We investigate the transport equationwith suitable boundary conditions through an equivalent integral equation. Assuming the incoming fluxes, the internal source term f(x,μ), the cross section c(x) and the parameter ξ to be nonnegative, we prove the existence of a unique dominant eigenvalue ξ=ξ0(τ) for which the homogeneous problem has a positive solution (critical case), the existence of a unique positive solution for ξ < ξ0(τ) (non-critical case), and the absence of positive solutions for ξ > ξ0(τ) (supercritical case). We show ξ0(τ) to decrease continuously from ∞ to some ξ0(∞)>0 whenever ξ increases from 0 to ∞ (monotonicity). The results are obtained by studying an operator that leaves invariant the cone of nonnegative functions in L∞(0,τ).