A note on global stability of three-dimensional Ricker models

Gyllenberg, Mats; Jiang, Jifa; Niu, Lei

doi:10.1080/10236198.2019.1566459

Cited by 16 publications

(22 citation statements)

References 15 publications

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“…Niu and Ruiz-Herrera [13] proved that every orbit converges to a fixed point when there is a unique positive fixed point whose index is −1; Balreira, Elaydi and Luís [14] and Gyllenberg, Jiang and Niu [15] provided criteria on the global stability of the unique positive fixed point under special conditions. However, it should be pointed out that how to prove there is no chaos in the 3D Leslie-Gower competitive map is still a very difficult problem, which is left for future research.…”

Section: Discussionmentioning

confidence: 99%

“…However, it was proved in [6] and [13] that only trivial dynamics can occur, that is every orbit converges to a fixed point, when there is no positive fixed point or there is a unique positive fixed point whose index is −1 by the topological results on homeomorphisms of the plane. Criteria on the global stability of the positive fixed point are provided in [14] and [15].…”

Section: Introductionmentioning

confidence: 99%

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Chaotic attractors in the four-dimensional Leslie–Gower competition model

Gyllenberg

Jiang

Niu

2020

Physica D: Nonlinear Phenomena

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We study the occurrence of the chaotic attractor in the four-dimensional classical Leslie-Gower competition model. We find that chaos can be generated by a cascade of quasiperioddoubling bifurcations starting from a supercritical Neimark-Sacker bifurcation of the positive fixed point in this model. The chaotic attractor is contained in the three-dimensional carrying simplex, that is a globally attracting invariant manifold. Biologically, the result implies that the invasion attempts by an invader into a trimorphic population under the Leslie-Gower dynamics can lead to chaos.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Chaotic attractors in the four-dimensional Leslie–Gower competition model

Gyllenberg

Jiang

Niu

2020

Physica D: Nonlinear Phenomena

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“…The following definition of equivalence appears to be unnecessarily pompous, but it prepares the way for the analogous definition in higher dimensions. for j = i, that is (see (32)) sgn(γ ij ) = sgn(γ σ(i)σ(j) ) for j = i.…”

Section: Classification Via Boundary Dynamicsmentioning

confidence: 99%

“…The following lemma is the 3D specialization of Theorem 2.4 in [9] (see also Theorem 1.2 in [32]), which can be used to establish our global stability for class 33. Figure 3.…”

Section: Stability and Permanencementioning

confidence: 99%

Permanence and universal classification of discrete-time competitive systems via the carrying simplex

Gyllenberg¹,

Jiang

Niu³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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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.Received xxxx 20xx; revised xxxx 20xx.

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“…Notice that from (14), α g is a curve that divides S into two connected components. Moreover, by the previous discussion, it is clear that α g joins {r 1 , r 2 }.…”

Section: Applications To Population Modelsmentioning

confidence: 99%

Linearization and invariant manifolds on the carrying simplex for competitive maps

Mierczyński

Niu

Ruiz-Herrera

2019

Journal of Differential Equations

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A result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface of codimension one which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. Our results give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics.

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A note on global stability of three-dimensional Ricker models

Cited by 16 publications

References 15 publications

Chaotic attractors in the four-dimensional Leslie–Gower competition model

Chaotic attractors in the four-dimensional Leslie–Gower competition model

Permanence and universal classification of discrete-time competitive systems via the carrying simplex

Linearization and invariant manifolds on the carrying simplex for competitive maps

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