Implementation Filters and Delay-Budget Instability in Coupled Replicator--Mutator Dynamics

We model an adaptive contest in which two antagonistically coupled populations continually reallocate effort among competing methods, but decisions are not fielded instantly.

Each side has an intended portfolio and a deployed portfolio: intended reallocations follow delayed observations of the opponent, while deployment follows intent through a first-order implementation filter.

Under barycentric balance and uniform exploration, the linearized scalar branches have a characteristic factor in which hard observation and deployment lags enter only through their total sum, whereas implementation rates enter through real filter factors that cannot be absorbed into selection or exploration.

In the strictly antagonistic class, negative spectral branches split into three regimes: weak branches have no positive-frequency crossing, intermediate branches lose stability through a delay-induced Hopf bifurcation, and strong branches are at or beyond the implementation-filter instability margin already at zero hard delay.

This gives an operational delay-budget rule: in the delay-induced window, reducing any hard lag has the same first-order stabilizing leverage at onset; in the filter-induced regime, hard-lag reduction alone cannot restore stability.

Balanced scalar performance observables generically show a mean shift and a second harmonic at twice the compositional frequency, and under strict antagonism the two performance signals are locked in antiphase with fixed amplitude ratio.

For a baseline branch, a finite-dimensional Hopf normal-form calculation gives a negative cubic coefficient, and direct simulations reproduce the predicted threshold, amplitude scaling, and observable signatures.

Motivating applications include cybersecurity and rapid technological countermeasure adaptation.