Complex competitive systems and competitive thermodynamics.

This publication reviews the framework of abstract competition, which is aimed at studying complex systems with competition in their generic form. Although the concept of abstract competition has been derived from a specific field--modelling of mixing in turbulent reacting flows--this concept is, generally, not attached to a specific phenomenon or application. Two classes of competition rules, transitive and intransitive, need to be distinguished. Transitive competitions are shown to be consistent (at least qualitatively) with thermodynamic principles, which allows for introduction of special competitive thermodynamics. Competitive systems can thus be characterized by thermodynamic quantities (such as competitive entropy and competitive potential), which determine that the predominant direction of evolution of the system is directed towards higher competitiveness. There is, however, an important difference: while conventional thermodynamics is constrained by its zeroth law and is fundamentally transitive, the transitivity of competitive thermodynamics depends on the transitivity of the competition rules. The analogy with conventional thermodynamics weakens as competitive systems become more intransitive, while strongly intransitive competitions can display types of behaviour associated with complexity: competitive cooperation and leaping cycles. Results of simulations demonstrating complex behaviour in abstract competitions are presented in the electronic supplementary material.