Selected mathematical and statistical methods in biology

Some specific methods with applications in biology, ecology and related natural sciences will be presented. The stress will be on less known methods with a wide potential in applications. In the first part, correspondences between two sets will be considered as a tool for investigating various connections among their elements. One of the sets is called the set of objects and another is the set of attributes and the correspondence shows which object posseses which attribute. In this context, an algebraic discipline called formal concept analysis will be presented. Furthermore, binary relations illustrating internal connections within one set will be elaborated. Equivalence relations and ordering relations have a special role in applications. Equivalence relations correspond to partitions of the set into disjoint classes. Classification of organisms is based on partitions of sets and equivalence relations and every taxon represents an equivalence class. Ordering relations often appear in applications as natural orderings on a set of objects. Two elements can be comparable or uncomparable under an ordering. Finally, some new methods connected to fuzzy sets and fuzzy relations will be presented. These concepts allow possibility to erase strong borders between sets, and instead of strong belongingness of elements to sets, grades of belongingness are introduced. In this context fuzzy cluster analysis are explained and software packages in which this method is implemented are listed.