Semi-global persistence and stability for a class of forced discrete-time population models

Abstract

We consider persistence and stability properties for a class of forced discrete-time difference equations with three defining properties: the solution is constrained to evolve in the non-negative orthant, the forcing acts multiplicatively, and the dynamics are described by so-called Lur'e systems, containing both linear and non-linear terms. Many discrete-time biological models encountered in the literature may be expressed in the form of a Lur'e system and, in this context, the multiplicative forcing may correspond to harvesting, culling or time-varying (such as seasonal) vital rates or environmental conditions. Drawing upon techniques from systems and control theory, and assuming that the forcing is bounded, we provide conditions under which persistence occurs and, further, that a unique non-zero equilibrium is stable with respect to the forcing in a sense which is reminiscent of input-to-state stability, a concept well-known in nonlinear control theory. The theoretical results are illustrated with several examples. In particular, we discuss how our results relate to previous literature on stabilization of chaotic systems by so-called proportional feedback control.

Boundedness, persistence and stability for classes of forced difference equations arising in population ecology

“…We discuss the generalization of Franco et al. (2017), and compare our results to those in Rebarber et al. (2012), Smith and Thieme (2013), Townley et al.…”

“…We provide some comments relating the above result to the persistency theory developed in Franco et al. (2017), and on the assumption (P1).…”

“…We comment that several persistence concepts and their properties have been studied in the literature (Franco et al. 2017; Schreiber 2012; Smith and Thieme 2011, 2013; Wen et al. 2016).…”

“…In the literature, the study of boundedness, persistence and stability properties of (1.1) does not usually include forcing [denoted by the terms u and v in (1.1)], with Franco et al. (2017) being an exception. As is well-known, classical (Lyapunov) stability notions are not applicable when exogenous, and potentially persistent, forcing is present, and forcing can have adverse, extreme and unintuitive effects on nonlinear dynamics, see, for example, Teel and Hespanha (2004).…”

Small-gain stability theorems for positive Lur’e inclusions

“…More generally, for arbitrary g : R + → R + , whenever y * > 0 is such that 21) in particular meaning that the estimates in both (ii) and (iii) do not hold, then x * := (I − A) −1 Bg(y * ) is a non-zero equilibrium of (3.18) and the zero equilibrium of (3.18) cannot then be globally asymptotically or exponentially stable. The papers [49,64] consider attractivity and stability properties of the non-zero equilibrium x * when m = p = 1 and under conditions on A, B, C and g which ensure that y * > 0 in (3.21) is unique.…”