Multiple scales and geometric structures: additional sources of randomness
In Jagers (2009), it was stated that “when physicists and mathematicians selfconfidently moved on to biology, they equipped themselves with the arsenal of Applied Mathematics [Mathematical Physics] in an attempt to repeat the same success story, as for Physics. At the beginning of the twentieth century Mathematical Biology became almost equivalent to deterministic approaches in terms of differential equations on the level of biological systems or collectives (typically populations).” In this article, I will reflect on how biology and medicine have exploited existing theoretical frameworks and stimulated the development of new technical weaponry. Until the 1970s, biomathematics and biostatistics, though naïve with respect to modern mathematical and computational methods, developed as bridges between biological data and mathematical modelling (see, e.g. Bailey 1975, and references therein). In contrast, the approach taken by mathematicians between the 1970s and the 1980s was often based on reaction–diffusion systems described in terms of deterministic PDE’s. Unfortunately the motivation for the mathematical modelling had shifted towards an approach that was at best discipline (biology) independent and, at worst, inappropriately trans-disciplinary (Capasso et al. 1985). Anyhow, research in biomathematics has played an important role in identifying relevant biological principles that are responsible for biological patterns, such as animal coating (see, e.g. Murray 2003 and references therein). Due to the high mathematical complexity, what is still missing, towards a full biological understanding, is the link between biological phenomena that occur at different scales; this is fundamental for a realistic control of the relevant phenomena, as required in Medicine, and Bioengineering. Clearly, PDE modelling is essentially macroscopic. Though a good source for insight in biological applications, such models smooth out the discrete and individual