PERMANENCE AND UNIVERSAL CLASSIFICATION OF DISCRETE-TIME COMPETITIVE SYSTEMS VIA THE CARRYING SIMPLEX

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Gyllenberg , M , Jiang , J , Niu , L & Yan , P 2020 , ' PERMANENCE AND UNIVERSAL CLASSIFICATION OF DISCRETE-TIME COMPETITIVE SYSTEMS VIA THE CARRYING SIMPLEX ' , Discrete and continuous dynamical systems , vol. 40 , no. 3 , pp. 1621-1663 . https://doi.org/10.3934/dcds.2020088

Title: PERMANENCE AND UNIVERSAL CLASSIFICATION OF DISCRETE-TIME COMPETITIVE SYSTEMS VIA THE CARRYING SIMPLEX
Author: Gyllenberg, Mats; Jiang, Jifa; Niu, Lei; Yan, Ping
Contributor organization: Department of Mathematics and Statistics
Date: 2020-03
Language: eng
Number of pages: 43
Belongs to series: Discrete and continuous dynamical systems
ISSN: 1078-0947
DOI: https://doi.org/10.3934/dcds.2020088
URI: http://hdl.handle.net/10138/328534
Description: We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of 33 stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes 1-25 and 33; there is always a heteroclinic cycle in class 27; Neimark-Sacker bifurcations may occur in classes 26-31 but cannot occur in class 32. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes 29,31,33 and those in class 27 with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.
Subject: 111 Mathematics
Permanence
population model
Neimark-Sacker bifurcation
heteroclinic cycle
phase portrait
fixed point index
classification
competitive system
carrying simplex
Peer reviewed: Yes
Usage restriction: openAccess
Self-archived version: acceptedVersion


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