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Cooperative systems theory and global stability of diffusion models

Cooperative systems theory and global stability of diffusion models Acta Applicandae Mathematicae 14 (1989), 49-57. 49 © 1989 by IIASA. Cooperative Systems Theory and Global Stability of Diffusion Models Y. Takeuchi Department of Applied Mathematics Faculty of Engineering Shizuoka University Hamamatsu 432, Japan AMS Subject Classification (1980): 92A17, 34D20 Key words: global stability, cooperative systems, diffusion 1. Introduction Many authors consider the effect of spatial factors, such as diffusion or migration among patches, in population dynamics. We suppose that the system is composed of several patches connected by diffusion and occupied by a single species. Furthermore, the species is supposed to be able to survive in all the patches at a positive globally stable equilibrium point if the patches are isolated, or if the diffusion among patches is neglected and the species is confined to each patch. The problem considered in this paper is whether the equilibrium point, the value of which can be changed according to the strength of diffusion, continues to be positive and globally stable, if we increase the rates of diffusion. Allen [1] proved by applying comparison techniques that the model of such a single species diffusion system remains weakly persistent if the strength of diffusion is small enough. The homotopy function technique was http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Cooperative systems theory and global stability of diffusion models

Acta Applicandae Mathematicae , Volume 14 (2) – Apr 30, 2004

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References (10)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/BF00046673
Publisher site
See Article on Publisher Site

Abstract

Acta Applicandae Mathematicae 14 (1989), 49-57. 49 © 1989 by IIASA. Cooperative Systems Theory and Global Stability of Diffusion Models Y. Takeuchi Department of Applied Mathematics Faculty of Engineering Shizuoka University Hamamatsu 432, Japan AMS Subject Classification (1980): 92A17, 34D20 Key words: global stability, cooperative systems, diffusion 1. Introduction Many authors consider the effect of spatial factors, such as diffusion or migration among patches, in population dynamics. We suppose that the system is composed of several patches connected by diffusion and occupied by a single species. Furthermore, the species is supposed to be able to survive in all the patches at a positive globally stable equilibrium point if the patches are isolated, or if the diffusion among patches is neglected and the species is confined to each patch. The problem considered in this paper is whether the equilibrium point, the value of which can be changed according to the strength of diffusion, continues to be positive and globally stable, if we increase the rates of diffusion. Allen [1] proved by applying comparison techniques that the model of such a single species diffusion system remains weakly persistent if the strength of diffusion is small enough. The homotopy function technique was

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Apr 30, 2004

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