2015
DOI: 10.1007/s00285-015-0920-1
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On heteroclinic cycles of competitive maps via carrying simplices

Abstract: We concentrate on the effects of heteroclinic cycles and the interplay of heteroclinic attractors or repellers on the boundary of the carrying simplices for low-dimensional discrete-time competitive systems. Based on the existence of the carrying simplex for the competitive mapping, we provide the criteria on stability of the heteroclinic cycle. This result can be seen as a discrete counterpart of that for the continuous-time systems. Several concrete discrete-time competition models are further analyzed, whic… Show more

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Cited by 30 publications
(48 citation statements)
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“…Under the assumption, one has Ω(∂Σ) = {q {1} , q {2} , q {3} }. It follows from the proof of Theorem 3 in [50] that there are real numbers ν 1 , ν 2 , ν 3 > 0 such that (18) holds if > 0, so T is permanent by Theorem 3.4; see Fig. 1.…”
Section: 2mentioning
confidence: 96%
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“…Under the assumption, one has Ω(∂Σ) = {q {1} , q {2} , q {3} }. It follows from the proof of Theorem 3 in [50] that there are real numbers ν 1 , ν 2 , ν 3 > 0 such that (18) holds if > 0, so T is permanent by Theorem 3.4; see Fig. 1.…”
Section: 2mentioning
confidence: 96%
“…The importance of the existence of a carrying simplex stems from the fact that it captures the relevant long-term dynamics. In particular, it contains all nontrivial fixed points, periodic orbits, invariant circles and heteroclinic cycles (see, for example, [48,50,49,34]). Therefore, the common approach in the study of these systems is to focus on the dynamics on the carrying simplex.…”
mentioning
confidence: 99%
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“…It is known (for example, in [17]) that T is injective on C and that (8) has a carrying simplex, but its geometry is not known.…”
Section: Examplesmentioning
confidence: 99%