Positive Constant Steady State 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.

Global Solution and Stability of a Haptotaxis Mathematical Model for Complex MAP

A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired complex MAP that is necessary to keep the nanoparticulate immobilized at any given time. In this paper, the global solution and stability of a parabolic–ordinary-parabolic haptotaxis system to complex MAP are studied. First, the existence of a local classical solution to system () has been observed using fixed point argument and parabolic Schauder estimates. Furthermore, some a priori estimates that can raise the regularity estimate of the solution for the relatively complicated first equation of system () from Lρ to L2ρ (ρ≥1) are given; then, the local classic solution can thus extend to the global classic solution when the space dimension N≤3. Lastly, by using various analytical methods, a threshold value ξ00(ξ00<0) is found, such that positive constant steady state (u*,v*,w*) becomes unstable when ξ<ξ00. Our results show that the haptotaxis plays a crucial role in determining the stability to the model (), that is, it can have a destabilizing effect.

Global dynamics of a time-delayed nonlocal reaction-diffusion model of within-host viral infections.

In this paper, we study a time-delayed nonlocal reaction-diffusion model of within-host viral infections. We introduce the basic reproduction number and show that the infection-free steady state is globally asymptotically stable when , while the disease is uniformly persistent when . In the case where all coefficients and reaction terms are spatially homogeneous, we obtain an explicit formula of and the global attractivity of the positive constant steady state. Numerically, we illustrate the analytical results, conduct sensitivity analysis, and investigate the impact of drugs on curtailing the spread of the viruses.

Global Hopf Bifurcation Of a Delayed Diffusive Gause-Type Predator-Prey System with the Fear Effect and Holling Type III Functional Response

In this paper, a delayed diffusive predator-prey system with the fear effect and Holling type III functional response is considered, and Neumann boundary condition is imposed on this system. First, we explore the stability of the unique positive constant steady state and the existence of local Hopf bifurcation. Then the global attraction domain G* of system (1.4) is obtained by the comparison principle and the iterative method. Through constructing the Lyapunov function, we investigate uniform boundedness of periodic solutions' periods. Finally, we prove the global continuation of periodic solutions by the global Hopf bifurcation theorem of Wu. Moreover, some numerical simulations that support the analysis results are given.

Spatiotemporal Dynamics of a Diffusive Immunosuppressive Infection Model with Nonlocal Competition and Crowley–Martin Functional Response

In this paper, we introduce the Crowley–Martin functional response and nonlocal competition into a reaction–diffusion immunosuppressive infection model. First, we analyze the existence and stability of the positive constant steady states of the systems with nonlocal competition and local competition, respectively. Second, we deduce the conditions for the occurrence of Turing, Hopf, and Turing–Hopf bifurcations of the system with nonlocal competition, as well as the conditions for the occurrence of Hopf bifurcations of the system with local competition. Furthermore, we employ the multiple time scales method to derive the normal forms of the Hopf bifurcations reduced on the center manifold for both systems. Finally, we conduct numerical simulations for both systems under the same parameter settings, compare the impact of nonlocal competition, and find that the nonlocal term can induce spatially inhomogeneous stable periodic solutions. We also provide corresponding biological explanations for the simulation results.

On the Schnakenberg model with crucial reversible reactions

This paper is devoted to the Schnakenberg model with crucial reversible reactions, under Neumann boundary conditions. Existence and uniqueness of the strong solution are obtained on basis of semigroup theory. It is found that the proposed system admits four possible positive constant steady states, and explicit conditions of the stability, Turing instability, steady‐state bifurcation, and Hopf bifurcation are determined. Besides, numerical simulations are given to show the theoretical results and depict the spatiotemporal patterns.

Stationary Patterns of a Cross-Diffusion Prey-Predator Model with Holling Type II Functional Response

In this paper, we consider positive steady-state solutions of a cross-diffusions prey-predator model with Holling type II functional response. We investigate sufficient conditions for the existence and the nonexistence of nonconstant positive steady state solutions. It is observed that nonconstant positive steady states do not exist with small cross-diffusion coefficients, and the constant positive steady state is global asymptotically stable without cross-diffusion. Furthermore, we show that if natural diffusion coefficient or cross-diffusion coefficient of the predator is large enough and other diffusion coefficients are fixed, then under some conditions, at least one nonconstant positive steady state exists.

