Complex competitive systems

competition [37] is suggested to study the principles of competition in their most generic form [8–10]. Consider a complex competitive system, which has a large number of autonomous elements engaged in competition with each other. The evolution of a competitive system involves a process of determining a winner and a loser for competition between any two elements of the system. The non-conservative properties (i.e. information) of the loser are lost while the winner duplicates its information into the resource previously occupied by the loser. The duplication process may involve random changes, which are customarily called mutations irrespective of the physical nature of the process. These mutations are predominantly negative or detrimental but can occasionally deliver a positive outcome. Interaction between the winner and loser may also involve redistribution of conservative properties, which is expected to be in favour of the winner (i.e. from the loser to the winner). It is easy to see that abstract competition can be represented by a system of Pope particles, provided conventional conservative mixing is replaced by competitive mixing. In the present work, the terms “elements” and “particles” are used synonymously with “elements” primarily referring to competing components of general nature and “particles” to their computational implementations. We mostly focus on the non-conservative properties, which are most interesting, while limiting our consideration of conservative properties to the particles themselves (i.e. the number of particles is preserved by mixing). We also restrict our analysis to competitive mixing of couples of particles although more complicated schemes may also be considered if needed. Let yp be the set of properties associated with particle p. If particle p appears to be a winner in competition with another particle q, we may write yq ≺ yp. On some occasions, the particles may have the same strength (i.e. yp ' yq) and no winner can be determined or the winner has to be selected randomly. Competitive exchange of information can be illustrated by the following effective reaction involving the wining particle p and the losing particle q yp + yq + E → yp + y′ p, yp yq (2.1) where y′ p = yp + m represents a mutated version of yp, yp is stronger than yq and E indicates existence of an external source of exergy that may be needed by these transformations. The mutations m are expected to be predominantly negative, which means that yp y′ p is much more likely than yp ≺ y′ p. Abstract competition deals with complex competitive systems (CCS), which share with complex adaptive systems (CAS, [6, 7]) their major premises: 1) working of a complex system is not trivially reducible to working of its elements and 2) complex systems of different physical origins should possess some in-depth similarity. CAS and CCS, however, tend to differ in the other respects. The elements of competitive systems compete rather than adapt and tend to move and mix instead of having fixed communication links typical for CAS. The long-standing Darwinian tradition of studying evolutionary systems of high complexity tends to give a higher Phil. Trans. R. Soc. A 2013 371 (1982) 20120244