arXiv:2611.14756v1 Announce Type: new Abstract: The multivariate breeder's equation $\Delta\bar z=G\nabla\ln\bar W$ is a widely used discrete-time modified-gradient system describing the evolution of continuous traits in organisms. This system could more conventionally be expressed as $x_{k+1}=x_k+P_k\nabla f(x_k)$. For its continuous-time counterpart $x'=P(t)\nabla f(x)$, it has been proven that an isolated local maximum of $f$ is uniformly stable whenever $P(t)$ is positive semidefinite, as is the case for the "$G$-matrix." Furthermore, the continuous time system is uniformly asymptotically stable when the maximum is an isolated critical point and $\int_0^\infty\lambda_1(P(t))\,dt=\infty$. We show that the literal discrete analog of this result is false: with constant positive definite $P$ the iterates can oscillate or diverge. The result is restored by a single additional hypothesis, a uniform bound on the largest eigenvalue of $P_k$, $\lambda_n$, such that $\lambda_n(P_k)\le 2/L$ where $L$ is a local Lipschitz constant of $\nabla f$. Under this bound the maximum is uniformly stable; under a stricter version together with $\sum_k\lambda_1(P_k)=\infty$, the maximum is uniformly asymptotically stable. Given these hypotheses, a level-set estimate of the basin of attraction carries over. Both bounds are sharp, and each remaining hypothesis is shown to be necessary. Biologically, the bound says that genetic variance in every direction must be small relative to the curvature of log mean fitness; outside this regime the mean phenotype can overshoot a fitness peak as hypothesized by Fisher's geometric model. As further application of our results, the same conditions apply to the mean update of Gaussian evolutionary optimization algorithms, in which the covariance of the search population plays the role of $G$.
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