Frequency–Phase Response of a Periodically Forced Two-Species Lotka–Volterra Competition Model

Periodic forcing is a standard feature of ecological competition models, but the response of a coexistence state is often summarized by numerical trajectories or by a high-frequency approximation. We derive two complementary diagnostics for a two-species Lotka–Volterra competition system with sinusoidal forcing of intrinsic growth rates and constant competition coefficients. First, every positive solution with the forcing period has cycle means exactly equal to the equilibrium of the unforced system; this follows from averaging the per-capita log-growth equations and is independent of forcing amplitude, frequency, and phase. Second, for small forcing, the complex population response is given by a resolvent, $(i\omega I-J)^{-1}$, acting on the species-specific forcing vector, where $J$ is the coexistence Jacobian. This formula predicts amplitudes and phase shifts, including the cross-species effects of competition. Numerical integrations for a positive-coexistence parameter set validate the response formula over a broad frequency range and show strong sensitivity of oscillation amplitude to the relative phase of the two growth cycles, even though cycle means remain fixed. The computed periodic orbit is locally attracting, with both Floquet multipliers inside the unit circle. The contribution is a compact frequency–phase response analysis and a mean-invariance result for a classical periodically forced competition model; it is not a claim that periodic Lotka–Volterra competition is new. The simulations are illustrative and are not fitted to empirical population data.