe pattern, and spot-strip pattern in terms of varying the two control parameters. To support the theoretical results, a series of numerical tests are carried out, especially, to illustrate the diverse patterns near the codimension- Turing–Hopf bifurcation point (δ21T,m(0)H) and the significant impact of δ21 and m on the distribution of the two species.

Turing–Hopf Bifurcation and Its Normal Form in a Diffusive Three-Species Food Chain System with Strong Allee Effect

In this paper, we consider a diffusive three-species food chain system with strong Allee effect subject to the Neumann boundary condition. The dynamics of the local system including stability of interior equilibria and Hopf bifurcation are discussed. For the diffusive system, we study the existence of non-negative solutions, stability of a positive homogeneous steady state, diffusion-driven Turing instability and the occurrence of Turing–Hopf bifurcation. Employing the multiple scale analysis, we derive the normal form of Turing–Hopf bifurcation, which helps us classify the dynamical behaviors of the diffusive system near the Turing–Hopf bifurcation point.

Dynamic analysis of a plant-water model with spatial diffusion

Vegetation pattern is one of the typical features in arid or semi-arid areas and thus revealing the mechanism of vegetation evolution is helpful to understand the behavior of ecosystem. To this end, we investigate the spatiotemporal dynamics of a diffusive plant-water model in an arid flat environment in this paper. By carrying out bifurcation analysis, we find that the system has the properties of Turing-Hopf bifurcation. Based on the normal form theory of reaction-diffusion equations, the dynamics of the diffusive plant-water model in the neighbourhood of the Turing-Hopf bifurcation point are exactly analyzed. Our results show that tiny changes of parameters can induce the switch of four different states including uniform state, time periodic state, spatially inhomogeneous steady state and spatially inhomogeneous periodic state. Moreover, there are different types of bistable phenomenon between desert state and other states, which will provide some insights into that whether ecosystems are vulnerable.

Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering two different prey behaviours’ transition

In this paper, we study the Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analysing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by calculating and analysing the normal form on the centre manifold associated with the Turing–Hopf bifurcation, we investigate the dynamical classification near the Turing–Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.

Spatiotemporal patterns in a diffusive predator–prey system with Leslie–Gower term and social behavior for the prey

In this paper, we deal with a new approximation of a diffusive predator–prey model with Leslie–Gower term and social behavior for the prey subject to Neumann boundary conditions. A new approach for a predator–prey interaction in the presence of prey social behavior has been considered. Our main topic in this work is to study the influence of the prey's herd shape on the predator–prey interaction in the presence of Leslie–Gower term. First of all, we examine briefly the system without spatial diffusion. By analyzing the distribution of the eigenvalues associated with the constant equilibria, the local stability of the equilibrium points and the existence of Hopf bifurcation have been investigated. Then, the spatiotemporal dynamics introduced by self‐diffusion was determined, where the existence of the positive solution, Hopf bifurcation, Turing‐driven instability, and Turing–Hopf bifurcation point have been derived. Further, the effect of the prey's herd shape rate on the prey and predator equilibrium densities as well as on the Hopf bifurcating points has been discussed. Finally, by using the normal form theory on the center manifold, the direction and stability of the bifurcating periodic solutions have also been obtained. To illustrate the theoretical results, some graphical representations are given.

Diffusion‐driven codimension‐2 Turing–Hopf bifurcation in the general Brusselator model

In this paper, spatiotemporal dynamics for a general reaction–diffusion system of Brusselator type under the homogeneous Neumann boundary condition is considered. It is shownthat the reaction–diffusion system has a unique steady state solution. For some suitable parameters, we prove that the steady state solution can be a codimension‐2 Turing–Hopf point. To understand the spatiotemporal dynamics in the vicinity of the Turing–Hopf bifurcation point, we calculate and analyze the normal form on the center manifold by analytical methods. A wealth of complex spatiotemporal dynamics near the degenerate point are obtained. It is proved that the system undergoes a codimension‐2 Turing–Hopf bifurcation. Moreover, several numerical simulations are carried out to illustrate the validity of our theoretical results.

Spatial patterns in a vegetation model with internal competition and feedback regulation

The vegetation patterns are a characteristic particularity of semiarid zones, which can be the future of modern ecology for its importance. This paper aims to study a diffusive vegetation model for the subject of studying the complex patterns generated by the presence of Turing-Hopf bifurcation. The main focus is on analyzing the effect of the locative internal rivalry between plants and feedback regulation on the pattern formations. The cross-diffusion produced by the positive feedback regulation generates surprising dynamics such as Hopf bifurcation, Turing bifurcation, Turing-Hopf bifurcation, which confirms the imbalance of the distribution of the vegetation in the semi-desert regions. For analyzing the spatiotemporal behavior near the Turing-Hopf bifurcation point, the Amplitude equation restricted at this point is used. Further, by using numerical simulations, the complex dynamics induced by the positive feedback redistribution and inner competition are explored.

Turing-Hopf patterns in a morphochemical model for electrodeposition with cross-diffusion

This paper focuses on the impact of cross-diffusion for Turing-Hopf instability in a morphochemical model for electrodeposition (DIB) and completes the analysis on the role of cross-diffusion on pattern formation in electrodeposition we recently carried out in [Lacitignola et al. 2018]. We derive and discuss conditions for the destabilization a´la Turing of the Hopf limit cycle in the neighborhood of the codimension-two Turing-Hopf bifurcation point, resulting in oscillatory Turing patterns. Numerical validation of the above theoretical findings is provided both in the case of Weak and Strong Turing-Hopf instability. Experimental evidence of this kind of phenomenology is then discussed also in relation to the charging and discharging process of a battery.

