Convex geometry of the carrying simplex for the May-Leonard map

Baigent, Stephen

doi:10.3934/dcdsb.2018288

Cited by 8 publications

(15 citation statements)

References 33 publications

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“…Repeating the above process we obtain x k (n) < 2 3 y k for all n ≥ m 1 , a contradiction to y ∈ ω(x). The contradictions in both cases above show that ω(x) ⊂ (∩ k ℓ=1 π ℓ ) ∩ Σ.…”

Section: (A) Under the Assumption Thatmentioning

confidence: 91%

On existence and uniqueness of a carrying simplex in Kolmogorov differential systems

Hou

2020

Nonlinearity

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For a C 1 map T from C = [0, +∞) N to C of the form Ti(x) = xifi(x), the dynamical system x(n) = T n (x) as a population model is competitive if ∂f i ∂x j ≤ 0 (i = j). A well know theorem for competitive systems, presented by Hirsch (J. Bio. Dyn. 2 (2008) 169-179) and proved by Ruiz-Herrera (J. Differ. Equ. Appl. 19 (2013) 96-113) with various versions by others, states that, under certain conditions, the system has a compact invariant surface Σ ⊂ C that is homeomorphic to ∆ N−1 = {x ∈ C : x1 + • • • + xN = 1}, attracting all the points of C \ {0}, and called carrying simplex. The theorem has been well accepted with a large number of citations. In this paper, we point out that one of its conditions requiring all the N 2 entries of the Jacobian matrix Df = (∂f i ∂x j) to be negative is unnecessarily strong and too restrictive. We prove the existence and uniqueness of a modified carrying simplex by reducing that condition to requiring every entry of Df to be nonpositive and each fi is strictly decreasing in xi. As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.

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Section: (A) Under the Assumption Thatmentioning

confidence: 91%

On existence and uniqueness of a carrying simplex in Kolmogorov differential systems

Hou

2020

Nonlinearity

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“…Some progress has been made here for continuous time competitive Lotka-Volterra systems by Zeeman and Zeeman [40]. Recent progress on the convexity or concavity of carrying simplices for competitive maps can be found in [5,3]. • a bifurcation analysis that links change in stability to change in geometry of the carrying simplex local to a fixed point.…”

Section: Discussionmentioning

confidence: 99%

“…Baigent [5] studied the existence of the planar Leslie-Gower model under similar conditions, and also showed that the carrying simplex was either a convex or concave curve. Recently, Baigent studied the 3-dimensional Leslie-Gower model [3] in May-Leonard form and established parameter regions where the carrying simplex was either convex or concave.…”

Section: The Leslie-gower Modelmentioning

confidence: 99%

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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“…Mierczyński proved in [21,22] that the carrying simplex is a C 1 submanifold-with-corners neatly embedded in the nonnegative orthant when it is convex. For the convexity of the carrying simplex and their influence on the global dynamics, we refer the reader to [23,24,8,25,26]. Whether the carrying simplex is smooth or not is still unknown when it is not convex.…”

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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Convex geometry of the carrying simplex for the May-Leonard map

Cited by 8 publications

References 33 publications

On existence and uniqueness of a carrying simplex in Kolmogorov differential systems

On existence and uniqueness of a carrying simplex in Kolmogorov differential systems

Carrying Simplices for Competitive Maps

Linearization and invariant manifolds on the carrying simplex for competitive maps

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