scholarly journals Stability and Stabilization of Ecosystem for Epidemic Virus Transmission Under Neumann Boundary Value Via Impulse Control

2021 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Impulse Control ◽  
Virus Transmission ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Control Technique ◽  
Prevention Measures ◽  
Unique Existence ◽  
Globally Exponential Stability ◽  
The Face ◽  
The Stability

In this paper, by using the variational method, a sufficient condition for the unique existence of the stationary solution of the reaction-diffusion ecosystem is obtained, which directly leads to the global asymptotic stability of the unique equilibrium point. Besides, employing impulse control technique derives the globally exponential stability criterion of delayed feedback ecosystem.And numerical examples illuminate the effectiveness of impulse control, which has a certain enlightening effect on the actual epidemic prevention work . That is, in the face of the epidemic situation, taking a certain frequency of positive and effective epidemic prevention measures is conducive to the stability and control of the epidemic situation. particularly, the newly-obtained theorems quantifies this feasible step.

scholarly journals Stability and Stabilization of Ecosystem for Epidemic Virus Transmission Under Neumann Boundary Value Via Impulse Control

2021 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Stability Criterion ◽  
Impulse Control ◽  
Boundary Value ◽  
Virus Transmission ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Prevention Measures ◽  
Unique Existence ◽  
Globally Exponential Stability ◽  
The Stability

In this paper, by using the variational method, a sufficient condition for the unique existence of the stationary solution of the reaction-diffusion ecosystem is obtained, which directly leads to the global asymptotic stability of the unique equilibrium point. Moreover, delayed feedback ecosystem with reaction-diffusion item is considered, and utilizing impulse control results in the globally exponential stability criterion of the delayed ecosystem. It is worth mentioning that the Neumann zero-boundary value that the infected and the susceptible people or animals should be controlled in the epidemic prevention area and not allowed to cross the border, which is a good simulation of the actual situation of epidemic prevention. And numerical examples illuminate the effectiveness of impulse control, which has a certain enlightening effect on the actual epidemic prevention work . That is, in the face of the epidemic situation, taking a certain frequency of positive and effective epidemic prevention measures is conducive to the stability and control of the epidemic situation. Particularly, the newly-obtained theorems quantifies this feasible step. Besides, utilizing Laplacian semigroup derives the $p$th moment stability criterion for the impulsive ecosystem.

scholarly journals Bifurcation Analysis in a Diffusive Mussel-Algae Model with Delay

2019 ◽  
Vol 29 (11) ◽  
pp. 1950144 ◽  
Author(s):  
Zuolin Shen ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Center Manifold Reduction ◽  
Existence Of Positive Solutions ◽  
Boundary Equilibrium ◽  
Functional Differential ◽  
The Stability

In this paper, we consider the dynamics of a delayed reaction–diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.

scholarly journals Stability and Bifurcation Analysis for a Class of Generalized Reaction-Diffusion Neural Networks with Time Delay

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Tianshi Lv ◽  
Qintao Gan ◽  
Qikai Zhu
Keyword(s):  
Neural Networks ◽  
Local Field ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Static Neural Networks ◽  
Stability And Bifurcation ◽  
Uniform Steady State ◽  
The Stability

Considering the fact that results for static neural networks are much more scare than results for local field neural networks and our purpose letting the problem researched be more general in many aspects, in this paper, a generalized neural networks model which includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks is built and the stability and bifurcation problems for it are investigated under Neumann boundary conditions. First, by discussing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed and the existence of Hopf bifurcations is shown. By using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae which determine the direction and stability of bifurcating periodic solutions are acquired. Finally, numerical simulations show the results.

scholarly journals Stability and Hopf Bifurcation in a Three-Component Planktonic Model with Spatial Diffusion and Time Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Kejun Zhuang ◽  
Gao Jia ◽  
Dezhi Liu
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Solution ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Time Behavior ◽  
Delayed Reaction ◽  
Constant Equilibrium ◽  
Homogeneous Neumann Boundary Condition ◽  
Long Time ◽  
The Stability

Due to the different roles that nontoxic phytoplankton and toxin-producing phytoplankton play in the whole aquatic system, a delayed reaction-diffusion planktonic model under homogeneous Neumann boundary condition is investigated theoretically and numerically. This model describes the interactions between the zooplankton and two kinds of phytoplanktons. The long-time behavior of the model and existence of positive constant equilibrium solution are first discussed. Then, the stability of constant equilibrium solution and occurrence of Hopf bifurcation are detailed and analyzed by using the bifurcation theory. Moreover, the formulas for determining the bifurcation direction and stability of spatially bifurcating solutions are derived. Finally, some numerical simulations are performed to verify the appearance of the spatially homogeneous and nonhomogeneous periodic solutions.

Stability and Hopf Bifurcation of a Reaction–Diffusion Neutral Neuron System with Time Delay

2017 ◽  
Vol 27 (14) ◽  
pp. 1750214 ◽  
Author(s):  
Tao Dong ◽  
Linmao Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neuron System ◽  
Center Manifold Theorem ◽  
Feedback Strength ◽  
The Stability ◽  
System With Time Delay ◽  
Zero Equilibrium

In this paper, a type of reaction–diffusion neutral neuron system with time delay under homogeneous Neumann boundary conditions is considered. By constructing a basis of phase space based on the eigenvectors of the corresponding Laplace operator, the characteristic equation of this system is obtained. Then, by selecting time delay and self-feedback strength as the bifurcating parameters respectively, the dynamic behaviors including local stability and Hopf bifurcation near the zero equilibrium point are investigated when the time delay and self-feedback strength vary. Furthermore, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form and the center manifold theorem for the corresponding partial differential equation. Finally, two simulation examples are given to verify the theory.

scholarly journals The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

2018 ◽  
Vol 23 (5) ◽  
pp. 749-770 ◽  
Author(s):  
Xin Wei ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Arbitrary Order ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Positive Equilibrium ◽  
Distribution Of Eigenvalues ◽  
The Stability

This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions.