Cross diffusion induced spatially inhomogeneous Hopf bifurcation for a three species Lotka–Volterra food web model with cycle

In this article, we consider a three species Lotka–Volterra food web reaction–diffusion model with cycle, in which the intensity of the rate of preying on species 1 by species 2, the rate of preying on species 2 by species 3 and the rate of preying on species 3 by species 1 are in general asymmetrical. By suitably choosing cross diffusion coefficients as the bifurcation parameter, a spatially inhomogeneous Hopf bifurcation at a positive constant steady state is proved to occur for a sequence of critical values of the bifurcation parameter. In addition, we demonstrate that these spatially inhomogeneous bifurcating periodic solutions are stable when diffusion coefficients are located in suitable ranges. Our results show that cross diffusion plays a key role in the formation of spatially inhomogeneous periodic oscillatory patterns.

Turing–Hopf bifurcation of a diffusive Holling–Tanner model with nonlocal effect and digestion time delay

In this paper, we study the codimension‐two Turing–Hopf bifurcation of a diffusive Holling–Tanner model with nonlocal effect and digestion time delay. The stability, Turing bifurcation, Hopf bifurcation, and Turing–Hopf bifurcation of this model are first investigated. Then, we derive the algorithm for calculating the normal form of Turing–Hopf bifurcation for this model. At last, we carry out some numerical simulations to verify our theoretical analysis results. The stable positive constant steady state and the stable spatially inhomogeneous periodic solutions are found. Furthermore, the evolution process from unstable spatially inhomogeneous steady states to stable positive constant steady state, the evolution process from unstable spatially inhomogeneous steady states to stable spatially inhomogeneous periodic solutions, and the evolution process from one unstable spatially inhomogeneous periodic solution to another stable spatially inhomogeneous periodic solution are also found.

Modeling the fear effect in the predator-prey dynamics with an age structure in the predators.

We incorporate the fear effect and the maturation period of predators into a diffusive predator-prey model. Local and global asymptotic stability for constant steady states as well as uniform persistence of the solution are obtained. Under some conditions, we also exclude the existence of spatially nonhomogeneous steady states and the steady state bifurcation bifurcating from the positive constant steady state. Hopf bifurcation analysis is carried out by using the maturation period of predators as a bifurcation parameter, and we show that global Hopf branches are bounded. Finally, we conduct numerical simulations to explore interesting spatial-temporal patterns.

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>

Turing–Hopf Bifurcation Analysis of the Sel’kov–Schnakenberg System

In this paper, we investigate the spatiotemporal dynamics of the Sel’kov–Schnakenberg system. The stability of the positive constant steady state is studied by the linear stability theory. Hopf bifurcation and Turing–Hopf bifurcation are generated by varying two parameters in the model. The normal form near the Turing–Hopf singularity is calculated to explore the complex dynamics of the system. Finally, numerical simulations are carried out to verify the theoretical results. Our results show that the Sel’kov–Schnakenberg system exhibits complex dynamics near the Turing–Hopf singularity, including the existence of inhomogeneous steady states, homogeneous periodic solutions and inhomogeneous periodic solutions.

Bifurcations in a diffusive resource-consumer model with distributed memory

Spatial memory is significant in modeling animal movement. For a diffusive consumer-resource model, a memory-based diffusion of consumer can result in richer and more realistic dynamics. In fact, memory-based diffusion is related to the resource distributions in past times because the memory decays over time. We originally propose a consumer-resource model with distributed memory, and then investigate the influence of the weak memory kernel on the stability of the positive constant steady state. When the memory-based diffusion coefficient is negative, the mean delay does not affect the stability of the positive constant steady state; however, when the memory-based diffusion coefficient is positive, the mean delay can lead to the spatially inhomogeneous periodic oscillation patterns. The direction and stability of Turing bifurcation induced by the memory-based diffusion coefficient are calculated by using the methods of Crandall and Rabinowitz, and the direction and stability of Hopf bifurcation induced by the mean delay are determined by the normal form theory.

Spatiotemporal dynamics and Turing patterns in an eco-epidemiological model with cannibalism

In this work, we have studied the spatiotemporal dynamics and pattern formation in an eco-epidemiological model with cannibalism, which are not explored in the existing literature. Cannibalism is a process of killing and consuming the full or a part of an individual of the same species (conspecifics). A cannibalistic reaction–diffusion system is proposed and analyzed. In order to observe the change in system stability, we have investigated local and global asymptotic stability for the constant positive steady state. The existence of Hopf bifurcation and Turing instability are proved to understand the change in the system dynamics and to observe the species movement. Numerical simulation is performed to observe spatiotemporal dynamics and formation of Turing patterns. The existence of Hopf bifurcation is verified numerically in the spatiotemporal system It is observed that cannibalism stabilizes the temporal system, which is shown by bifurcation diagrams, phase portraits and time-series plots. Moreover, its increased cost promotes the occurrence of Turing patterns in the spatiotemporal system (i.e., checked numerically that for an increased cost of cannibalism, Turing instability holds). The movement of prey population also plays an important role in forming Turing patterns, as a significant change in movement results in various types of interesting two-dimensional Turing patterns such as hot-spot, stripe and labyrinth patterns. We see that both predator species show the same species distribution arrangements under cannibalistic pressure. Hence cannibalism can be seen as an essential factor for the significant change in spatiotemporal dynamics and species distribution arrangement.

