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Langa, J. A.; Robinson, J. C.; Suárez, A.

Pullback permanence in a non-autonomous competitive Lotka–Volterra model. (English) Zbl 1027.35017

J. Differ. Equations 190, No. 1, 214-238 (2003).
The goal of the paper is to study the long-time behaviour of the non-autonomous competitive system \[ \begin{aligned} u_t-\Delta u=u(\lambda -a(t)u-bv) &\quad\text{in } Q\times (s,+\infty),\\ v_t-\Delta v=v(\mu -cv-du) &\quad\text{in } Q\times (s,+\infty), \end{aligned} \] where \(Q\subset \mathbb{R}^N\) is a smooth bounded domain, \(b, c, d\) and \(a(t)\) are positive. Both the forwards and pullback behaviour of system are analysed. The notion of ”pullback permanence” is introduced. The existence of a non-autonomous attractor and conditions for pullback permanence are estblished. The main tools in the paper are the theory of attractors for non-autonomous differential equations, the sub-supersolution method and the spectral theory for linear elliptic equations.
Reviewer: Evgeniy A.Kalita (Donetsk)

Cited in 5 Documents

MSC:

35B41 Attractors
35K57 Reaction-diffusion equations

Keywords:

attractors for non-autonomous differential equations; competitive diffusion system; pullback attractor; sub-supersolution method
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References:

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