Analysis of a certain class of replicator equations

Motivated by a problem in the evolution of sensory systems where gains obtained by improvements in detection are offset by increased costs, we prove several results about the dynamics of replicator equations with an n × n game matrix of the form: A_ij = a_ib_j - c _i. First, we show that, generically, for this class of game matrix, all equilibria must be on the 1-skeleton of the simplex, and that all interior solutions must limit to the boundary. Second, for the particular ordering, a₁ < a₂ < ⋯ < a_n and b ₁ > b₂ > ⋯ > b_n, which is most natural in the study of the evolution of sensory systems, we show that topological restrictions require a unique local attractor in every face of the simplex. We conjecture that the unique local attractor for the full simplex is, in fact, a global attractor, and prove this for n ≤ 5. In a separate argument supporting the conjecture, we show that there can be no chain recurrent invariant set entirely contained in the 1-skeleton of the simplex. Finally, we discuss the special, non-generic case and give a local description of the dynamics when there is an interior equilibrium.