{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:DJGXNB2FB24MNECKKJD6BVNYT6","short_pith_number":"pith:DJGXNB2F","canonical_record":{"source":{"id":"1612.05774","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-17T14:26:17Z","cross_cats_sorted":[],"title_canon_sha256":"d29992623f8d7f5553c5b67b4d529b25270ab777825bf965bd582c8b462f091f","abstract_canon_sha256":"38b343f054ae58a0609c2eef100a1e8e8119daa45848da2af714cb3b0df9951d"},"schema_version":"1.0"},"canonical_sha256":"1a4d7687450eb8c6904a5247e0d5b89fa9c9902a834b3e8c81786cc34776febe","source":{"kind":"arxiv","id":"1612.05774","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.05774","created_at":"2026-05-18T00:37:57Z"},{"alias_kind":"arxiv_version","alias_value":"1612.05774v3","created_at":"2026-05-18T00:37:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.05774","created_at":"2026-05-18T00:37:57Z"},{"alias_kind":"pith_short_12","alias_value":"DJGXNB2FB24M","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DJGXNB2FB24MNECK","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DJGXNB2F","created_at":"2026-05-18T12:30:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:DJGXNB2FB24MNECKKJD6BVNYT6","target":"record","payload":{"canonical_record":{"source":{"id":"1612.05774","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-17T14:26:17Z","cross_cats_sorted":[],"title_canon_sha256":"d29992623f8d7f5553c5b67b4d529b25270ab777825bf965bd582c8b462f091f","abstract_canon_sha256":"38b343f054ae58a0609c2eef100a1e8e8119daa45848da2af714cb3b0df9951d"},"schema_version":"1.0"},"canonical_sha256":"1a4d7687450eb8c6904a5247e0d5b89fa9c9902a834b3e8c81786cc34776febe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:37:57.567060Z","signature_b64":"eNfP+tt/q8ZIuRwbUbsMDQg4ieTKWV/XhHSSez1iB+/WdYE4qxDRjj4SSOhyo4/9GcWg73yFICZxEFExp7aJAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a4d7687450eb8c6904a5247e0d5b89fa9c9902a834b3e8c81786cc34776febe","last_reissued_at":"2026-05-18T00:37:57.566440Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:37:57.566440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.05774","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:37:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6dXb2ngfr9f8bZ4zLvsBDUdClNqY9e5RYpqXFCdHY+hQK9WmwXMXJ6NRU5v8B5HkHuoJV5iUJk0ahpC4HJH/Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T07:58:25.940060Z"},"content_sha256":"86bf871d73cd4b30231af1bb0a6409bd8938eef7fae790127b5cdd17cddf7b7b","schema_version":"1.0","event_id":"sha256:86bf871d73cd4b30231af1bb0a6409bd8938eef7fae790127b5cdd17cddf7b7b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:DJGXNB2FB24MNECKKJD6BVNYT6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Non-cooperative Fisher--KPP systems: traveling waves and long-time behavior","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"L\\'eo Girardin (LJLL)","submitted_at":"2016-12-17T14:26:17Z","abstract_excerpt":"This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation. These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem. Non-cooperative KPP systems can model various phenomena where the followin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05774","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:37:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Te4EcclpGw5L345Pzp0UZtT80EJmRtbsYIip7rvODijBpXK9XcSdXbn/d0awMLC2/3AZ0/w/ihMHiwPdcGiwAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T07:58:25.940664Z"},"content_sha256":"469c6b402fe3ea9eae251a3893e28c10e66dce10eed34fdb687e2970bceccc27","schema_version":"1.0","event_id":"sha256:469c6b402fe3ea9eae251a3893e28c10e66dce10eed34fdb687e2970bceccc27"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DJGXNB2FB24MNECKKJD6BVNYT6/bundle.json","state_url":"https://pith.science/pith/DJGXNB2FB24MNECKKJD6BVNYT6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DJGXNB2FB24MNECKKJD6BVNYT6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-01T07:58:25Z","links":{"resolver":"https://pith.science/pith/DJGXNB2FB24MNECKKJD6BVNYT6","bundle":"https://pith.science/pith/DJGXNB2FB24MNECKKJD6BVNYT6/bundle.json","state":"https://pith.science/pith/DJGXNB2FB24MNECKKJD6BVNYT6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DJGXNB2FB24MNECKKJD6BVNYT6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:DJGXNB2FB24MNECKKJD6BVNYT6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"38b343f054ae58a0609c2eef100a1e8e8119daa45848da2af714cb3b0df9951d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-17T14:26:17Z","title_canon_sha256":"d29992623f8d7f5553c5b67b4d529b25270ab777825bf965bd582c8b462f091f"},"schema_version":"1.0","source":{"id":"1612.05774","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.05774","created_at":"2026-05-18T00:37:57Z"},{"alias_kind":"arxiv_version","alias_value":"1612.05774v3","created_at":"2026-05-18T00:37:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.05774","created_at":"2026-05-18T00:37:57Z"},{"alias_kind":"pith_short_12","alias_value":"DJGXNB2FB24M","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DJGXNB2FB24MNECK","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DJGXNB2F","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:469c6b402fe3ea9eae251a3893e28c10e66dce10eed34fdb687e2970bceccc27","target":"graph","created_at":"2026-05-18T00:37:57Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation. These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem. Non-cooperative KPP systems can model various phenomena where the followin","authors_text":"L\\'eo Girardin (LJLL)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-17T14:26:17Z","title":"Non-cooperative Fisher--KPP systems: traveling waves and long-time behavior"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05774","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:86bf871d73cd4b30231af1bb0a6409bd8938eef7fae790127b5cdd17cddf7b7b","target":"record","created_at":"2026-05-18T00:37:57Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"38b343f054ae58a0609c2eef100a1e8e8119daa45848da2af714cb3b0df9951d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-17T14:26:17Z","title_canon_sha256":"d29992623f8d7f5553c5b67b4d529b25270ab777825bf965bd582c8b462f091f"},"schema_version":"1.0","source":{"id":"1612.05774","kind":"arxiv","version":3}},"canonical_sha256":"1a4d7687450eb8c6904a5247e0d5b89fa9c9902a834b3e8c81786cc34776febe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1a4d7687450eb8c6904a5247e0d5b89fa9c9902a834b3e8c81786cc34776febe","first_computed_at":"2026-05-18T00:37:57.566440Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:37:57.566440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eNfP+tt/q8ZIuRwbUbsMDQg4ieTKWV/XhHSSez1iB+/WdYE4qxDRjj4SSOhyo4/9GcWg73yFICZxEFExp7aJAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:37:57.567060Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.05774","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86bf871d73cd4b30231af1bb0a6409bd8938eef7fae790127b5cdd17cddf7b7b","sha256:469c6b402fe3ea9eae251a3893e28c10e66dce10eed34fdb687e2970bceccc27"],"state_sha256":"b84d623b29378b2306131559d73691c22f93b7ab3f79b92999dbd0b9bf1f517e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZCzJsTSp+HpHqnokTU+YDy+bHqMcdkU8WdmssUco1Bi1bVdEGdirJ9JUtBoNXaQOD+GdviHJa6q/xuED7nF4Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T07:58:25.943907Z","bundle_sha256":"f1158dd8a276bdbe59d221727bdd291a8789f93ee1edf885c0158a69473842ac"}}