Global attractors in competitive systems. (English) Zbl 0724.92024

Franke, John E.; Yakubu, Abdul-Aziz

Nonlinear Anal., Theory Methods Appl. 16, No. 2, 111-129 (1991).

This paper deals with the 2-dimensional discrete dynamical system \[ x_ i'=x_ i\lambda_ i(x_ 1+x_ 2),\quad i=1,2, \] on \({\mathbb{R}}^ 2_+=[0,\infty)\times [0,\infty)\). The per capita growth rates \(\lambda_ i\) are assumed to be continuous maps from \({\mathbb{R}}^ 1_+\) to \({\mathbb{R}}^ 1_+\). The resulting map \(F:\;{\mathbb{R}}^ 2_+\to {\mathbb{R}}^ 2_+\) defined by \(F(x_ 1,x_ 2)=(x_ 1\lambda_ 1(x_ 1+x_ 2)\), \(x_ 2\lambda_ 2(x_ 1+x_ 2))\) is shown to have a global attractor under a condition called 2-dominance. In fact, under this condition the first species goes extinct.
In order to define the 2-dominance condition, first assume that F restricted to each axis, \(f_ i(x_ i)=x_ i\lambda_ i(x_ i)\), has zero and infinity repellors. Biologically this models pioneer species with crowding and ecosystem constraints. These conditions produce an attractor \(T_ i\) on each axis. When max \(T_ 1 < \min T_ 2\), F is said to be 2-dominant. Under these conditions, if p is in the interior of \({\mathbb{R}}^ 2_+\) and \(q=(q_ 1,q_ 2)\) is in the \(\omega\)-limit set of p, then \(q_ 1=0\).

Reviewer: John E.Franke

MSC:

92D40	Ecology
37N99	Applications of dynamical systems
92D25	Population dynamics (general)

Keywords:

extinction; recurrent points; omega-limt set; discrete dynamical system; per capita growth rates; global attractor; 2-dominance; repellors; pioneer species

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