The invasion of a novel host species can create a mismatch in host choice and offspring survival (performance) when native parasitoids attempt to exploit the invasive host without being able to circumvent its resistance mechanisms. Invasive hosts can therefore act as evolutionary trap reducing parasitoids' fitness and this may eventually lead to their extinction. Yet, escape from the trap can occur when parasitoids evolve behavioral avoidance or a physiological strategy compatible with the trap host, resulting in either host-range expansion or a complete host-shift. We developed an individual based model to investigate which conditions promote parasitoids to evolve behavioral preference that matches their performance, including host-trap avoidance, and which conditions lead to adaptations to the unsuitable hosts. The model was inspired by solitary endo-parasitoids attacking larval host stages. One important aspect of these conditions was reduced host survival during incompatible interaction, where a failed parasitization attempt by a parasitoid resulted not only in death of her offspring but also in host killing. This non-reproductive host mortality had a strong influence on the likelihood of establishment of novel host-parasitoid relationship, in some cases constraining adaptation to the trap host species. Moreover, our model revealed that host-search efficiency and genetic variation in host-preference play a key role in the likelihood that parasitoids will include the suboptimal host in their host range, or will evolve behavioral avoidance resulting in specialization and host-range conservation, respectively. Hence, invasive species might change the evolutionary trajectory of native parasitoid species, which is important for predicting biocontrol ability of native parasitoids towards novel hosts.