Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Permanence and universal classification of discrete-time competitive
  systems via the carrying simplex   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.

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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio