Exclusion and dominance in discrete population models via the carrying simplex

Ruiz-Herrera, Alfonso

doi:10.1080/10236198.2011.628663

Cited by 44 publications

(84 citation statements)

References 16 publications

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“…Lemma 4.21 (Theorem 2.2 in [74] and Theorem 3.1 in [69]). Consider the threedimensional map T given by (8) which satisfies the conditions Υ1), Υ2) and Υ3) in Lemma 3.3.…”

Section: Stability and Permanencementioning

confidence: 99%

“…The two-dimensional Leslie-Gower model is thoroughly analyzed by Cushing et al [14]. The higher dimensional case was analyzed in [40,74,50,49].…”

Section: 1mentioning

confidence: 99%

“…, n. The discrete-time dynamical system induced by such T has been much used to describe the interactions of n species with non-overlapping generations; see [63,47,42,20,70,60,53,40]. The statement of Hirsch's Theorem in [40] was rigorously proved by Ruiz-Herrera [74] under similar assumptions to Hirsch's. Other criteria on the existence of a carrying simplex for Kolmogorov mappings were also established in [5,50,49], and we refer the readers to the paper [49] for a review.…”

mentioning

confidence: 99%

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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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“…Lemma 4.21 (Theorem 2.2 in [74] and Theorem 3.1 in [69]). Consider the threedimensional map T given by (8) which satisfies the conditions Υ1), Υ2) and Υ3) in Lemma 3.3.…”

Section: Stability and Permanencementioning

confidence: 99%

“…The two-dimensional Leslie-Gower model is thoroughly analyzed by Cushing et al [14]. The higher dimensional case was analyzed in [40,74,50,49].…”

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

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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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“…In a manner similar to that used before, we can prove that there exists a neighborhood U (6) ⊂ U (5) such that DP −l (y)u ≤ σ −l u for any y ∈ U (6) and any u ∈ R n whose direction is sufficiently close to W . Now we can take a convex neighborhood U (7) ⊂ U (6) that satisfies…”

Section: Notation and Definitionsmentioning

confidence: 99%

“…In order to analyze the global dynamics of such discrete-time systems, it suffices to study the dynamics of the systems restricted to this invariant hypersurface. In particular, one can use the topological results on the homeomorphisms of the plane such as the translation arc and degree (Ruiz-Herrera [7], Jiang and Niu [11] and Niu and Ruiz-Herrera [15]) for three-dimensional competitive maps with a carrying simplex.…”

Section: Introductionmentioning

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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Carrying Simplices for Competitive Maps

Baigent¹

2019

Springer Proceedings in Mathematics &Amp; Statistics

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The carrying simplex is a finite-dimensional, attracting Lipschitz invariant manifold that is commonly found in both continuous and discrete-time competition models from Ecology. It can be studied using the graph transform and cone conditions often applied to study attractors in continuous-time finite and infinitedimensional models from applied mathematics, including chemical reaction networks and reaction diffusion equations. Here we show that the carrying simplex can also be studied from the point of view of the graph transform and cone conditions. However, unlike many of the models mentioned above, we do not use-at least directly-a gap condition that is often used to establish existence of a globally and exponentially attracting manifold. Instead we use contraction of phase volume to 'suck' hypersurfaces together uniformly, and ultimately onto the carrying simplex. We give a proof of the existence of the carrying simplex for a class of competitive maps, viewed here as also normally monotone maps. The result is not new, but is carried out in the framework of the graph transform to indicate how the carrying simplex relates to other well-known classes of invariant manifolds. We also discuss the relation between hypersurfaces with positive normals, unordered hypersurfaces and also the type of maps that preserve these types of hypersurfaces. Finally we review several examples from models in Ecology where the carrying simplex is known to exist.

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Exclusion and dominance in discrete population models via the carrying simplex

Cited by 44 publications

References 16 publications

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

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

Linearization and invariant manifolds on the carrying simplex for competitive maps

Carrying Simplices for Competitive Maps

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