Consider a map T: X--> X, where X is a subset of a Banach space endowed with a partial order which is induced by a cone C. T is called monotone if it preserves this partial order. There is a rich history to understand the asymptotic behavior of the sequences generated by monotone maps. If a further condition on T is imposed, namely that of sublinearity, and if T maps the cone C into itself, then these sequences have exactly one of three possible fates.
In this talk I will present a proof of this trichotomy, which is due to Krause and Ranft in the case C= R^n_+, and to Krause and Nussbaum in the more general infinite dimensional case.
In a companion mathematical biology seminar on 2/27, I will discuss a recently proposed larval dispersal model whose dynamics can be understood in part by applying this trichotomy.