Exclusion and dominance in discrete population models via the carrying simplex. (English) Zbl 1303.92107
Summary: This paper is devoted to show that M. W. Hirsch’s results [J. Biol. Dyn. 2, No. 2, 169–179 (2008; Zbl 1152.92026)] on the existence of a carrying simplex are a powerful tool to understand the dynamics of Kolmogorov models. For two and three species, we prove that there is exclusion for our models if and only if there are no coexistence states. The proof of this result is based on a result in planar topology due to J. Campos et al. [J. Differ. Equations 138, No. 1, 157–170, Art. No. DE973265 (1997; Zbl 0886.34025)]. For an arbitrary number of species, we will obtain dominance criteria following the notions of J. E. Franke and A.-A. Yakubu [Nonlinear Anal., Theory Methods Appl. 21, No. 5, 369–378 (1993; Zbl 0788.34043)]. In this scenario, the crucial fact will be that the carrying simplex is an unordered manifold. Applications in concrete models are given.
MSC:
| 92D25 | Population dynamics (general) |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
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