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The system starts in the state concentrated on type 1, the state of low fitness. We investigate this system for $N \\to \\infty$ on the original and large time scales.\n  We show that for some $\\alpha \\in (0,s)$ at times $\\alpha^{-1} \\log N+t, t \\in \\R, N \\to \\infty$ the emergence of type 2 (positive global type-2 i"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1104.0253","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-04-01T20:41:12Z","cross_cats_sorted":[],"title_canon_sha256":"206d62f30da8968546762c590e4ed4c0a32b9b84b7d665e184170e181360705b","abstract_canon_sha256":"c2c56d0a771201797a68caa3bd80127b3bc3ad9c359d9dd06a9295ce5e8a5dd2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:04.353158Z","signature_b64":"dFtNmejG6yH+ILclu03HlrkTqkpM034y9jlxZw3J4QeOZXdShI/bkYJNKaxpHIQYJkLjxQ1SzcO812AkFOPaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f31d14c225dc27689bd51f35b2cc3ebeb191977d370098a53802d5b117df4c57","last_reissued_at":"2026-05-18T04:25:04.352725Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:04.352725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invasion by rare mutants in a spatial two-type Fisher-Wright system with selection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Andreas Greven, Donald A. 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