We consider an extension of the Poisson hail model where the service speed is either 0 or ∞ at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process, and on the geometrical properties of the zero speed set.

The $q$-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of its moments in terms of matchings, where $q$ follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case $q=0$ of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier–Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

Decisions on the choice of appropriate tactical and strategic control techniques require an understanding of the seasonal distribution and ecology of the target species. To address this need, data were collected from 1997 to 2000 using crop surveys, field sampling and pheromone trapping to monitor the population trends of Helicoverpa armigera (Hübner) and H. punctigera (Wallengren) in the Riverina region of south-eastern Australia. The data gathered are compared to predictions from the HElicoverpa Armigera and Punctigera Simulation (HEAPS) model, that simulates Helicoverpa population dynamics to assist in identifying the drivers of regional populations and provide a framework to make informed decisions. The results highlight the contrasting seasonal dynamics of H. punctigera and H. armigera within the region. Helicoverpa punctigera populations peak in the spring, driven by large scale spring migration into the region and subsequently decline. Helicoverpa armigera populations, conversely, are characterized by a period of spring recruitment of adults from overwintering pupae within the region, followed by a within-season population increase leading to severe late season pressure. The within-season increase of H. armigera is a result of the succession of crop and non-crop hosts with the main driver being sequentially sown, unsprayed maize. Tactical and strategic management options are discussed in light of these findings.