Fitness is asymptotically irrelevant under uniform competition in phenotype-structured populations
We prove that, in phenotype-structured populations under uniform competition, the long-term phenotypic distribution does not depend on the proliferation rates: it is the stationary distribution of the switching dynamics alone. The density u(x,t) of cells with phenotype x evolves by advection and diffusion in phenotype space (switching) and by proliferation at a phenotype-dependent rate r(x,t), modulated by a factor g(U) common to all phenotypes, where U is the total population. This extends to continuous phenotypes a recent result for compartmental models. On a bounded interval with no-flux boundary conditions, the total population converges monotonically to the carrying capacity U*, the density converges in L1 to U*psi, where psi is the stationary density of the switching dynamics, and the nonlinear model is asymptotically equivalent to the linear one; by an explicit estimate, the imprint of fitness decays at the spectral gap of the switching dynamics. On the whole real line we prove the same convergence under two natural conditions, confinement and no escape to infinity, covering Ornstein-Uhlenbeck dynamics, multi-well landscapes, and heavy-tailed densities. If g'(U*) < 0 and psi satisfies a Poincare inequality, convergence is exponential in a weighted L2 norm, while no uniform L1 rate holds for Ornstein-Uhlenbeck dynamics. The proof follows the compartmental one: under uniform competition the reaction term has one sign, so its size is the growth rate of the total population, integrable in time, and the switching dynamics contracts the zero-mass perturbations it produces. Since it uses only the Markov semigroup generated by switching, the proof extends to general mechanisms; we apply it to nonlocal switching by jumps and to diffusions with non-gradient drift in bounded domains of R^d, whose stationary density is not of Boltzmann form and carries a circulating flux.