Positive Constant Steady-state Solution Research Articles

Hopf bifurcation in a delay-diffusion housefly equation with Dirichlet boundary conditions

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

Dynamics analysis of a diffusional immunosuppressive infection model with Beddington-DeAngelis functional response

<abstract><p>This paper introduces diffusion into an immunosuppressive infection model with virus stimulation delay and Beddington-DeAngelis functional response. First, we study the stability of positive constant steady state solution and show that the Hopf bifurcation will exist under certain conditions. Second, we derive the normal form of the Hopf bifurcation for the model reduced on the center manifold by using the multiple time scales (MTS) method. Moreover, the direction and stability of the bifurcating periodic solution are investigated. Finally, we present numerical simulations to verify the results of theoretical analysis and provide corresponding biological explanations.</p></abstract>

Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme

<p style='text-indent:20px;'>A depletion-type reaction-diffusion Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme and the homogeneous Neumann boundary conditions is introduced and investigated in this paper. Firstly, the boundedness of positive solution of the parabolic system is given, and the constant steady state solutions of the model are exhibited by the Shengjin formulas. Through rigorous theoretical analysis, the stability of the corresponding positive constant steady state solution is explored. Next, a priori estimates, the properties of the nonconstant steady states, non-existence and existence of the nonconstant steady state solution for the corresponding elliptic system are investigated by some estimates and the Leray-Schauder degree theory, respectively. Then, some existence conditions are established and some properties of the Hopf bifurcation and the steady state bifurcation are presented, respectively. It is showed that the temporal and spatial bifurcation structures will appear in the reaction-diffusion model. Theoretical results are confirmed and complemented by numerical simulations.</p>

Stability and bifurcation of a delayed diffusive predator-prey system with food-limited and nonlinear harvesting.

Based on ecological significance, a delayed diffusive predator-prey system with food-limited and nonlinear harvesting subject to the Neumann boundary conditions is investigated in this paper. Firstly, the sufficient conditions of the stability of nonnegative constant steady state solutions of system are derived. The existence of Hopf bifurcation is obtained by analyzing the associated characteristic equation and the conditions of Turing instability are derived when the system has no delay. Furthermore, the occurrence conditions the Hopf bifurcation are discussed by regarding delay expressing the gestation time of the predator as the bifurcation parameter. Secondly, by using upper-lower solution method, the global asymptotical stability of a unique positive constant steady state solution of system is investigated. Moreover, we also give the detailed formulas to determine the direction, stability of Hopf bifurcation by applying the normal form theory and center manifold reduction. Finally, numerical simulations are carried out to demonstrate our theoretical results.

Dynamics of a Leslie-Gower predator-prey system with cross-diffusion

A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating timeperiodic solutions are investigated and a normal form of Bogdanov–Takens bifurcation is determined as well.

Qualitative analysis for a diffusive predator–prey model with hunting cooperative

This paper investigates a predator–prey model that adds a cooperation term to the attack rate of the predator population. For the non-spatial model, the existence and stability of non-negative equilibrium points, and bifurcations are studied by choosing cooperation coefficient as control parameter. Our analytical results show that hunting cooperation can be beneficial to the predator population. However, as the cooperation coefficient increases, hunting cooperation can also destabilize the model and promote a sudden collapse of the predator population. For the spatial model, the stability of positive constant steady state solution, Hopf bifurcation and Turing instability are discussed. It is obtained that when the predation diffusion is not smaller than the prey diffusion, the spatial model can reserve the stable stability of the positive constant steady state solution. It is noted that the model without hunting cooperative does not generate Turing instability, while the model with hunting cooperative may generate Turing instability.

Bifurcation and Stability Analysis of a Diffusive Predator-Prey Model with Ratio-Dependent Type III Functional Response

This paper is concerned with a diffusive Leslie-Gower predator-prey system with ratio dependent Holling type III functional response subject to Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, local stability, existence of a Hopf bifurcation at the coexistence of the equilibrium and stability of bifurcating periodic solutions of the system in the absence of diffusion are studied. Furthermore, Turing instability and Hopf bifurcation analysis for the system with diffusion are studied. Finally, numerical simulations are provided in order to verify our theoretical results.

Spatiotemporal dynamics and spatial pattern in a diffusive intraguild predation model with delay effect

Based on biological meaning, a kind of diffusive intraguild predation (IGP: resource, IG prey and IG predator) model with delay effect is investigated in this paper. The model has Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type II functional response between IG prey and IG predator. We first give sufficient conditions on the stability of possible nonnegative constant steady-state solutions for the proposed model, which give us a complete picture of the global dynamics. Then we investigate Hopf bifurcation near the unique positive constant steady-state solution by taking delay as bifurcation parameter and derive the Hopf bifurcation threshold. It is shown that the delay can induce three types of bistability (node-node bistability, node-cycle bistability and cycle-cycle bistability), periodic oscillations and irregular oscillations triggering spatiotemporal chaos in the diffusive IGP model. Numerical simulations are performed to illustrate our theoretical results and suggest that delay can even trigger the emergence of self-organised spatiotemporal patterns, which evolve from spiral patterns to irregular spatial patterns via spatiotemporal Hopf bifurcation. In addition, the impact of diffusion on the model’s dynamics under certain time delay are also explored.

