Logarithmic corrections and interacting fronts in shifting environments and competition systems

We investigate refined large-time propagation and front interaction in a two-species Lotka–Volterra competition–diffusion system on the real line. When the faster competitor spreads into unoccupied habitat, its invasion front generates a moving heterogeneous environment for the trailing competitor, whose front position may be influenced by the leading front’s Bramson delay. We first study a scalar Fisher–KPP equation in a shifting environment. Under weighted integrability assumptions on the growth profile, we determine the logarithmic front corrections up to a bounded error and identify the corresponding limiting wave profiles, thereby explicitly demonstrating how the logarithmic displacement of the environment modifies the classical Bramson correction. The resulting asymptotics distinguish locally pulled propagation, nonlocal pulling, and two critical transition regimes. In the forced-wave regime, the invasion front remains locked to the moving environment, and the dynamics select a unique stationary profile from a one-parameter family. In the nonlocal pulling regime, the favourable habitat ahead accelerates the invasion front even though the front lags behind the environmental shift. At the threshold of nonlocal pulling, the interaction modifies only the logarithmic correction and, at a further critical parameter value, produces an additional $\ln \ln t$ term. We then apply these results to the competition system. In the nonlocally pulled regime, we determine the logarithmic displacement of the trailing invasion front, explicitly quantifying the transmission of the leading species’ Bramson delay, and prove convergence to the corresponding travelling-wave profile. Finally, we establish a Liouville-type rigidity principle for a class of monotone parabolic systems. This yields uniqueness of monostable travelling waves in both competition regimes.