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Appendix S1. The limit inline image.

Appendix S2. Computer simulation details.

Appendix S3. Tests of accuracy.

Figure C1. Samples of simulated invasion probabilities, PE (y-axis, eq. 8a), of the continuous time example model (eqs. 19, 20).

Figure C2. The average relative error (eq. C1) of the predicted P0 (eq. 13a) (grey shading and contour lines), depicted as a function of the mean fitness advantage inline image (x-axis) and the carrying capacity K (y-axis).

Figure C3. The average relative error (eq. C1) of the predicted P0 (eq. 13a) (grey shading and black contour lines), depicted as a function of the variance (x-axis) and correlation time (y-axis) of the external environment (ɛt, see eqs. 19, 20).

Figure C4. Same as Figure C1, but for the discrete time model (eq. 21).

Figure C5. The average relative error (eq. C1) of the predicted P0 of the discrete time model (eq. 21), depicted as a function of the mean fitness advantage inline image (x-axis) and the carrying capacity K (y-axis).

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