BIFURCATION AND SPATIOTEMPORAL PATTERNS IN A HOMOGENEOUS DIFFUSION-COMPETITION SYSTEM WITH DELAYS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250049 ◽  
Author(s):  
JIA-FANG ZHANG ◽  
WAN-TONG LI ◽  
XIANG-PING YAN
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Hopf Bifurcations ◽  
Asymptotically Stable ◽  
Interspecies Competition ◽  
Globally Asymptotically Stable ◽  
Two Delays ◽  
Delayed System ◽  
The Stability

A competitive Lotka–Volterra reaction-diffusion system with two delays subject to Neumann boundary conditions is considered. It is well known that the positive constant steady state of the system is globally asymptotically stable if the interspecies competition is weaker than the intraspecies one and is unstable if the interspecies competition dominates over the intraspecies one. If the latter holds, then we show that Hopf bifurcation can occur as the parameters (delays) in the system cross some critical values. In particular, we prove that these Hopf bifurcations are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as Hopf bifurcation of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system generates the spatially nonhomogeneous Hopf bifurcation. Furthermore, we derive conditions for determining the direction of spatially nonhomogeneous Hopf bifurcations and the stability of bifurcating periodic solutions. These results indicate that the diffusion plays an important role for deriving the complex spatiotemporal dynamics.

Spatiotemporal Patterns in a Delayed Reaction–Diffusion Mussel–Algae Model

2019 ◽  
Vol 29 (12) ◽  
pp. 1950164 ◽  
Author(s):  
Zuolin Shen ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Dimensionless Parameter ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Patterns ◽  
Delayed Reaction ◽  
Distribution Of Eigenvalues ◽  
Positivity Of Solutions ◽  
The Stability

We study the spatiotemporal patterns of a delayed reaction–diffusion mussel–algae system subject to Neumann boundary conditions. This paper is a continuation of our previous studies on the mussel–algae model. We prove the global existence and positivity of solutions. By analyzing the distribution of eigenvalues, we obtain the stability conditions for the positive constant steady state, the existence of Hopf bifurcation and the Turing instability. We show the dynamic classification near the Turing–Hopf singularity in the dimensionless parameter space and observe a transiently spatially nonhomogeneous periodic solution in simulations. Both theoretical and numerical results reveal that the Turing–Hopf bifurcation can enrich the diversity of the spatial distribution of populations.

Em transe-to

2020 ◽  
Vol 2 (1) ◽  
pp. 32-55
Author(s):  
Natalia Christofoletti Barrenha
Keyword(s):  
The Face ◽  
The Stability ◽  
The City

This text seeks to explore the Argentine films Castro (Alejo Moguillansky, 2009) and El asaltante (Pablo Fendrik, 2007) from within the displacement of their characters through the city. This transit configures the organising element of the plots, determining the direction and rhythm of events. The escape motto will structure the film analyses, which are also twinned by the sensory apprehension that comes from the spaces they travel through. The notion of escape, as explored by Esteban Dipaola in Argentine cinema of the 1990s, continues to throb in mid-to-late 2000s production, and in these films represents the means by which the protagonists deploy critical attitudes—sometimes radical and explosive, sometimes silent—in the face of fixed notions, suggesting some scepticism about the “stability” and “order” that they (dis)encounter in normality. RESUMEN Este texto busca explorar los largometrajes argentinos Castro (Alejo Moguillansky, 2009) y El asaltante (Pablo Fendrik, 2007) a partir del desplazamiento de sus personajes por la ciudad. El transitar se configura como elemento organizador de las tramas, determinando la dirección y el ritmo de los acontecimientos. El tema de la fuga irá estructurando los análisis de las películas, las cuales también están relacionadas por la aprehensión sensorial que hacen de los espacios que recorren. La noción de fuga, tal y como fue explorada por Esteban Dipaola en el cine argentino de los años 90, continúa vigente en la producción de mediados/fines de la primera década del siglo XXI, y en estas películas es el recurso por medio del cual los protagonistas despliegan actitudes críticas – a veces radicales y explosivas, y a veces silenciosas – frente a nociones convencionales, lo cual hace pensar que existe un cierto escepticismo con relación a la “estabilidad” y al “orden” que ellos (des)encuentran en la normalidad. RESUMO Este texto busca explorar os longas-metragens argentinos Castro (Alejo Moguillansky, 2009) e El asaltante (Pablo Fendrik, 2007) a partir do deslocamento de seus personagens pela cidade. O transitar configura-se como elemento organizador das tramas, determinando a direção e o ritmo dos acontecimentos. O mote da fuga estruturará as análises dos filmes, os quais também se irmanam pela apreensão sensorial que fazem dos espaços que percorrem. A noção de fuga, conforme explorada por Esteban Dipaola no cinema argentino da década de 1990, continua a pulsar na produção de meados/fins dos anos 2000, e é, nestes filmes, o recurso através do qual os protagonistas desdobram atitudes críticas – às vezes radicais e explosivas, às vezes silenciosas – diante de noções fixas, sugerindo certo ceticismo em relação à “estabilidade” e à “ordem” que eles (des)encontram na normalidade.

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