scholarly journals Stabilization for a Periodic Predator-Prey System

2007 ◽  
Vol 2007 ◽  
pp. 1-17
Author(s):  
Sebastian Aniţa ◽  
Carmen Oana Tarniceriu
Keyword(s):  
Internal Control ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Predator Prey ◽  
Periodic Environment ◽  
Prey System

A reaction-diffusion system modelling a predator-prey system in a periodic environment is considered. We are concerned in stabilization to zero of one of the components of the solution, via an internal control acting on a small subdomain, and in the preservation of the nonnegativity of both components.

On a reaction–diffusion system modelling infectious diseases without lifetime immunity

2021 ◽  
pp. 1-25
Author(s):  
HONG-MING YIN
Keyword(s):  
Diffusion Coefficients ◽  
A Priori ◽  
Main Tool ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Sobolev Embedding ◽  
System Modelling ◽  
Strongly Coupled ◽  
Elliptic And Parabolic Equations

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

Nonlinear stability and numerical simulations for a reaction–diffusion system modelling Allee effect on predators

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Florinda Capone ◽  
Maria Francesca Carfora ◽  
Roberta De Luca ◽  
Isabella Torcicollo
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Stability ◽  
Allee Effect ◽  
Complex Dynamics ◽  
Stability Result ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Oscillatory Pattern

Abstract A reaction–diffusion system governing the prey–predator interaction with Allee effect on the predators, already introduced by the authors in a previous work is reconsidered with the aim of showing destabilization mechanisms of the biologically meaning equilibrium and detecting some aspects for the eventual oscillatory pattern formation. Extensive numerical simulations, depicting such complex dynamics, are shown. In order to complete the stability analysis of the coexistence equilibrium, a nonlinear stability result is shown.

Stabilizing A Reaction-Diffusion System Via Feedback Control

2013 ◽  
Vol 59 (1) ◽  
pp. 191-200
Author(s):  
Smaranda C. Dodea
Keyword(s):  
Feedback Control ◽  
Principal Eigenvalue ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Necessary Condition ◽  
System Modelling ◽  
Stabilizing Control ◽  
Internal Stabilizability ◽  
Predator System

Abstract A two-component reaction-diffusion system modelling a prey-predator system is considered. A necessary condition and a sufficient condition for the internal stabilizability to zero of one the two components of the solution while preserving the nonnegativity of both components have been established by Aniţa. In case of stabilizability, a feedback stabilizing control of harvesting type has been indicated. The rate of stabilization corresponding to the indicated feedback control depends on the principal eigenvalue of a certain elliptic operator. A large principal eigenvalue leads to a fast stabilization. The first goal of this paper is to approximate this principal eigenvalue. The second goal is to derive a conceptual iterative algorithm to improve at each iteration the position of the support of the stabilizing control in order to get a faster stabilization.

Mathematical and computational studies of fractional reaction-diffusion system modelling predator-prey interactions

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Kolade M. Owolabi ◽  
Edson Pindza
Keyword(s):  
Stability Analysis ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Computational Studies ◽  
Mathematical Basis ◽  
Transform Method ◽  
Reaction Diffusion Systems ◽  
Fractional Reaction

Abstract This paper provides the essential mathematical basis for computational studies of space fractional reaction-diffusion systems, from biological and numerical analysis perspectives. We adopt linear stability analysis to derive conditions on the choice of parameters that lead to biologically meaningful equilibria. The stability analysis has a lot of implications for understanding the various spatiotemporal and chaotic behaviors of the species in the spatial domain. For the solution of the full reaction-diffusion system modelled by the fractional partial differential equations, we introduced the Fourier transform method to discretize in space and advance the resulting system of ordinary differential equation in time with the fourth-order exponential time differencing scheme. Numerical results.

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