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.     

Stationary patterns of a predator-prey model with spatial effect

In this paper, spatial patterns of predator-prey model with cross diffusion are investigated. The Hopf and Turing bifurcation critical line in a spatial domain are obtained by using mathematical theory. Moreover, exact Turing space is given in two parameters space. Our results reveal that cross diffusion can induce stationary patterns, which may be useful to help us better understand the dynamics of the real ecosystems.     

Pattern formation driven by cross-diffusion in a 2D domain

In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.     

Spatiotemporal Dynamic Analysis in a Time-space Discrete Brusselator Mode

In this paper, we study the spatiotemporal patterns of a Brusselator model with discrete time-space by using the coupled mapping lattice (CML) model. The existence and stability conditions of the equilibrium point are obtained by using linear stability analysis. Then, applying the center manifold reduction theorem and the bifurcation theory, the parametric conditions of the flip and the Neimark-Sacker bifurcation are described respectively. Under space diffusion, the model admits the Turing instability at stable homogeneous solutions under some certain conditions. Two nonlinear mechanisms, including flip-Turing instability and Neimark-Sacker-Turing instability, are presented. Through numerical simulation, periodic windows, invariant circles, chaotic phenomenon and some interesting spatial patterns are found.     

Stability analysis in a diffusional immunosuppressive infection model with delayed antiviral immune response

In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Analysis of spatiotemporal patterns in a single species reaction–diffusion model with spatiotemporal delay

Employing the theories of Turing bifurcation in the partial differential equations, we investigate the dynamical behavior of a single species reaction–diffusion model with spatiotemporal delay. The linear stability and the conditions for the occurrence of Turing bifurcation in this model are obtained. Moreover, the amplitude equations which represent different spatiotemporal patterns are also obtained near the Turing bifurcation point by using multiple scale method. In Turing space, it is found that the spatiotemporal distributions of the density of this researched species have spots pattern and stripes pattern. Finally, some numerical simulations corresponding to the different spatiotemporal patterns are given to verify our theoretical analysis.     

Hopf bifurcation in a reaction–diffusion model with Degn–Harrison reaction scheme

In this paper, we consider a chemical reaction–diffusion model with Degn–Harrison reaction scheme under homogeneous Neumann boundary conditions. The existence of Hopf bifurcation to ordinary differential equation (ODE) and partial differential equation (PDE) models are derived, respectively. Furthermore, by using the center manifold theory and the normal form method, we establish the bifurcation direction and stability of periodic solutions. Finally, some numerical simulations are shown to support the analytical results, and to reveal new phenomenon on the Hopf bifurcation.     

Dynamics in a diffusive plankton system with delay and toxic substances effect

The dynamics of a reaction–diffusion plankton system with delay and toxic substances effect is considered. Existence and priori bound of a solution for a model without delay are shown. Global asymptotic stability of the axial equilibrium is obtained. The stability of the positive equilibrium and the existence of Hopf bifurcation are investigated by analyzing the distribution of eigenvalues. And the properties of Hopf bifurcation are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Some numerical simulations are carried out for illustrating the theoretical results.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

A class of coupled cell–bulk ODE–PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius \(\epsilon \ll 1\) are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell–bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE–PDE cell–bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell–bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator–inhibitor reaction–diffusion systems in the limit of long-range inhibition and short-range activation.     

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.     

Diffusion-driven instability and bifurcation in the Lengyel–Epstein system

Lengyel–Epstein reaction–diffusion system of the CIMA reaction is considered. We derive the precise conditions on the parameters so that the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become Turing unstable or diffusively unstable. We also perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.     

一类反应扩散系统的分歧与斑图

本文研究了一类发生在密闭容器中的不可激活的高次自催化反应扩散系统．在适当的条件下,用渐进近似的方法讨论了系统平衡态的稳定范围;用多重尺度的方法证明了当扩散系数λ充分小时,系统出现两种类型的斑图,一类是由Hopf分歧引出的驻波斑图;另一类是由 Pitchfork分歧引出的定波斑图．进一步还讨论了,在分歧点附近,对于大于空间或等于空间波数的小扰动,斑图是局部稳定的,而小于自身空间波数的小扰动,斑图是不稳定的．     

Spatio-temporal dynamics of a reaction–diffusion–advection food-limited system with nonlocal delayed competition and Dirichlet boundary condition

Spatio-temporal dynamics of a reaction–diffusion–advection food-limited population model with nonlocal delayed competition and Dirichlet boundary condition are considered. Existence and stability of the positive spatially nonhomogeneous steady state solution are shown. Existence and direction of the spatially nonhomogeneous steady-state-Hopf bifurcation are proved. Stable spatio-temporal patterns near the steady-state-Hopf bifurcation point are numerically obtained. We also investigate the joint influences of some important parameters including advection rate, food-limited parameter and nonlocal delayed competition on the dynamics. It is found that the effect of advection on Hopf bifurcation is opposite with the corresponding no-flux system. The theoretical results provide some interesting highlights in ecological protection in streams or rivers.     

Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity

In this paper, we have investigated the phenomena of Turing pattern formation in a predator-prey model with habitat complexity in presence of cross diffusion. Using the linear stability analysis, the conditions for the existence of stationary pattern and the existence of Hopf bifurcation are obtained. It is shown analytically that the presence of cross diffusion in the system supports the formation of Turing pattern. Two parameter bifurcation analysis are done analytically and corresponding bifurcation diagrams are presented numerically. A series of simulation results are plotted for different biologically meaningful parameter values. Effects of variation of habitat complexity and the predator mortality rate and birth rate of prey on pattern formation are also reported. It is shown that cross-diffusion can lead to a wide variety of spatial and spatiotemporal pattern formation. It is found that the model exhibits spot and stripe pattern, and coexistence of both spot and strip patterns under the zero flux boundary condition. It is observed that cross-diffusion, habitat complexity, birth rate of prey and predator’s mortality rate play a significant role in the pattern formation of a distributed population system of predator-prey type.     

Mixed quadratic-cubic autocatalytic reaction–diffusion equations: Semi-analytical solutions

Semi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a one-dimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction–diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semi-analytical results are show to be highly accurate, in comparison to numerical solutions of the governing partial differential equations.     

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.     

Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions

We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of inequalities, introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable. Moreover, this parameter domain is rather different from the known case when also Dirichlet conditions are assumed. In particular, bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know.     

Pattern formation of a predator–prey model

In this paper, we analyze the spatial pattern of a predator–prey system. We get the critical line of Hopf and Turing bifurcation in a spatial domain. In particular, the exact Turing domain is given. Also we perform a series of numerical simulations. The obtained results reveal that this system has rich dynamics, such as spotted, stripe and labyrinth patterns, which shows that it is useful to use the reaction–diffusion model to reveal the spatial dynamics in the real world.     

一类自催化可逆生化反应模型的Hopf分支及其稳定性

在齐次Neumann边界条件下,研究一类自催化可逆三分子生化反应模型.首先对常微分系统,给出Hopf分支的存在性及稳定性.其次对偏微分系统,建立由扩散系数引起的Turing不稳定性,同时给出Hopf分支的存在性,并利用规范型理论和中心流形定理建立Hopf分支的方向和稳定性.最后,借助Matlab软件进行数值模拟,验证补...