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Once a site is occupied, there is no breeding from outside the patch and the probability $c(i)$ of success of an intra-patch breeding decreases with the size $i$ of the population in the site. We prove the existence of a critical value $\\lambda_{cr}(\\phi, c, N)$ and a critical value $\\phi_{cr}(\\lambda, c, N)$. We con"},"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":"0811.1279","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-11-08T17:23:04Z","cross_cats_sorted":[],"title_canon_sha256":"86b289d618a7a5c42cb812bd5837e0074a15a4bd8047b143b7dd833145fa3c9f","abstract_canon_sha256":"2aeea1c2c7cd2e1c094301f3429cc50eda763aa362d13d5c9eb637dd6ceba04a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:01.217349Z","signature_b64":"BQ4q6slBpRa6YBP4N9lndhH508BDeSO0dT4AFMIzwemo07XfF12wkt54tLXJI4d9rFdxwAMIuasq3CDj71TnAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8601800d32116f626aaa671369a60e174cf7d9ec8ee5f27d1f2e96796bb7f8b3","last_reissued_at":"2026-05-18T04:02:01.216731Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:01.216731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A self-regulating and patch subdivided population","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniela Bertacchi, Fabio Zucca, Lamia Belhadji","submitted_at":"2008-11-08T17:23:04Z","abstract_excerpt":"We consider an interacting particle process on a graph which, from a macroscopic point of view, looks like $\\Z^d$ and, at a microscopic level, is a complete graph of degree $N$ (called a patch). 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