Mathematics and biology: a Kantian view on the history of pattern formation theory

Driesch’s statement, made around 1900, that the physics and chemistry of his day were unable to explain self-regulation during embryogenesis was correct and could be extended until the year 1972. The emergence of theories of self-organisation required progress in several areas including chemistry, physics, computing and cybernetics. Two parallel lines of development can be distinguished which both culminated in the early 1970s. Firstly, physicochemical theories of self-organisation arose from theoretical (Lotka 1910–1920) and experimental work (Bray 1920; Belousov 1951) on chemical oscillations. However, this research area gained broader acceptance only after thermodynamics was extended to systems far from equilibrium (1922–1967) and the mechanism of the prime example for a chemical oscillator, the Belousov–Zhabotinski reaction, was deciphered in the early 1970s. Secondly, biological theories of self-organisation were rooted in the intellectual environment of artificial intelligence and cybernetics. Turing wrote his The chemical basis of morphogenesis (1952) after working on the construction of one of the first electronic computers. Likewise, Gierer and Meinhardt’s theory of local activation and lateral inhibition (1972) was influenced by ideas from cybernetics. The Gierer–Meinhardt theory provided an explanation for the first time of both spontaneous formation of spatial order and of self-regulation that proved to be extremely successful in elucidating a wide range of patterning processes. With the advent of developmental genetics in the 1980s, detailed molecular and functional data became available for complex developmental processes, allowing a new generation of data-driven theoretical approaches. Three examples of such approaches will be discussed. The successes and limitations of mathematical pattern formation theory throughout its history suggest a picture of the organism, which has structural similarity to views of the organic world held by the philosopher Immanuel Kant at the end of the eighteenth century

Mathematical models for Isoptera (Insecta) mound growth

In this research we proposed two mathematical models for Isoptera mound growth derived from the Von Bertalanffy growth curve, one appropriated for Nasutitermes coxipoensis, and a more general formulation. The mean height and the mean diameter of ten small colonies were measured each month for twelve months, from April, 1995 to April, 1996. Through these data, the monthly volumes were calculated for each of them. Then the growth in height and in volume was estimated and the models proposed

Progress in modelling herring populations:An individual-based model of growth

tock assessment may gain from taking into account individual variations in growth, because size is a key predictor of survival and reproduction. In trying to understand patterns in empirical observations, a major challenge is to model the changes in the size dis- tribution of a cohort with age. We introduce an individual-based growth model that is founded on the use of a stochastic class of processes called subordinators. This modelling approach has several desirable features, because it (i) can take account of both indi- vidual and environmental sources of random variations, (ii) has the property of letting size increase monotonically, and (iii) ensures that the mean size-at-age follows the widely accepted von Bertalanffy equation. The parameterization of the model is tested on two Atlantic herring (Clupea harengus) datasets collected from the stocks of North Sea autumn spawners (ICES Divisions IVa, IVb, and IVc) and western Baltic spring spawners (ICES Subarea III). The size distributions obtained from the subordinator model largely match the observed size distributions, suggesting that this approach might be successfully implemented to support the assessment of commercial fish stocks, such as when modelling of size variability is required