Chapter 2: Geometry of the Fitness Surface and Trajectory Dynamics of Replicator Systems

Original Abstract

We study the geometry of the mean fitness surface of replicator systems and its relationship to evolutionary trajectory dynamics. Using the symmetric--antisymmetric decomposition of the fitness landscape matrix, we derive an explicit formula for the rate of change of mean fitness and establish necessary conditions for its monotonicity along trajectories. In general, replicator trajectories do not reach the maximum of the fitness surface, even in the presence of a unique asymptotically stable equilibrium. We characterise, in terms of the symmetric and antisymmetric parts of the fitness matrix, the precise conditions under which an equilibrium coincides with a local extremum of the fitness surface. Circulant matrices are identified as a natural and nontrivial class satisfying these conditions. We establish a two-way connection between fitness surface maxima and evolutionarily stable states: evolutionary stability implies a local fitness maximum, and the converse holds under the identified structural conditions. When the unique asymptotically stable equilibrium is a local maximum, it is evolutionarily stable and realises the global maximum of the fitness surface; an unstable equilibrium forces the global maximum to the boundary of the simplex. The framework is extended to general Lotka--Volterra systems, where an analogue of mean fitness is shown to share the same extremal properties. Results are illustrated through six examples spanning autocatalytic and hypercyclic replication, a parametric family exhibiting Andronov--Hopf bifurcation and heteroclinic cycles, and the Eigen quasispecies model.