<p>In this paper, the structure of the set of positive solutions of a semilinear interface elliptic problem is studied in detail. In this kind of problem, the domain is the union of two different regions, separated by a membrane or interface where it happens an interchange of flux. In each region, we assume a nonlinear reaction term: a sublinear function in one of them, and a logistic function in the other. The combination of both terms, as well as the interface, give an interesting structure to the set of positive solutions. In particular, we will show that the existence of the interface provides the model a cooperative effect that makes populations from both regions coexist in situations where without the interface they would disappear.</p>