  We introduce an asymmetric noisy voter model to study the joint effect of
immigration and a competition-dispersal tradeoff in the dynamics of two species
competing for space in regular lattices. Individuals of one species can invade
a nearest-neighbor site in the lattice, while individuals of the other species
are able to invade sites at any distance but are less competitive locally,
i.e., they establish with a probability $g \le 1$. The model also accounts for
immigration, modeled as an external noise that may spontaneously replace an
individual at a lattice site by another individual of the other species. This
combination of mechanisms gives rise to a rich variety of outcomes for species
competition, including exclusion of either species, mono-stable coexistence of
both species at different population proportions, and bi-stable coexistence
with proportions of populations that depend on the initial condition.
Remarkably, in the bi-stable phase, the system undergoes a discontinuous
transition as the intensity of immigration overcomes a threshold, leading to a
half loop dynamics associated to a cusp catastrophe, which causes the
irreversible loss of the species with the shortest dispersal range.
