Carrying simplices in discrete competitive systems and age-structured semelparous populations

Abstract

For discrete competitive dynamical systems, amenable general conditions are presented to guarantee the existence of the carrying simplex and then these results are applied to age-structured semelparous population models, as well as to an annual plant competition model.

Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems

“…The conditions above are stated for R 2 + , but are a special case of the n−dimensional problem as treated in [12,18] (under less restrictive conditions on differentiability of F, G). Other related conditions for the n−dimensional problem appear in [19,15,7] and the recent paper [16].…”

“…Under conditions C1 -C4 there exists a unique carrying simplex; that is, a Lipschitz invariant manifold passing through all non-trivial fixed points (i.e. those other than O) of (1) and that attracts R 2 + \ O [12,18,7]. Many of the techniques utilised to study competitive discrete-time maps owe their origins to P. de Mottoni and Shiaffino's study of the attractors and repellers of the competitive Poincaré map of the periodic competitive Lotka-Volterra equations [6].…”

Linearization and invariant manifolds on the carrying simplex for competitive maps

“…Noticing that the spectral radius of the restriction DP −1 (0)| W is ν −1 , we deduce that there is l ∈ N such that DP −l (0)| W < σ −l . 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…”

“…for all y ∈ M 2 ∩ U (5) . Thus, if y ∈ M 2 ∩ U (5) with P (y) ∈ M 2 ∩ U (5) , we have that P (L y ) ⊂ L P (y) . It remains to prove (1).…”

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

“…x i < y i provided y i > 0. Since the early work of Hirsch [1] and Smith [2], it is well known that most competitive maps admit an invariant hypersurface Σ of codimension one, known as the carrying simplex, which attracts all nontrivial orbits (see, for instance, [3,4,5,6,7,8,9]). The origin 0 is a repeller for T , and Σ is the boundary in R n + of the basin of repulsion of the origin which satisfies the following properties:…”