Carrying simplices in nonautonomous and random competitive Kolmogorov systems

Abstract

The purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems via the skew-product flows approach. It is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skewproduct flow associated with a nonautonomous (random) competitive Kolmogorov system.

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

“…Corollary 7.1 is motivated by the results by Zanolin [48] and Zhao [50] on the timeperiodic Kolmogorov competitive systems. Now, by Theorem 0, Propositions 7.1-7.3 and Corollary 7.1, there exists a carrying simplex S (see also for the similar conditions in [38] that guarantee the existence of S for nonautonomous/random competitive Kolmogorov systems).…”

Floquet Bundles for Tridiagonal Competitive-Cooperative Systems and the Dynamics of Time-Recurrent Systems

“…In particular, Corollary 4.6 generalizes the results of de Mottoni and Schiaffino [16] and Hale and Somolinos [6], who proved that all solutions of two-dimensional T -periodic competitive or cooperative systems are asymptotic to T -periodic solutions. See also [9] and [24] for extensions of this work.…”

“…Moreover, Theorem 4.7 also extends all the results for the two-dimensional case mentioned above to higher dimensions (n ≥ 3). We refer to [14] and [29] for other related extensions of the two-dimensional case.…”

Dynamics of Kolmogorov systems of competitive type under the telegraph noise

“…A systematic review has been given in [1]. Recently, [16] considers the influence of both Markov switching and white noise on system (1.1); A. Bobrowski et al in [8] use the Markov semigroup to study the stability of the stationary distribution of random systems (1.1); W. Shen, Y. Wang in [19] study the random competitive Kolmogorov systems via the skew-product flows approach. .…”