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

Abstract

Boundedness, persistence and stability properties are considered for a class of nonlinear, possibly infinite-dimensional, forced difference equations which arise in a number of ecological and biological contexts. The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes), disturbances induced by seasonal or environmental variation, or migration. We provide sufficient conditions under which the states of these models are bounded and persistent uniformly with respect to the forcing terms. Under mild assumptions, the models under consideration naturally admit two equilibria when unforced: the origin and a unique non-zero equilibrium. We present sufficient conditions for the non-zero equilibrium to be stable in a sense which is strongly inspired by the input-to-state stability concept well-known in mathematical control theory. In particular, our stability concept incorporates the impact of potentially persistent forcing. Since the underlying state-space may be infinite dimensional, our framework enables treatment of so-called integral projection models (IPMs). The theory is applied to a number of examples from population dynamics.

Persistence and Stability for a Class of Forced Positive Nonlinear Delay-Differential Systems

“…Of particular interest will be d * -persistence when d = c. We now introduce two assumptions on the nonlinearity f . Obviously, the interpretation of (N1) and (N2) depends on the particular context, but we refer the reader to [14,Sect. 3] for more discussion of these types of conditions for a class of ecological population models.…”

A switching feedback control approach for persistence of managed resources

“…Assumptions (NL1) and (NL2) together entail that solutions of the system of nonlinear difference equations (2.9) for initial condition x 0 ∈ R n + are nonnegative for each h ∈ q. We note that if g h is bounded for every h ∈ q, then the affine linear bound in (NL2) is satisfied, and the conjunction of (NL1) and (NL2) entails that solutions of (2.9) are bounded by [14,Theorem 4.4,statement (a)]. The assumption that g h (0) = 0 implies that (x, s) = (0, s 0 ) is a constant solution of (2.4), for any s 0 > 0.…”

Stabilization by Adaptive Feedback Control for Positive Difference Equations with Applications in Pest Management

“…Systems of positive Lur'e difference equations have recently been proposed and considered as models in ecology in, for example, [9,11,14,15,39,42,44]. Briefly, as Lur'e difference equations contain both linear and nonlinear components, they are often an appropriate framework for modelling density-independent (that is, linear) and density-dependent (that is, non-linear) vital-or transition-rates.…”

“…The inclusion of density-dependence permits modelling Allee [8], competition or crowding effects. Consequently, (1.1) admits a range of realistic and non-trivial dynamic behaviour including boundedness of solutions [15], attractive zero and non-zero equilibria, as well as exhibiting fluctuating and even chaotic solutions. Another appealing facet is that Lur'e systems are reasonably well-understood mathematically and amenable to analysis.…”