In-evolution operators in Lotka–Volterra coalgebras

In this paper, we study properties of in-evolution operators in Lotka–Volterra coalgebras. First, we cope with the notion of coalgebra character, and classify characters in special cases as, for instance, those of two-dimensional Lotka–Volterra coalgebras and of normalized duals of Lotka–Volterra algebras. Then, we study the in-evolution operators defined by nonzero characters of Lotka–Volterra coalgebras, focusing on the case when the in-evolution operators are stochastic maps. For such operators, we study the usual equilibrium issues, paying special attention to idempotent in-evolution operators, and the information that their idempotency provides on the structure of Lotka–Volterra coalgebras.