Stabilization strategies for some reaction–diffusion systems (original) (raw)
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Internal stabilizability for a reaction–diffusion problem modeling a predator–prey system
Nonlinear Analysis: Theory, Methods & Applications, 2005
In this work we consider a 2× 2 system of semilinear partial differential equations of parabolic-type describing interactions between a prey population and a predator population, featuring a Holling-type II functional response to predation. We address the question of stabilizing the predator population to zero, upon using a suitable internal control supported on a small subdomain of the whole spatial domain , and acting on predators. We give necessary and sufficient conditions for this stabilizability result to hold.
Stabilization of a Predator-Prey System with Nonlocal Terms
Mathematical Modelling of Natural Phenomena, 2015
We investigate the zero-stabilizability for the prey population in a predator-prey system via a control which acts in a subregion ω of the habitat Ω, and on the predators only. The dynamics of both interacting populations is described by a reaction-diffusion system with nonlocal terms describing migrations. A necessary condition and a sufficient condition for the zero-stabilizability of the prey population are derived in terms of the sign of the principal eigenvalues to certain non-selfadjoint operators. In case of stabilizability, a constant stabilizing control is indicated. The rate of stabilization corresponding to such a stabilizing control is dictated by the principal eigenvalue of a certain operator. A large principal eigenvalue leads to a fast stabilization to zero of the prey population. A method to approximate all these principal eigenvalues is presented. Some final comments concerning the relationship between the stabilization rate and the properties of ω and Ω are given as well.
Stabilization for a Periodic Predator-Prey System
Abstract and Applied Analysis, 2007
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.
Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations
Journal of Mathematical Biology, 1996
Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.
arXiv (Cornell University), 2018
A reaction-diffusion system exhibiting Turing's diffusion driven instability is considered. The equation for an activator is supplemented by unilateral terms of the type s − (x)u − , s + (x)u + describing sources and sinks active only if the concentration decreases below and increases above, respectively, the value of the basic spatially constant solution which is shifted to zero. We show that the domain of diffusion parameters in which spatially non-homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms. It is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources. The case of mixed (Dirichlet-Neumann) boundary conditions as well as that of pure Neumann conditions is described.
A predator-prey reaction-diffusion system with nonlocal effects
Journal of Mathematical Biology, 1996
Weconsider apredator-prey systemin theform of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurcations can occur in the degenerate case when the nonlocal term is in fact local.
Instabilities in reaction-diffusion systems
Applied mathematical sciences, 2014
In this paper we discuss the solution stability of two-component reaction-diffusion systems with constant diffusion coefficients. Linear stability analysis is performed near the steady state solution of the system discussing the dependence of the system stability on its parameters. A comprehensive linear stability analysis results in three types of instabilities: (1) Stationary periodic, (2) Oscillatory uniform and (3) Stationary uniform. Precise parameter regimes are identified for each. Mathematics Subject Classification: 35B36, 35K57, 70K50
On the stability of diffusion systems with chemical reactions
Mathematical Biosciences, 1974
The stability problem for diffusion systems with chemical reaction is treated by the methods of semigroup theory and functional analysis. The general theory is reviewed and is applied in two ways to a biological system with internal heat and mass transfer. Examples of the method are given for the case where the reaction kinetics are treated as an irreversible, first-order reaction.
Physical Review E, 2004
We study the problem of stabilization of a homogeneous solution in a two-variable reaction-convectiondiffusion one-dimensional system with oscillatory kinetics, in which moving or stationary patterns emerge in the absence of control. We propose to use a formal spatially weighted feedback control to suppress patterns in an absolutely or convectively unstable system and pinning control for a convectively unstable system. The latter approach is very effective and may require only one actuator to adjust feed conditions. In the former approach, the positive diagonal elements of the appropriate dynamics matrix are shifted to the left-hand part of the complex plane to ensure linear (asymptotic) stability of the system according to Gershgorin criterion. Moreover, we construct a controller that (with many actuators) will approach the global stability of the solution, according to Liapunov's direct method. We apply two alternative approaches to reveal the unstable modes: an approximate one that is based on linear stability analysis of an unbounded system, and an exact one that uses a traditional eigenstructure analysis of bounded systems. The number of required actuators increases dramatically with system size and with the distance from the bifurcation point. The methodology is developed for a system with learning cubic kinetics and is tested on a more realistic cross-flow reactor model.