## Global attractors in competitive systems.(English)Zbl 0724.92024

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)
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### References:

 [1] Block, L.; Franke, J.E., The chain recurrent set, attractors and explosions, Ergodic dynam. sys., 5, 321-327, (1985) · Zbl 0572.54037 [2] Collet, P.; Eckmann, J.P., Iterated maps on the interval as dynamical system, (1980), Birkhauser Boston · Zbl 0441.58011 [3] Comins, H.N.; Hassell, M.P., Predation in multi-prey communities, J. theor. biol., 62, 93-114, (1976) [4] Devaney, R.L., An introduction to chaotic dynamical systems, (1987), Addison-Wesley Redwood City, CA [5] Franke, J.E.; Selgrade, J.F., Hyperbolicity and cycles, Trans. am. math. soc., 245, 251-262, (1978) · Zbl 0396.58023 [6] Franke J.E. & Yakubu A., Mutual exclusion versus coexistence for discrete competitive systems, preprint. · Zbl 0735.92023 [7] Franke J.E. & Yakubu A., Geometry of mutual exclusion in competitive systems, preprint. · Zbl 0735.92023 [8] Hallam, T.G.; Svoboda, L.J.; Gard, T.C., Persistence and extinction in three species Lotka-Volterra competitive systems, Math. biosci., 46, 117-124, (1979) · Zbl 0413.92013 [9] Hassell, M.P.; Comins, H.N., Discrete time models for two species competition, Theor. population biol., 9, 202-221, (1976) · Zbl 0338.92020 [10] Hofbauer, J., A unified approach to persistence, Acta applicandae mathematicae, 14, 11-22, (1989) · Zbl 0669.92020 [11] Hofbauer, J.; Hutson, V.; Jansen, W., Coexistence for systems governed by difference equations of Lotka-Volterra type, J. math. biol., 25, 553-570, (1987) · Zbl 0638.92019 [12] Li, T.; Yorke, J.A., Period three implies chaos, Am. math. monthly, 82, 985-992, (1975) · Zbl 0351.92021 [13] Lotka, A.J., Elements of mathematical biology, (1976), Dover New York [14] Nitecki, Z., Differential dynamics: an introduction to the orbit structure of diffeomorphism, (1971), M.I.T. Press Cambridge, MA [15] So J.W.-H. & Hofbauer J., Uniform persistence and repellors for maps, preprint. · Zbl 0678.58024
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