math.DS — Dynamical Systems
Convergence Criteria for Measure-Valued Population Dynamics with Mutation and Horizontal Transfer
We study a measure-valued population equation on a compact trait space with positive baseline growth and competition, an additional mutation birth gain, and frequency-dependent horizontal transfer with an antisymmetric net kernel. We give sufficient conditions for convergence to a unique nonzero equilibrium. Separating total mass from trait frequencies yields positive mass bounds. Without mutation, maximal time-averaged fitness on the initial support tends to zero, and the endpoint measure in each joint subsequential limit is supported on the zero set of the limiting averaged fitness. Potential transfer and recipient-independent competition reduce the frequency dynamics to a normalized positive semigroup. For general transfer, a common-reference balance gives an entropy criterion allowing arbitrarily strong circulation. An adjoint-weighted entropy identity includes singular population components and the normalization correction in transfer. Mutation dissipates deviations in weighted composition, while a quadratic-form condition controls the remaining mass deviation; in finite dimensions, mutation can also compensate for negative directions of that form. A separate contraction criterion compares arbitrary trajectories using only coefficient bounds. Invariant resident laws yield invasion indices for recurrent environments. Examples exhibit bistability under complete reset mutation and periodic populations whose time averages are not equilibria. For rank-one mutation, we classify stationary frequencies, prove a critical lower bound ruling out exponential convergence from an initial point mass at a maximizer, and determine the condensate when the initial population assigns positive mass to the maximizing set.