Appendix S1. Matlab code for calculating transient elsasticities.

Appendix S2. Stage projection matricies for Cynoglossum from one native (top panel) and one introduced (bottom panel) population.

Appendix S3, Fig. S1. dominant and subdominant eigenvalues for Cynoglossum in the complex plane (top panel) native range, with and without insects and (bottom panel) introduced range, with and without herbivores.

Appendix S3, Fig. S2. dominant and subdominant eigenvalues for Calathea  in the complex plane (a) natural insects (b) without Eurybia, with ants, and (c) with Eurybia, without ants.

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