Stability Properties of Solutions to Systems of Reaction-Diffusion Equations

Casten, Richard G.; Holland, Charles J.

doi:10.1137/0133023

Cited by 98 publications

(71 citation statements)

References 9 publications

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“…It is noteworthy that other approaches have been used in the past for studying the stability properties of solutions to reaction-diffusion equations. For example, Casten & Holland (1977) have described a method to determine the local asymptotic stability of a nonlinear reaction-diffusion system with zero Neumann boundary conditions based on the eigenvalues of a matrix arising from the linearization of the system of equations.…”

Section: Derivation Of the Modelmentioning

confidence: 99%

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

Perumpanani¹,

Sherratt²,

Maini³

1995

IMA J Appl Math

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The authors study the effect of advection on reaction-diffusion patterns. It is shown that the addition of advection to a two-variable reaction-diffusion system with periodic boundary conditions results in the appearance of a phase difference between the patterns of the two variables which depends on the difference between the advection coefficients. The spatial patterns move like a travelling wave with a fixed velocity which depends on the sum of the advection coefficients. By a suitable choice of advection coefficients, the solution can be made stationary in time. In the presence of advection a continuous change in the diffusion coefficients can result in two Turing-type bifurcations as the diffusion ratio is varied, and such a bifurcation can occur even when the inhibitor species does not diffuse. It is also shown that the initial mode of bifurcation for a given domain size depends on both the advection and diffusion coefficients. These phenomena are demonstrated in the numerical solution of a particular reaction-diffusion system, and finally a possible application of the results to pattern formation in Drosophila larvae is discussed.

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Section: Derivation Of the Modelmentioning

confidence: 99%

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

Perumpanani¹,

Sherratt²,

Maini³

1995

IMA J Appl Math

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show abstract

“…In this case, eigenvalues with a positive real part may be created (depending on the form of the fitness function). The spatially homogeneous solution may therefore become unstable [51,39,10] and the new equilibrium solution is no longer spatially homogeneous. If this is the case, spatial patterns may occur.…”

Section: Conclusion 2 If An Ess Has Been Found For a One-species Poimentioning

confidence: 99%

“…[31]. It implies that the linearized stability of P E around the zero solution of P E is also the local stability of the non-linear problem P E [10].…”

Section: Ecologically Stable Equilibriummentioning

confidence: 99%

“…Systems of the form (2.2) are known as reaction-diffusion equations [51,47,42,10,25]. Traditionally, the boundary of the spatial domain is denoted by ∂Ω (here ∂Ω should be understood as a single symbol; ∂ is not an operator).Ω designates the spatial domain, including its boundary.…”

Section: Problem Statementmentioning

confidence: 99%

“…It can be shown [34,10,25] that with zero flux boundary conditions, there is an infinite number of stability matrices of the linearized problem (2.8), namely 10) where λ k are eigenvalues that obey:…”

Section: Ecologically Stable Equilibriummentioning

confidence: 99%

See 2 more Smart Citations

Evolutionary Games in Space

Kronik

Cohen

2009

Math. Model. Nat. Phenom.

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Abstract. The G-function formalism has been widely used in the context of evolutionary games for identifying evolutionarily stable strategies (ESS). This formalism was developed for and applied to point processes. Here, we examine the G-function formalism in the settings of spatial evolutionary games and strategy dynamics, based on reaction-diffusion models. We start by extending the point process maximum principle to reaction-diffusion models with homogeneous, locally stable surfaces. We then develop the strategy dynamics for such surfaces. When the surfaces are locally stable, but not homogenous, the standard definitions of ESS and the maximum principle fall apart. Yet, we show by examples that strategy dynamics leads to convergent stable inhomogeneous strategies that are possibly ESS, in the sense that for many scenarios which we simulated, invaders could not coexist with the exisiting strategies.

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Linear and nonlinear stability thresholds for a diffusive model of pioneer and climax species interaction

Buonomo

Rionero

2008

Math Methods in App Sciences

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SUMMARYIn this paper a reaction-diffusion model describing two interacting pioneer and climax species is considered. The role of diffusivity and forcing (stocking or harvesting of the species) on the nonlinear stability of a coexistence equilibrium is analysed. The study is performed in the context of a new approach to nonlinear L 2 -stability based on the analysis of stability of the zero solution of a suitable linear system of ordinary differential equations. Theorems concerning the effect of forcing and diffusivity on the dynamics are established and stability-instability thresholds for the system are obtained. An example to illustrate the practical use of the results is also provided.

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Stability Properties of Solutions to Systems of Reaction-Diffusion Equations

Cited by 98 publications

References 9 publications

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

Evolutionary Games in Space

Linear and nonlinear stability thresholds for a diffusive model of pioneer and climax species interaction

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