Trivial dynamics in discrete-time systems : carrying simplex and translation arcs

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http://hdl.handle.net/10138/301293

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Niu , L & Ruiz-Herrera , A 2018 , ' Trivial dynamics in discrete-time systems : carrying simplex and translation arcs ' , Nonlinearity , vol. 31 , no. 6 , pp. 2633–2650 . https://doi.org/10.1088/1361-6544/aab46e

Title: Trivial dynamics in discrete-time systems : carrying simplex and translation arcs
Author: Niu, Lei; Ruiz-Herrera, Alfonso
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2018-06
Language: eng
Number of pages: 18
Belongs to series: Nonlinearity
ISSN: 0951-7715
URI: http://hdl.handle.net/10138/301293
Abstract: In this paper we show that the dynamical behavior in R-+(3) (first octant) of the classical Kolmogorov systems T(x(1), x(2), x(3)) = (x(1)F(1)(x), x(2)F(2)(x), x(3)F(3)(x)) of competitive type admitting a carrying simplex can be sometimes determined completely by the number of fixed points on the boundary and the local behavior around them. Roughly speaking, T has trivial dynamics (i.e. the omega limit set of any orbit is a connected set contained in the set of fixed points) provided T has exactly four hyperbolic nontrivial fixed points {p(1), p(2), p(3), p(4)} in partial derivative R-+(3) with {p(1), p(3)} local attractors on the carrying simplex and {p(2), p(4)} local repellers on the carrying simplex; and there exists a unique hyperbolic fixed point in IntR(+)(3). Our results are applied to some classical models including the Leslie-Gower models, Atkinson-Allen systems and Ricker maps.
Subject: 111 Mathematics
trivial dynamics
carrying simplex
translation arcs
whole dynamics
fixed point index
COMPETITIVE-SYSTEMS
EQUIVALENT CLASSIFICATION
DIFFERENTIAL-EQUATIONS
PERIODIC-ORBITS
FIXED-POINTS
MODELS
BOUNDARY
MAPS
UNIQUENESS
SIMPLICES
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