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THE CONVERGENCE OF A CLASS OF QUASIMONOTONE REACTION–DIFFUSION SYSTEMS

Published online by Cambridge University Press:  30 October 2001

YI WANG  and
JIFA JIANG
YI WANG
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China; wangyi@mail.ustc.edu.cn, jiangjf@ustc.edu.cn
JIFA JIANG
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China; wangyi@mail.ustc.edu.cn, jiangjf@ustc.edu.cn
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Abstract

It is proved that every solution of the Neumann initial-boundary problem

[formula here]

converges to some equilibrium, if the system satisfies (i) ∂Fi/∂uj [ges ] 0 for all 1 [les ] i ≠ j [les ] n, (ii) F(u * g(s)) [ges ] h(s) [midast ] F(u) whenever u ∈ ℝn+ and 0 [les ] s [les ] 1, where x * y = (x1y1, …, xnyn) and g, h : [0, 1] → [0, 1]n are continuous functions satisfying gi(0) = hi(0) = 0, gi(1) = hi(1) = 1, 0 < gi(s); hi(s) < 1 for all s ∈ (0, 1) and i = 1, 2, …, n, and (iii) the solution of the corresponding ordinary differential equation system is bounded in ℝn+. We also study the convergence of the solution of the Lotka–Volterra system

[formula here]

where ri > 0, α [ges ] 0, and aij [ges ] 0 for i ≠ j.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 395 - 408
Copyright
The London Mathematical Society 2001

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