Asymptotic dynamics in a two-species chemotaxis model with non-local terms

In this study, we consider an extended attraction two species chemotaxis system of parabolic-parabolic-elliptic type with nonlocal terms under homogeneous Neuman boundary conditions in a bounded domain $Ω \subset \mathbb{R}^n(n≥1)$ with smooth boundary. We first prove the global existence of non-negative classical solutions for various explicit parameter regions. Next, under some further explicit conditions on the coefficients and on the chemotaxis sensitivities, we show that the system has a unique positive constant steady state solution which is globally asymptotically stable. Finally, we also find some explicit conditions on the coefficient and on the chemotaxis sensitivities for which the phenomenon of competitive exclusion occurs in the sense that as time goes to infinity, one of the species dies out and the other reaches its carrying capacity. The method of eventual comparison is used to study the asymptotic behavior.

Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics

In this paper, a reaction-diffusion system known as an autocatalytic reaction model is considered. The model is characterized by a system of two differential equations which describe a type of complex biochemical reaction. Firstly, some basic characterizations of steady-state solutions of the model are presented. And then, the stability of positive constant steady-state solution and the non-existence, existence of non-constant positive steady-state solutions are discussed. Meanwhile, the bifurcation solution which emanates from positive constant steady-state is investigated, and the global analysis to the system is given in one dimensional case. Finally, a few numerical examples are provided to illustrate some corresponding analytic results.

Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack

This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the stability and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results.

Dynamical analysis of a logistic equation with spatio-temporal delay

This paper is concerned with a logistic equation with spatio-temporal delay. The local asymptotic behavior of positive constant steady-state solution of the equation is considered. In particular, by using the iterative technique, sufficient conditions are established for the existence of traveling wave front solution connecting the zero and the positive constant steady-state solution. Finally, numerical simulations supporting the theoretical analysis are also included.

Deciphering Dynamics of Epidemic Spread: The Case of Influenza Virus

In this paper, we have proposed and analyzed a simple model of Influenza spread with an asymptotic transmission rate. Existence and uniqueness of solutions are established and shown to be uniformly bounded for all non-negative initial values. We have also found a sufficient condition which ensures the persistence of the model system. This implies that both susceptible and infected will always coexist at any location of the inhabited domain. This coexistence is independent of values of the diffusivity constants for two subpopulations. The global stability of the endemic equilibrium is established by constructing a Lyapunov function. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation, conditions for Hopf and Turing bifurcations are obtained. We have also studied the criteria for diffusion-driven instability caused by local random movements of both susceptible and infective subpopulations. Turing patterns selected by the reaction–diffusion system under zero flux boundary conditions have been explored.Numerical simulations show that contact rate, β which is related to the reproduction number [Formula: see text], plays an important role in spatial pattern formation. It was found that diffusion has appreciable influence on spatial spread of epidemics. The wave of chaos appears to be a dominant mode of disease dispersal. This suggests a bidirectional spread for influenza epidemics. The epidemic propagates in the form of nonchaotic and chaotic waves as observed in H1N1 incidence data of positive tests in 2009 in the United States. We have conducted numerical simulations to confirm the analytic work and observed interesting behaviors. This suggests that influenza has a complex dynamics of spatial spread which evolves with time.

Bifurcation analysis of the Gierer–Meinhardt system with a saturation in the activator production

The reaction–diffusion Gierer–Meinhardt system with a saturation in the activator production is considered. Stability of the unique positive constant steady state solution is analysed, and associated Hopf bifurcations and steady state bifurcations are obtained. A global bifurcation diagram of non-trivial periodic orbits and steady state solutions with respect to key system parameters is obtained, which improves the understanding of dynamics of Gierer–Meinhardt system with a saturation in different parameter regimes.

Spatiotemporal Complexity of a Leslie-Gower Predator-Prey Model with the Weak Allee Effect

We investigate a diffusive Leslie-Gower predator-prey model with the additive Allee effect on prey subject to the zero-flux boundary conditions. Some results of solutions to this model and its corresponding steady-state problem are shown. More precisely, we give the stability of the positive constant steady-state solution, the refineda prioriestimates of positive solution, and the nonexistence and existence of the positive nonconstant solutions. We carry out the analytical study for two-dimensional system in detail and find out the certain conditions for Turing instability. Furthermore, we perform numerical simulations and show that the model exhibits a transition from stripe-spot mixtures growth to isolated spots and also to stripes. These results show that the impact of the Allee effect essentially increases the model spatiotemporal complexity.

Global Stability of a Reaction-diffusion System of a Competitor-Competitor-Mutualist Model

In this paper, we study a reaction-diffusion system of a competitorcompetitor-mutualist model with Neumann boundary condition. Using iteration method, we investigate the global asymptotic stability of the unique positive constant steady-state solution under some assumptions. We also give some sufficient conditions under which there are no nonconstant positive steady-state solution exist.

Hopf Bifurcations in a Predator-Prey Diffusion System with Beddington-DeAngelis Response

This paper is concerned with a two-species predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and subject to homogeneous Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, the asymptotic stability of the positive constant steady-state solution and the existence of local Hopf bifurcations are investigated. Also, it is shown that the appearance of the diffusion and homogeneous Neumann boundary conditions can lead to the appearance of codimension two Bagdanov-Takens bifurcation. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Finally, numerical simulations supporting the theoretical analysis are also included.