An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach

We summarise the properties and the fundamental mathematical results associated with basic models which describe coagulation and fragmentation processes in a deterministic manner and in which cluster size is a discrete quantity (an integer multiple of some basic unit size). In particular, we discuss Smoluchowski's equation for aggregation, the Becker-Döring model of simultaneous aggregation and fragmentation, and more general models involving coagulation and fragmentation.

Hexagonal patterns in finite domains

In many mathematical models for pattern formation, a regular hexagonal pattern is stable in an infinite region. However, laboratory and numerical experiments are carried out in finite domains, and this imposes certain constraints on the possible patterns. In finite rectangular domains, it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational. In practice, it is found experimentally that in a rectangular region, patterns of irregular hexagons are often observed. This work analyses the geometry and dynamics of irregular hexagonal patterns. These patterns occur in two different symmetry types, either with a reflection symmetry, involving two wavenumbers, or without symmetry, involving three different wavenumbers. The relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case. It is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls. Numerical simulations of a model partial differential equation are also presented to illustrate the behaviour.

Does the abundance of hoverfly mimics (Syrphidae) depend on the numbers of their hymenopteran models ?

We tested the prediction that, if hoverflies are Batesian mimics, this may extend to behavioral mimicry such that their numerical abundance at each hour of the day (the daily activity pattern) is related to the numbers of their hymenopteran models. After accounting for site, season, microclimatic responses and for general hoverfly abundance at three sites in north-west England, the residual numbers of mimics were significantly correlated positively with their models 9 times out of 17, while 16 out of 17 relationships were positive, itself a highly significant non-random pattern. Several eristaline flies showed significant relationships with honeybees even though some of them mimic wasps or bumblebees, perhaps reflecting an ancestral resemblance to honeybees. There was no evidence that good and poor mimics differed in their daily activity pattern relationships with models. However, the common mimics showed significant activity pattern relationships with their models, but the rarer mimics did not. We conclude that many hoverflies show behavioral mimicry of their hymenopteran models.

Artificial immune systems

The human immune system has numerous properties that make it ripe for exploitation in the computational domain, such as robustness and fault tolerance, and many different algorithms, collectively termed Artificial Immune Systems (AIS), have been inspired by it. Two generations of AIS are currently in use, with the first generation relying on simplified immune models and the second generation utilising interdisciplinary collaboration to develop a deeper understanding of the immune system and hence produce more complex models. Both generations of algorithms have been successfully applied to a variety of problems, including anomaly detection, pattern recognition, optimisation and robotics. In this chapter an overview of AIS is presented, its evolution is discussed, and it is shown that the diversification of the field is linked to the diversity of the immune system itself, leading to a number of algorithms as opposed to one archetypal system. Two case studies are also presented to help provide insight into the mechanisms of AIS; these are the idiotypic network approach and the Dendritic Cell Algorithm.