Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents

Abstract

This paper extends the work of Salceanu and Smith [12, 13] where Lyapunov exponents were used to obtain conditions for uniform persistence ina class of dissipative discrete-time dynamical systems on the positive orthant of R(m), generated by maps. Here a united approach is taken, for both discrete and continuous time, and the dissipativity assumption is relaxed. Sufficient conditions are given for compact subsets of an invariant part of the boundary of R(m+) to be robust uniform weak repellers. These conditions require Lyapunov exponents be positive on such sets. It is shown how this leads to robust uniform persistence. The results apply to the investigation of robust uniform persistence of the disease in host populations, as shown in an application.

Competitive exclusion and coexistence in ann-species Ricker model

“…This follows from Theorem 3.2 in [9], because the assumptions (H1) − (H3), as well as condition (20) in [9] are satisfied with the choice of B as above. For simplicity, we have formulated all persistence results in this paper in the form of uniform persistence.…”

“…The uniform persistence of the total population (as given in the above lemma), as well as all the persistence results given in Section 2.2 are, in fact, robust (the persistence is uniform with respect to small changes in parameters; see, for example, [4,9] for more details on this concept). This follows from Theorem 3.2 in [9], because the assumptions (H1) − (H3), as well as condition (20) in [9] are satisfied with the choice of B as above.…”

“…Although, for our purposes, a weaker version of this lemma would suffice, we state it here in a more general form. The set-up and idea of proof follows [9,10]. Thus, consider a difference equation of the form…”

Robust uniform persistence in discrete and continuous nonautonomous systems

“…Let φ(t, z) be the solution semiflow generated by (1.1), for t = n ∈ Z + (Z + denotes the set of non-negative integers), or by (1.2), for t ∈ R + (R + denotes the set of non-negative real numbers). It was shown in [14] that, if there exists a closed set B that absorbs all trajectories (corresponding to a fixed ξ = ξ 0 ) and whose restriction to an arbitrary neighborhood of the set X = {z = (x, y) ∈ R p + × R q + | y = 0} is compact, and if all Lyapunov exponents λ(z, η) corresponding to z in Ω(M) (the union of omega limit sets of points in M = B ∩ X ) and to nonnegative unit vectors η in R q + , are positive, then M is a robust uniform weak repeller (∃ ε > 0 and Ξ a neighborhood of ξ 0 such that lim sup t→∞ d(φ(t, z, ξ ), M) > ε, ∀z ∈ R p+q + \ X , ξ ∈ Ξ ) and the system is robustly uniformly persistent (∃ ε > 0 and Ξ a neighborhood of ξ 0 such that lim inf t→∞ d(φ(t, z, ξ ), X ) > ε, z ∈ R p+q + \ X , ξ ∈ Ξ ).…”

“…In this paper we build on the previous work in [14], where the author used Lyapunov exponents to obtain sufficient conditions for compact subsets of the boundary of the positive cone R m + to be robust uniform weak repellers and then used this to obtain robust persistence results for systems of difference and differential equations of the form …”