Spatially inhomogeneous periodic patterns induced by distributed memory in the memory-based single population model

In this paper, we investigate the influence of the distributed delay on the stability and spatiotemporal dynamics in the memory-based diffusion single population model. Under the assumption that the positive equilibrium of the corresponding ordinary differential equation is locally asymptotically stable, it has been shown that the weak kernel does not affect the stability of this positive constant steady state, but for the strong kernel, the combined effect of the diffusion rate of the memory diffusion and the distributed delay can affect the stability and lead to the rich dynamics. For the strong kernel, the theoretical conditions for the stability, Hopf bifurcation and double Hopf bifurcation are explicitly determined.

Threshold dynamics of a nonlocal and time-delayed West Nile virus model with seasonality

In this paper, we study a nonlocal periodic reaction–diffusion system modeling the West Nile virus transmission between mosquitoes and birds. For the subsystem of mosquito growth, we establish a global stability result on the extinction and persistence in terms of mosquito reproduction number RM, which is defined as the cone spectral radius of a monotone homogeneous map. For the full model, we introduce the basic reproduction number R0 and show that the disease transmission dynamics is determined by the sign of R0−1 under the assumption that RM>1. Moreover, we obtain the global attractivity of the positive constant steady state in the case where all the coefficients are constants. We also conduct numerical simulations to reveal the effects of diffusion, heterogeneity and periodic delay on R0.

Bifurcations in the diffusive Bazykin model

Dangerous tipping points and catastrophic transitions in ecosystems have recently been popular for detecting early warning signals in ecology. B-tipping is induced by bifurcation such as spatial pattern formation resulting from Turing instability. As one of the most important models in predator-prey interactions, we extend the Bazykin model to incorporate diffusive movement under homogeneous Neumann boundary conditions. For the local model, we provide some preliminary analysis on stability and Hopf bifurcation. For the reaction-diffusion model, we first improve some sufficient conditions for the local and global stability of a semi-trivial constant steady state or a unique positive constant steady state in Du and Lou (2001) [11]. Next we obtain the sufficient and necessary conditions for Turing instability, show the existence of Turing bifurcation, Hopf bifurcation, Turing-Turing bifurcation, Turing-Hopf bifurcation and Turing-Turing-Hopf bifurcation, and the nonexistence of triple-Turing bifurcation. Our results reveal that the model can exhibit complex spatial, temporal and spatiotemporal patterns, including complex regime shifts and critical transitions at bifurcation points, transient states (spatially inhomogeneous periodic solutions), tristability (a pair of non-constant steady states and a spatially homogeneous periodic solution), heteroclinic orbits (connecting a spatially inhomogeneous periodic solution to a non-constant steady state or a spatially homogeneous periodic solution, connecting a spatially homogeneous periodic solution to non-constant steady states and vice versa). Finally, numerical simulations illustrate complex dynamics and verify our theoretical results.

Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain

<p style='text-indent:20px;'>In this article, Turing instability and the formations of spatial patterns for a general two-component reaction-diffusion system defined on 2D bounded domain, are investigated. By analyzing characteristic equation at positive constant steady states and further selecting diffusion rate <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula> and diffusion ratio <inline-formula><tex-math id="M2">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> as bifurcation parameters, sufficient and necessary conditions for the occurrence of Turing instability are established, which is called the first Turing bifurcation curve. Furthermore, parameter regions in which single-mode Turing patterns arise and multiple-mode (or superposition) Turing patterns coexist when bifurcations parameters are chosen, are described. Especially, the boundary of parameter region for the emergence of single-mode Turing patterns, consists of the first and the second Turing bifurcation curves which are given in explicit formulas. Finally, by taking diffusive Schnakenberg system as an example, parameter regions for the emergence of various kinds of spatially inhomogeneous patterns with different spatial frequencies and superposition Turing patterns, are estimated theoretically and shown numerically.</p>

Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

<p style='text-indent:20px;'>We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.</p>

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>