scholarly journals Global stability and persistence in diffusive food chains

2001 ◽  
Vol 43 (2) ◽  
pp. 247-268 ◽  
Author(s):  
Yang Kuang
Keyword(s):  
Dynamical System ◽  
Steady State ◽  
Global Stability ◽  
Invariance Principle ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Food Chains ◽  
Liapunov Functional ◽  
Positive Steady State

AbstractIn this paper, the results of Freedman and So [13] on global stability and persistence of simple food chains are extended to general diffusive food chains. For global stability of the unique homogeneous positive steady state, our approach involves an application of the invariance principle of reaction-diffusion equations and the construction of a Liapunov functional. For persistence, we use the dynamical system results of Dunbar et al. [11] and Hutson and Moran [29].

Map dynamics versus dynamics of associated delay reaction–diffusion equations with a Neumann condition

2010 ◽  
Vol 466 (2122) ◽  
pp. 2955-2973 ◽  
Author(s):  
Taishan Yi ◽  
Xingfu Zou
Keyword(s):  
Steady State ◽  
Global Stability ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Neumann Condition ◽  
Main Concern ◽  
Homogeneous Neumann Boundary Condition ◽  
Globally Stable

In this paper, we consider a class of delay reaction–diffusion equations (DRDEs) with a parameter ε >0. A homogeneous Neumann boundary condition and non-negative initial functions are posed to the equation. By letting , such an equation is formally reduced to a scalar difference equation (or map dynamical system). The main concern is the relation of the absolute (or delay-independent) global stability of a steady state of the equation and the dynamics of the nonlinear map in the equation. By employing the idea of attracting intervals for solution semiflows of the DRDEs, we prove that the globally stable dynamics of the map indeed ensures the delay-independent global stability of a constant steady state of the DRDEs. We also give a counterexample to show that the delay-independent global stability of DRDEs cannot guarantee the globally stable dynamics of the map. Finally, we apply the abstract results to the diffusive delay Nicholson blowfly equation and the diffusive Mackey–Glass haematopoiesis equation. The resulting criteria for both model equations are amazingly simple and are optimal in some sense (although there is no existing result to compare with for the latter).

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