Patterns in Systems with Hysteresis
Pavel Gurevich, Free University of Berlin

In the talk, we illustrate pattern-formation mechanisms based on the so-called hysteresis phenomenon. Generally speaking, hysteresis refers to systems whose behavior depends not only on the current state of the system but also on the prehistory in a specific way. We give three examples.

First, we explain in more detail what hysteresis is and show how one can use it to automatically control temperature in chemical reactors. We will show that one can achieve periodic change of temperature around the desired one. However, complications can arise, e.g., a large number of switchings within short periods of time.

Our next example describes a colony of bacteria growing in response to a diffusing front of nutrient and forming a pattern of concentric circles. The main feature here is that the cell’s growth has a hysteretic dependence on the amount of nutrient and acid present.
The last example concerns developmental biology. We consider a mathematical model which might explain a mechanism of formation of a new head in hydra (fresh-water polyp) as a result of tissue transplantation. The hypothesis is that the nondiffusive cells differentiate according to “positional information”, which is (indirectly) related to the number of bound receptors of the cells. The receptors get bound by a special diffusive substance - so-called ligands. The production of ligands is, in turn, triggered by bound receptors and also depends on the number of ligands themselves according to the hysteresis law.

The results presented are obtained in collaboration with Willi Jaeger, Anna Marciniak-Czochra, and Sergey Tikhomirov.

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