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The latter, in form of so-called integral projection models, are often employed to model populations with a continuous structure or stage-parameter, in which case the state space X of ( 1.1) is naturally a space of functions. In fact, the case \(\dim \,X = \infty \) is included in our development, and therefore, ( 1.1) can be used to model certain partial-difference and integro-difference equations. Since we seek to model a variety of structured populations, see for instance Caswell ( 2001), Cushing ( 1998), the dimension of the state-space X is not constrained. Stability is, of course, a fundamental consideration in all fields where dynamic modelling occurs and pertains to both the qualitative and quantitative long-term behaviour of solutions. Informally, boundedness and persistence are opposite properties, concerned with populations becoming neither too big, or too small, respectively. We comment that several persistence concepts and their properties have been studied in the literature (Franco et al. The notion of persistence relates to the survival of a population (or survival of certain stage-classes) and is relevant, for example, in the context of providing lower bounds for predicted yield in agriculture and horticulture. Boundedness is a necessary property of a sensible biological model and, moreover, is a key concept when seeking to understand the potential effects of an invasive species or mutant in a novel environment, see, for example, Eager et al. There is clear biological relevance for these three properties depending on the context. Our motivation for studying ( 1.1), and particularly the properties of boundedness, persistence and stability, is the potential for applications to population biology and theoretical ecology where models of the form ( 1.1) often arise. Depending on the context, u and v are interpreted as a control, input or disturbance. Here \(b \in X\) and \(c^* \in X^*\), the dual space of X, and f is a (typically nonlinear) real-valued function that may depend on a variable u which, along with v, denotes forcing. The difference equation ( 1.1) comprises a linear component Ax, where A is a bounded linear operator on the state-space X, assumed to be a Banach space, and a nonlinear component \(b f(u,c^* x)\). Where \(x^\nabla \) denotes the image of x under the left shift operator, that is, \(x^\nabla (t) = x(t+1)\) for all nonnegative integers t. The theory is applied to a number of examples from population dynamics. Since the underlying state-space may be infinite dimensional, our framework enables treatment of so-called integral projection models (IPMs). In particular, our stability concept incorporates the impact of potentially persistent forcing. 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. Under mild assumptions, the models under consideration naturally admit two equilibria when unforced: the origin and a unique non-zero equilibrium. We provide sufficient conditions under which the states of these models are bounded and persistent uniformly with respect to the forcing terms. 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. 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.