Generalized Transition Fronts and Spectral Speed Asymptotics in Heterogeneous Cooperative Systems

We study generalized transition fronts for cooperative reaction--diffusion systems with spatially periodic and time-dependent coefficients. We obtain sufficient conditions for the existence of fronts and necessary constraints on their propagation speeds in terms of generalized principal eigenvalues. In the almost-periodic case, we identify the two spectral thresholds and examine the attainment of the critical speed under additional hypotheses on the normalized linear evolution. We then investigate the large-advection limit and construct a time-periodic example showing that a stationary spectral characterization need not extend to time-dependent flows. For symmetric cooperative systems in stationary periodic shears with equal diffusion matrices, we obtain explicit correction terms for the minimal speed in both the small- and large-diffusion limits. The two expansions follow from periodic eigenvalue perturbation, localization near a maximal streamline, and minimization over the exponential weight. We also establish a higher-order large-advection expansion and present numerical illustrations independent of the analytic proofs.