Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

In this paper, a reaction‐diffusion predator–prey system that incorporates the Holling‐type II and a modified Leslie‐Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Mathematical and numerical analysis of a three-species predator-prey model with herd behavior and time fractional-order derivative

In this paper, we investigate with a time fractional-order derivative in a three-species predator-prey model with the presence of prey social behavior. A new approximation for predator-prey interaction in the presence of prey social behavior has been considered. For the model analysis, the study has been divided into two principal parts. First of all, we study the local stability of the equilibria and the existence of Hopf bifurcation. Then, for the numerical analysis, the Caputo fractional derivative operator is utilized to approximate the numerical solution of the model. An excellent agreement is seen between the numerical results and the theoretical predictions.     

Stability and instability analysis for a ratio-dependent predator–prey system with diffusion effect

In this paper, we consider a ratio-dependent predator–prey system with diffusion. And we mainly discuss the following problems: (1) stability and Hopf bifurcation analysis of the positive equilibrium for the reduced ODE system; (2) Diffusion-driven instability of the equilibrium solution; (3) Hopf bifurcations for the corresponding diffusion system with homogeneous Neumann boundary conditions. In order to verify our theoretical results, some numerical simulations are also included, respectively.     

Effect of herd shape in a diffusive predator-prey model with time delay

In this paper, we deal with the effect of the shape of herd behavior on the interaction between predator and prey. The model analysis was studied in three parts. The first, The analysis of the system in the absence of spatial diffusion and the time delay, where the local stability of the equilibrium states, the existence of Hopf bifurcation have been investigated. For the second part, the spatiotemporal dynamics introduce by self diffusion was determined, where the existence of Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been proved. Further, the order of Hopf bifurcation points and regions of the stability of the non trivial equilibrium state was given. In the last part of the paper, we studied the delay effect on the stability of the non trivial equilibrium, where we proved that the delay can lead to the instability of interior equilibrium state, and also the existence of Hopf bifurcation. A numerical simulation was carried out to insure the theoretical results.     

Qualitative analysis for a Wolbachia infection model with diffusion

We consider a reaction-diffusion model which describes the spatial Wolbachia spread dynamics for a mixed population of infected and uninfected mosquitoes. By using linearization method, comparison principle and Leray-Schauder degree theory, we investigate the influence of diffusion on the Wolbachia infection dynamics. After identifying the system parameter regions in which diffusion alters the local stability of constant steady-states, we find sufficient conditions under which the system possesses inhomogeneous steady-states. Surprisingly, our mathematical analysis, with the help of numerical simulations, indicates that diffusion is able to lower the threshold value of the infection frequency over which Wolbachia can invade the whole population.     

Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, dynamic behavior of the proposed model system with and without discrete time delay is investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; Furthermore, local stability of the model system with discrete time delay is studied. It reveals that the discrete time delay has a destabilizing effect in the population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Finally, numerical simulations are carried out to show the consistency with theoretical analysis obtained in this paper.     

Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition

In this paper, we study the influence of the nonlocal interspecific competition of the prey population on the dynamics of the diffusive predator-prey model with prey social behavior. Using the linear stability analysis, the conditions for the positive constant steady state at which undergoes Hopf bifurcation, T-H bifurcation (Turing-Hopf bifurcation) are investigated. The Turing patterns occur in the presence of the nonlocal competition and cannot be found in the original system. For determining the dynamical behavior near T-H bifurcation point, the normal form of the T-H bifurcation has been used. Some graphical representations are provided to illustrate the theoretical results.     

Positivity and time behavior of a linear reaction–diffusion system,non-local in space and time

We consider a general linear reaction–diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle and positivity of the solution and investigate its asymptotic behavior. Moreover, we give an explicit expression of the limit of the solution for large times. In order to obtain these results, we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution to a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction–diffusion system. Using this and the facts that the diffusion equation on manifolds satisfies the maximum principle and its solution converges to a easily calculated constant, we can obtain analogous properties for the original system. Copyright © 2008 John Wiley & Sons, Ltd.     

Stability,Steady‐State Bifurcations,and Turing Patterns in a Predator–Prey Model with Herd Behavior and Prey‐taxis

In this paper, we consider a predator–prey model with herd behavior and prey‐taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey‐tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey‐taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots–strip pattern, strip pattern, stable nonconstant steady‐state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.     

Local discontinuous Galerkin approximation of non-Fickian diffusion model in viscoelastic polymers

In this paper, we investigate a fully discrete local discontinuous Galerkin approximation of a non-linear non-Fickian diffusion model in viscoelastic polymers. For the spatial discretization, we adopt local discontinuous Galerkin finite element method and for the time discretization we use backward Euler method. We derive the stability estimate and a priori error estimate for the discrete scheme. Numerical examples are given to verify the theoretical findings.     

