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Our results fully characterise the long-term behavior for a broad class of$\Lambda$-Wright--Fisher processes with frequency-dependent and environmentalselection. In particular, we reveal a rich variety of such parameter-dependentbehaviors and provide explicit criteria to discriminate between them. Thatincludes the situation in which both boundary points are asymptoticallyinaccessible {\DH} a new phenomenon in this context. Such a boundary behaviorhas significant biological implications, because it exhibits a class ofstochastic population models in which selection alone maintains coexistence. Ifa boundary point is asymptotically accessible, we derive polynomial/exponentialdecay rates for the probability of not being polynomially/exponentially closeto the boundary, depending on some weak/strong integrability conditions.Moreover, we provide a handy representation of the fixation probability.Siegmund duality plays a key role in our proofs. The dual process can besandwiched at the boundary in between two transformed L{\'e}vy processes. Inthis way we relate the boundary behavior of the dual to fluctuation propertiesof the L{\'e}vy processes and shed new light on previously establishedaccessibility conditions. Our method allows us to treat models that so farcould not be analyzed by means of moment or Bernstein duality. This closes anexisting gap in the literature.

$\Lambda$-Wright--Fisher processes with general selection and opposing environmental effects: fixation and coexistence