How they defeat this Astrophage?

They beat Astrophage by reverse‑engineering why another star system (Tau Ceti) was unaffected, then breeding an Astrophage‑resistant strain of taumoeba and sending that biological countermeasure back to the Solar System. Ryland Grace figures out the key biology, Rocky helps with engineering and experimentation, and together they develop and ship the resistant organisms. Grace then gives up his chance to return to Earth so the cure reaches home and Rocky can save his species.

If you want a step‑by‑step of the experiments or the specific biological mechanism they exploited, I don’t have those details in the supplied context—you can add passages or notes from the book.