Homeostatic Turnover, Repair, and the Demographic Form of Haldane's Principle

Haldane's principle relates equilibrium mutation load to deleterious mutational input rather than to selection strength. We study this balance in a homeostatically regulated, damage-structured population where mutation occurs during reproduction, repair restores individuals toward lower-damage states, and nonlinear demographic feedback maintains constant population size. An exact demographic identity shows that load at every time equals excess reproductive effort weighted by mean relative fecundity. Without repair, the equilibrium branch with positive pristine-class frequency is exactly solvable, and along it the load is exactly the mutation probability times the realized per-capita reproductive turnover rate; when the terminal cap is removed, this is precisely the realized deleterious input, for every mutation probability below one. For rare mutation it reduces to the mutation probability times the pristine turnover rate, with the local fitness contrast canceling at leading order. With a finite terminal class the branch instead reaches a closure-dependent boundary collision. Repair modifies the rare-mutation balance by partitioning first-order removal from the first damaged class between selection and restoration, multiplying the leading load by the selective share of that removal. An exact binary solution extends these results to finite mutation pressure: the unrepaired interior equilibrium collides nonhyperbolically with the fully damaged boundary and converges algebraically at the threshold, whereas any positive repair rate restores a unique interior equilibrium for positive mutation; at the no-repair threshold, its displacement from the boundary scales as the square root of the repair rate as repair vanishes. These results separate the familiar competing-transition structure of mutation--selection balance from the specifically homeostatic effects of turnover-dependent mutation input and nonlinear demographic feedback.