Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows

Abstract

Several results of uniform persistence above and below a minimal set of an abstract monotone skew-product semiflow are obtained. When the minimal set has a continuous separation the results are given in terms of the principal spectrum. In the case that the semiflow is generated by the solutions of a family of non-autonomous differential equations of ordinary, delay or parabolic type, the former results are strongly improved. A method of calculus of the upper Lyapunov exponent of the minimal set is also determined.

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

“…We now recall the notion of uniform persistence for a monotone skew-product semiflow based on the properties of the order, in the region situated strongly above a compact τ -invariant set K, as already defined in Novo et al [36], and then introduce the concept of strict persistence. When K = Ω × {0} our definition of uniform persistence agrees with Definition 3.1 of uniform persistence given in Mierczyński et al [31] or with Definition 3.1 of uniform (strong) ρ-persistence in Smith and Thieme [49] for an adequate ρ depending on the space X.…”

“…The reader is referred to [36] for further details. We are now in a position to state the main result in this section.…”

“…The objective of this paper is to continue the study of the dynamical theory of persistence in monotone skew-product semiflows generated by non-autonomous differential equations, initiated in Novo et al [36], as well as to show the applicability of this theory in the description of some mathematical models widely investigated in the literature, which are locally cooperative in small regions of the phase space. In particular we give a complete characterization of persistence for non-autonomous n-dimensional Nicholson systems with a patchy structure, which are able to model the temporal changes of the environment.…”

“…The papers by Faria and Röst [12], Freedman and Ruan [15], Garay and Hofbauer [16], Hetzer and Shen [20], Hirsch et al [21], Langa et al [26], Magal and Zhao [28], Mierczyński and Shen [30], Mierczyński et al [31], Novo et al [36], Salceanu and Smith [43], Schreiber [44,45], Thieme [51,52], Wang and Zhao [54], and references therein, provide a long but not complete list of works on this topic.…”

“…In [36] Novo et al investigate the presence of uniform persistence in the areas situated strongly above (or below) a minimal set K of the phase space of a nonautonomous family of cooperative differential equations. It is proved that after a convenient permutation of the variables, the coefficient matrix (or matrices in the delay case) of the linearized semiflow over K has a block lower triangular structure such that the linear semiflows generated by the lower-dimensional diagonal blocks admit a continuous separation.…”