Asymptotic stability of reaction-diffusion systems in chemical reactor and combustion theory

Some coupled reaction-diffusion systems arising from chemical diffusion processes and combustion theory are analyzed. This analysis includes the existence and uniqueness of positive time-dependent solutions, upper and lower bounds of the solution, asymptotic behavior and invariant sets, and the stability of steady-state solutions, including an estimate of the stability region. Explicit conditions for the asymptotic behavior and the stability of a steady-state solution are given. These conditions establish some interrelationship among the physical parameters of the diffusion medium, the reaction mechanism, the initial function and the type of boundary condition. Under the same set of physical parameters and reaction function, a comparison between the Neumann type and Dirichlet or third type boundary condition exhibits quite different asymptotic behavior of the solution. For the general nonhomogeneous system, multiple steady-state solutions may exist and only local stability results are obtained. However, for certain models it is possible to obtain global stability of a steady-state solution by either increasing the diffusion coefficients or decreasing the size of the diffusion medium. This fact is demonstrated by a one-dimensional tubular reactor model commonly discussed in the literature.     

Asymptotic behavior of an SEI epidemic model with diffusion

An SEI epidemic model with constant recruitment and infectious force in the latent period is investigated. This model describes the transmission of diseases such as SARS. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions are investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and by the method of upper and lower solutions and its associated monotone iterations. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small.     

Global behavior of a dynamic model with biodegradation of Microcystins

Considering the biodegradation pathway of Microcystins, in this paper, we propose a model described by a system of ordinary differential equations. We firstly investigate the local stability of the positive equilibrium and the existence of Hopf bifurcations. Then, the global stability of the positive equilibrium and the permanence of the model are considered. Finally, numerical simulations are carried out to illustrate the obtained results and we also consider the control strategy by changing the parameters in the model.     

Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

A five‐dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self‐diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans‐critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease‐free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a diffusive predator–prey model with predator saturation and competition

This article discusses a predator–prey system with predator saturation and competition functional response. The local stability, existence of a Hopf bifurcation at the coexistence equilibrium and stability of bifurcating periodic solutions are obtained in the absence of diffusion. Further, we discuss the diffusion-driven instability, Hopf bifurcation for corresponding diffusion system with zero flux boundary condition and Turing instability region regarding the parameters are established. Finally, numerical simulations supporting the theoretical analysis are also included.     

Asymptotic behaviors of a size-structured population model

In this paper we devote ourselves to the study of the asymptotic behavior of a size-structured population dynamics with random diffusion and delayed birth process. Within a semigroup framework, we discuss the local stability and asynchrony respectively for the considered population system under some conditions. We use for our discussion the techniques of operator matrices, Hille-Yosida operators, positivity, spectral analysis as well as Perron-Frobenius theory.     

Herd behavior in a predator–prey model with spatial diffusion: bifurcation analysis and Turing instability

We consider in this paper an ecological model, in a predator–prey interaction with the presence of a herd behavior. For the analysis of the model, the existence of positive solution and also the existence Hopf bifurcation, Turing driven instability, and Turing–Hopf bifurcation point have bee proved. Then by calculating the normal form, on the center of the manifold associated to the Hopf bifurcation points, the stability of the periodic solution has been proved. In the last part of the paper, numerical simulations has been given to illustrate our theoretical analysis.     

The interpersonal diffusion mechanism of unethical behavior in groups: a social network perspective

Previous studies examining the impact of the unethical behavior of a group of colleagues on an individuals unethical behavior have typically employed social learning theory as a theoretical foundation. In this research, we extend these rich yet defective examinations by addressing the largely ignored relationship perspective. Drawing on the social network perspective, we posit that the structure of relationships significantly influences the process of unethical behavior diffusion. Consistent with the theoretically derived hypotheses, our agent-based model simulation results provide general support for our theoretical model: colleagues unethical behaviors positively affect an individuals unethical behaviors, and this influence is positively moderated by group network density, group network closeness centrality and group size. This paper also discusses theoretical contributions, practical values, limitations and directions for future research.     

Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model

In this paper, we have investigated a homogeneous reaction–diffusion bimolecular model with autocatalysis and saturation law subject to Neumann boundary conditions. We mainly consider Hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive equilibrium solution of the system. Our results suggest the existence of spatially non-homogeneous periodic orbits and non-constant positive steady state solutions, which implies the possibility of rich spatiotemporal patterns in this diffusive biomolecular system. Numerical examples are presented to support our theoretical analysis.     

Avian–human influenza epidemic model with diffusion,nonlocal delay and spatial homogeneous environment

In this paper, an avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment is investigated. This model describes the transmission of avian influenza among poultry, humans and environment. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions is investigated. By means of linearization method and spectral analysis the local asymptotical stability is established. The global asymptotical stability for the poultry sub-system is studied by spectral analysis and by using a Lyapunov functional. For the full system, the global stability of the disease-free equilibrium is studied using the comparison Theorem for parabolic equations. Our result shows that the disease-free equilibrium is globally asymptotically stable, whenever the contact rate for the susceptible poultry is small. This suggests that the best policy to prevent the occurrence of an epidemic is not only to exterminate the asymptomatic poultry but also to reduce the contact rate between susceptible humans and the poultry environment. Numerical simulations are presented to illustrate the main results.