On smoothness of carrying simplices

Abstract

Abstract. We consider dissipative strongly competitive systemsẋ i = x i f i (x) of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface Σ, called the carrying simplex. In this note we give an amenable condition for Σ to be a C 1 submanifold-with-corners. We also provide conditions, based on a recent work of M. Benaïm (On invariant hypersurfaces of strongly monotone maps,

Invariant manifolds of Competitive Selection–Recombination dynamics

“…To write the asymptotic dynamics on Σ M , we would ideally like Σ M to be at least of class C 1 , so that the standard tools of dynamical systems on differentiable manifolds, such as linear stability analysis, bifurcation theory, and so on, can be applied. If the study of the smoothness of the codimension-one carrying simplex of continuous-and discrete-time competitive population models is indicative [46,47,48,49,50], and bearing in mind that our boundary conditions of Σ M are particularly simple, we might expect that when the TLTA model is K * M −competitive for some polyhedral cone K M , Σ M is generically C 1 , but this remains an interesting open problem.…”

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

“…If Σ is continuously differentiable, then the unorderedness of Σ translates into its normal bundle being contained in C + [1]. It is an open question as to exactly when Σ is differentiable on its interior, but much progress has been made obtaining sufficient conditions for Σ to satisfy various smoothness properties [22,21,5,4,23,12]. In two recent articles [24,20] Mierczyński has shown that convex carrying simplices are C 1 .…”

Linearization and invariant manifolds on the carrying simplex for competitive maps

“…On the other hand, and more importantly, the smoothness of the carrying simplex allows us to apply the tools coming from Differential Geometry, especially, the Grobman-Hartman theorem. To the best of our knowledge, the available results on the smoothness of the carrying simplex are the following: Jiang, Mierczyński and Wang in [16] gave equivalent conditions, expressed in terms of inequalities between Lyapunov exponents, for the carrying simplex to be a C 1 submanifold-with-corners, neatly embedded in the nonnegative orthant (for sufficient conditions in the case of ordinary differential equations, see Brunovský [17], Mierczyński [18], or, for the C k property in discrete time systems, Benaïm [19] and, in ordinary differential equations, Mierczyński [20]). 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.…”