A mathematical framework for the statistical interpretation of biological growth models.

Biological entities are inherently dynamic. As such, various ecological disciplines use mathematical models to describe temporal evolution. Typically, growth curves are modelled as sigmoids, with the evolution modelled by ordinary differential equations.

Modelling ethnogenesis.

One of the fundamental problems of contemporary history is to understand the processes governing the rise and fall of polities. The universality of boom-and-bust dynamics associated with the life-cycle of polities tempts to treat the problem mathematically and thus brings it to the framework of cliodynamics. Here we introduce a mathematical model of evolving polity under assumption that its evolution is associated with interactions of certain groups of people, forming the polity and differing by their psycho-ethic characteristics.

Toxin-mediated competition in weakly motile bacteria.

Many bacterial species produce toxins that inhibit their competitors. We model this phenomenon by extending classic two-species Lotka-Volterra competition in one spatial dimension to incorporate toxin production by one species. Considering solutions comprising two adjacent single-species colonies, we show how the toxin inhibits the susceptible species near the interface between the two colonies.

Scaling of Morphogenetic Patterns.

Mathematical studies of morphogenetic pattern formation are commonly performed by using reaction-diffusion equations that describe the dynamics of morphogen concentration. Various features of the modeled patterns, including their ability to scale, are analyzed to justify constructed models and to understand the processes responsible for these features in nature. In this chapter, we introduce a method for evaluation of scaling for patterns arising in mathematical models and demonstrate its use by applying it to a set of different models.

Dynamic principle for ensemble control tools.

Dynamical equations describing physical systems in contact with a thermal bath are commonly extended by mathematical tools called "thermostats." These tools are designed for sampling ensembles in statistical mechanics. Here we propose a dynamic principle underlying a range of thermostats which is derived using fundamental laws of statistical physics and ensures invariance of the canonical measure.