2018
DOI: 10.3934/dcds.2018027
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On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex

Abstract: We propose the generalized competitive Atkinson-Allen mapwhich is the classical Atkson-Allen map when r i = 1 and c i = c for all i = 1, ..., n and a discretized system of the competitive Lotka-Volterra equations.It is proved that every n-dimensional map T of this form admits a carrying simplex Σ which is a globally attracting invariant hypersurface of codimension one. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ on the space of all such three-dimensional m… Show more

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Cited by 20 publications
(54 citation statements)
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“…By Table 1 (27) we know that T [c3] belongs to class 27 for all 0 < c 3 < 1, whose unique positive fixed point p = ( , DT [c3] (p) has a pair of complex conjugate eigenvalues of modulus 1 which do not equal ±1, ±i, (−1 ± √ 3i)/2. By numerical calculation [28,57,34], we get the first Lyapunov coefficient l 1 ≈ −1.814×10 −2 < 0. Therefore, a supercritical Neimark-Sacker bifurcation occurs at c 3 = c * 3 , i.e., a stable invariant closed curve bifurcates from the fixed point p. On the other hand, it follows from (41) that ≈ 0.0026 > 0 for c 3 = c * 3 , so the heteroclinic cycle ∂Σ of T [c3] is repelling, i.e.…”
Section: 2mentioning
confidence: 96%
See 3 more Smart Citations
“…By Table 1 (27) we know that T [c3] belongs to class 27 for all 0 < c 3 < 1, whose unique positive fixed point p = ( , DT [c3] (p) has a pair of complex conjugate eigenvalues of modulus 1 which do not equal ±1, ±i, (−1 ± √ 3i)/2. By numerical calculation [28,57,34], we get the first Lyapunov coefficient l 1 ≈ −1.814×10 −2 < 0. Therefore, a supercritical Neimark-Sacker bifurcation occurs at c 3 = c * 3 , i.e., a stable invariant closed curve bifurcates from the fixed point p. On the other hand, it follows from (41) that ≈ 0.0026 > 0 for c 3 = c * 3 , so the heteroclinic cycle ∂Σ of T [c3] is repelling, i.e.…”
Section: 2mentioning
confidence: 96%
“…Proof. By Table 1, the map T ∈ DCS(3, f ) is in class 33 if the parameters satisfy the following inequalities (i) γ 12 > 0, γ 13 > 0, γ 21 > 0, γ 23 > 0, γ 31 > 0, γ 32 > 0; [48,34] and the Leslie-Gower model [49]; see Section 5 for details.…”
Section: Permanence and Classification For Discrete-time Systems 25mentioning
confidence: 99%
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“…The conditions (A1)-(A3) hold for the Leslie-Gower model (15) and the Atkinson-Allen model (16), so they admit a carrying simplex S; see [10,12]. The Ricker model (17)…”
Section: Applications To Population Modelsmentioning
confidence: 99%