Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

Abstract

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C 1 submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the C r (r 1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C 1 but not neatly embedded, the other in which S is not of class C 1 .

Linearization and invariant manifolds on the carrying simplex for competitive maps

“…These results solve some open problems suggested in [10,11,12,13,15]. (15), (16) and (17), where α ij := a ii −a ji , β ij := a jj −a ij a ii a jj −a ij a ji for i, j = 1, 2, 3 and i = j. The carrying simplex S with corresponding parameters in each class is given by a representative element in that class, i.e.…”

“…The conditions (A1)-(A3) hold for the Leslie-Gower model (15) and the Atkinson-Allen model (16), so they admit a carrying simplex S; see [10,12]. The Ricker model (17)…”

“…Whether the carrying simplex is smooth or not is still unknown when it is not convex. Mierczyński in [27,27] in the case of ordinary differential equations and Jiang, Mierczyński and Wang in [16] in the case of maps do provide examples which show that the carrying simplex at a boundary fixed point can be far from smooth. However, no examples are known of the lack of smoothness in the interior of a carrying simplex.…”

Exclusion and dominance in discrete population models via the carrying simplex

“…Firstly, we prove the existence of xð2NÞ for all N [ N. This claim follows from the arguments of Proposition 2.1 in Ref. [10].…”

Carrying simplices in nonautonomous and random competitive Kolmogorov systems

“…Based on this, by applying Takáč's invariant order decomposition methods (see [35,36]), Wang and Jiang [37,38] generalized the well-known result of Hirsch and verified the existence of the carrying simplex for Kolmogorov competitive mappings and the time-periodic competitive system (1.1). The geometry, smoothness and dynamics of carrying simplices have been widely investigated for autonomous cases (see [8,[22][23][24]39,40]) and time-periodic cases (see [3,20,26,32,38]). The theory of the carrying simplex has also been applied to many mathematical models such as Lotka-Volterra model [39,40], the age-structured semelparous populations [7], the growth of phytoplankton in a chemostat [33,34], and type-K competitive systems [10,21], etc.…”