Evolution coalgebras

Evolution coalgebras on chicken populations

“…Here we attempt to extend that point of view considered in [13] to chicken populations. Chicken populations were described by Labra, Ladra and Rozikov in [4] (see also [6]) as a simplified example of bisexual populations [5] with only one existing male type.…”

“…On the other hand, although genetic populations not obeying Mendel's laws, but based on the self-replication of their individuals, have been till now mainly described in terms of evolution algebras (we refer the reader to Tian's foundation of evolution algebras given in [16]), it has been recently shown that considering such populations from the coalgebraic viewpoint is also possible [13].…”

“…Such a reduction, from bisexual to chicken populations, is mainly motivated by the complexity of the cubic structure matrices that appear attached to bisexual populations. These structure matrices become much simpler square matrices under the Keeping in mind that cubic matrices arise naturally attached to genetic coalgebras [8] and considering chicken populations as a preliminary stage towards the coalgebraic representation of bisexual populations, we recover here the definition of evolution coalgebra of a chicken population, already sketched in [13], to deepen into their main algebraic properties. Knowing which the main coalgebraic properties of chicken populations are, is also the first stage to later consider when such coalgebras can be endowed with genetic realization [17] and therefore to study their connection to Markov processes, as done in [9] for those coalgebras arising from Mendelian populations.…”

“…After this introductory section, the second section recalls the notion of evolution algebra of the chicken population (for short, EACP) [4,6] and briefly summarizes the main results known for EACPs. The third section focuses first on general coalgebras [1,15] to recall then some already known coalgebraic structures with genetic significance, as for instance, coalgebras with genetic realization introduced by Tian and Li [17] and evolution coalgebras introduced in [13]. In the fourth section, we recover the notion of evolution coalgebra of a chicken population (for short ECCP) sketched in [13], to study then the algebraic properties of ECCPs.…”

“…The third section focuses first on general coalgebras [1,15] to recall then some already known coalgebraic structures with genetic significance, as for instance, coalgebras with genetic realization introduced by Tian and Li [17] and evolution coalgebras introduced in [13]. In the fourth section, we recover the notion of evolution coalgebra of a chicken population (for short ECCP) sketched in [13], to study then the algebraic properties of ECCPs. We also settle the relationship between ECCPs and EACPs through their dual structures.…”