Analytical classification of dispersal effects on total biomass in the presence of a weak Allee effect

Abstract

We investigate the impact of the weak Allee effect on the total biomass of a single species inhabiting a two-patch environment connected by asymmetric dispersal. In the proposed model, the first patch exhibits logistic growth, while the second patch follows a weak Allee threshold. We establish the global existence, nonnegativity, and boundedness of solutions, and prove that the system admits a unique positive equilibrium that is globally asymptotically stable. Analytical results are derived for the asymptotic regime of high dispersal rates using singular perturbation theory, revealing an emergent weak Allee effect at the metapopulation scale. We conduct a complete classification of the parameter space to determine when dispersal increases, decreases, or has a unimodal effect on total equilibrium biomass, and provide geometric and monotonicity analyses of the equilibrium. Numerical simulations illustrate the theoretical trichotomy and highlight the ecological implications of dispersal management in fragmented landscapes. Our results generalize previous studies on logistic models to include the weak Allee effect, offering new insights into the interplay between dispersal intensity, asymmetry, and population persistence.