BIFURCATION ANALYSIS OF A DIFFUSIVE PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE

<abstract> In this paper, we consider a diffusive predator-prey model with Beddington-DeAngelis functional response. The Turing instability and Hopf bifurcation of the coexisting equilibrium are investigated. We also use bifurcation parameters $ m, {d_2} $ to study the Turing-Hopf bifurcation. In addition, we compute the normal form for the Turing-Hopf bifurcation. On the basis of the corresponding normal form, there exists complex spatiotemporal dynamics near Turing-Hopf bifurcation point. Finally, Some numerical simulations are given to illustrate our theoretical results. </abstract>

Turing-Hopf bifurcation analysis in a diffusive Gierer-Meinhardt model

The reaction-diffusion Gierer-Meinhardt system in one dimensional bounded domain is considered in the present paper. The Hopf bifurcation is investigated, which is found to be degenerate. With the aid of Maple, the normal form associated with the degenerate Hopf bifurcation is obtained to determinate the existence of Bautin bifurcation. We get the universal unfolding for the Bautin bifurcation so that we can identify the stability of periodic solutions. Then, the existence of the codimension-two Turing-Hopf bifurcation is further investigated. To research the spatiotemporal dynamics of the model near the Turing-Hopf bifurcation point, the method of the multiple time scale analysis is adopted to derive the amplitude equations. It is noted that the Gierer-Meinhardt model may show the spatial, temporal or the spatiotemporal patterns, such as the nonconstant steady state, spatially homogeneous periodic solutions and the spatially inhomogeneous periodic solutions. Finally, some numerical simulations are presented to demonstrate the applicability of the theoretical results.

Spatiotemporal patterns induced by cross-diffusion in predator–prey model with prey herd shape effect

In this paper, we investigate a predator–prey model with herd behavior and cross-diffusion subject to the zero flux boundary conditions. First, the temporal behavior of the model has been investigated, where Hopf bifurcation has been obtained. Then, by analyzing the characteristic equation it has been proved that the cross-diffusion generate a complex dynamics such as Hopf bifurcation, Turing instability, even Turing–Hopf bifurcation. Further, the impact of the prey herd shape on the spatiotemporal patterns has been discussed. Furthermore, by computing and analyzing the normal form associated with the Turing–Hopf bifurcation point, the spatiotemporal dynamics near the Turing–Hopf bifurcation point has been discussed and also justified by some numerical simulations.

Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system

The Turing and Turing-Hopf bifurcations of a Leslie-Gower type predator-prey system with ratio-dependent Holling III functional response are investigated in this paper. Complex and interesting patterns induced by the bifurcations are identified theoretically and numerically. First the existence conditions of the Turing instability and the Turing-Hopf bifurcation are established from the theoretical analysis, respectively. Then by employing the technique of weakly nonlinear analysis, amplitude equations generated near the Turing instability critical value are derived. Various spatiotemporal patterns, such as homogeneous stationary state patterns, hexagonal patterns, coexisting patterns, stripe patterns, and their stability are determined via analyzing the obtained amplitude equations. Numerical simulations are presented to illustrate the theoretical analysis. Especially, the analogous-spiral and symmetrical wave patterns can be found near the codimension-two Turing-Hopf bifurcation point. A security center of the prey species can be found as well. These spatiotemporal patterns are explained from the perspective of the predators and prey species.

Turing-Hopf bifurcation of reaction-diffusion neural networks with leakage delay

In artificial neural networks, the diffusion phenomenon of electrons exists inevitably, due to the electromagnetic field of neural networks is heterogeneous. In this paper, we study the spatio-temporal dynamical behaviors of a reaction-diffusion neural network with leakage delay. By analyzing the corresponding characteristic equation, the sufficient and necessary conditions of Turing instability are obtained and the existence of Turing, Hopf, and Turing-Hopf bifurcations is also established. Furthermore, the truncated normal form up to third order is derived to understand and classify the spatio-temporal dynamics close to the Turing-Hopf bifurcation point. By numerical simulations, we find a pair of spatially inhomogeneous periodic solutions and illustrate the effects of time delays and spatial diffusion on the spatio-temporal dynamics of the model.

Spatiotemporal Dynamics in Reaction–Diffusion Neural Networks Near a Turing–Hopf Bifurcation Point

Since the electromagnetic field of neural networks is heterogeneous, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate the existence of Turing–Hopf bifurcation in a reaction–diffusion neural network. By the normal form theory for partial differential equations, we calculate the normal form on the center manifold associated with codimension-two Turing–Hopf bifurcation, which helps us understand and classify the spatiotemporal dynamics close to the Turing–Hopf bifurcation point. Numerical simulations show that the spatiotemporal dynamics in the neighborhood of the bifurcation point can be divided into six cases and spatially inhomogeneous periodic solution appears in one of them.

Turing\u2013Hopf bifurcation of a ratio-dependent predator-prey model with diffusion

In this paper, the Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion and Neumann boundary condition is considered. Firstly, we present a kind of double parameters selection method, which can be used to analyze the Turing–Hopf bifurcation of a general reaction-diffusion equation under Neumann boundary condition. By analyzing the distribution of eigenvalues, the stable region, the unstable region (including Turing unstable region), and Turing–Hopf bifurcation point are derived in a double parameters plane. Secondly, by applying this method, the Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion is investigated. Finally, we compute normal forms near Turing–Hopf singularity and verify the theoretical analysis by numerical simulations.