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We consider the σ-finite measure-valued diffusion corresponding to the evolution equation ut = Lu + β(x)u - ƒ(x,u), where ƒ (x,u) = α (x)u2 + ∫∞0 (e-ku-1+ku)n(x,dk) and n is a smooth kernel satisfying an integrability condition. We assume that β,α ∈ Cη(ℝd) with η∈(0,1], and α > 0. Under appropriate spectral theoretical assumptions we prove the existence of the random measure lim e-λct Xt (dx) t↑∞ (with respect to the vague topology), where λc is the principal eigenvalue of L + β on ℝd and it is assumed to be finite and positive, completing a result of Pinsky on the expectation of the rescaled process. Moreover we prove that this limiting random measure is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator L + β. When β is bounded from above, X is finite measure-valued. In this case, under an additional assumption on L + β, we prove the existence of the previous limit with respect to the weak topology. As a particular case, we show that if L corresponds to a positive recurrent diffusion Y and β is a positive constant, then lim e-βt Xt (dx) t↑∞ exists and equals to a nonnegative nondegenerate random multiple of the invariant measure for Y. Taking L = ½ Δ on ℝ and replacing β by δ0 (super-Brownian motion with a single point source), we prove a similar result with λc replaced by ½ and with the deterministic measure e-IxIdx. The proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups.