Kirchhoff Migration, Slow--Fast Reduction, and Volterra--Lyapunov Functions in Spatial Fisheries, as Applications of Jacobian Triangularity on Siphon Faces

Animal movement among fishing grounds, protected areas and habitat fragments is often fast compared with growth and harvesting. We study multi-patch models with fast migration, described by irreducible Kirchhoff matrices. The main results are: (1) invasion exponents at boundary equilibria equal Perron-weighted averages of local growth rates to leading order, with explicit second-order corrections, so invasion signs and local stability transfer from the reduced to the full system (singular perturbation and Perron--Frobenius theory, organised by siphon faces); (2) for Schaefer fisheries, the connected maximum sustainable yield is bounded by the sum of isolated yields, and optimal migration under corridor-rate limits is a convex quadratic program; (3) connecting patches can let a predator persist although it cannot persist on any isolated patch, and saturating predation can destabilize coexistence while invasion remains possible; (4) for several predator stocks sharing one prey, a single Volterra identity decides which stock survives and bounds the catch of the losing stocks, including at every migration speed for balanced migration; (5) an extension to consumers sharing recycled resources, building on the microbial consumer--resource model of Faul, Richard, Van der Meer and Mazza.