On Smoothness of Carrying Simplices
نویسندگان
چکیده
We consider dissipative strongly competitive systems ẋi = xifi(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 C1 submanifold-with-corners. We also provide conditions, based on a recent work of M. Benäım (On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 136 (1997), 302–319), guaranteeing that Σ is of class Ck+1.
منابع مشابه
Smoothness of Carrying Simplices for Three-dimensional Competitive Systems: a Counterexample
For a dissipative totally competitive system of ODEs ẋ = xf (x), ∂f /∂x < 0, i, j = 1, 2, 3, on the nonnegative octant K in R for which 0 is a repeller, M. W. Hirsch proved the existence of an invariant unordered Lipschitz surface (the carrying simplex) attracting all points in K \{0}. We give an example (of a Lotka–Volterra type) showing that the carrying simplex need not be of class C. AMS (M...
متن کاملGeometry of carrying simplices of 3-species competitive Lotka-Volterra systems
We investigate the existence, uniqueness and Gaussian curvature of the invariant carrying simplices of 3 species autonomous totally competitive Lotka-Volterra systems. Explicit examples are given where the carrying simplex is convex or concave, but also where the curvature is not single-signed. Our method monitors the curvature of an evolving surface that converges uniformly to the carrying sim...
متن کاملCubature formulae for spheres, simplices and balls
We obtain in explicit form the unique Gaussian cubature for balls (spheres) in Rn based on integrals over balls (spheres), centered at the origin, that integrates exactly all m-harmonic functions. In particular, this formula is exact for all polynomials in n variables of degree 2m − 1. A Gaussian cubature for simplices is also constructed. Upper bounds for the errors for certain smoothness clas...
متن کاملA characterization theorem of multivariate splines in blossoming form
It is known that polynomials in m variables of total degree n are equivalent to symmetric polynomials in n variables that are linear in each single variable. This principle, called the blossoming principle, has applied to the study of multivariate splines in this paper. For any spline on a simplicial partition 4, a smoothness condition on its polynomial pieces on any two simplices of 4 which ma...
متن کاملOn peaks in carrying simplices
A necessary and sufficient condition is given for the carrying simplex of a dissipative totally competitive system of three ordinary differential equations to have a peak singularity at an axial equilibrium. For systems of Lotka–Volterra type that result translates into a simple condition on the coefficients.
متن کامل