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We study the following impulsive reaction-diffusion model for any $m\\in \\mathbb Z^+$ \\begin{eqnarray*}\\label{}\n  \\ \\ \\ \\ \\ \\left\\{ \\begin{array}{lcl}\n  u^{(m)}_t = div(A\\nabla u^{(m)}-a u^{(m)}) + f(u^{(m)}) \\quad \\text{for} \\ \\ (x,t)\\in\\Omega\\times (0,1]\n  u^{(m)}(x,0)=g(N_m(x)) \\quad \\text{for} \\ \\ x\\in \\Omega\n  N_{m+1}(x):=u^{(m)}(x,1) \\quad \\text{for} \\ \\ x\\in \\Omega \\end{array}\\right.\n  \\end{eqnarray*} for functions $f,g$, a drift $a$ and a diffusion mat"},"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":"1511.00743","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-02T23:34:54Z","cross_cats_sorted":["q-bio.PE"],"title_canon_sha256":"3b3709b333d23feedb73a450190d06bf3203a3cc3dfbb949c8fbc0dee2f3839a","abstract_canon_sha256":"7a6db963ce126499d72ec702cb740c297f994a50a7a650ec82981d4896d6008a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:04.392672Z","signature_b64":"RXq6I95UP4Pnr5p4zjneLW/3neZYil7tzFFgEllJBE4E7rE5p9wBcoOAWc2Hy5HklI55jE1Kz0S/dq2Gh+GcDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b7ed796eaa0a346df9fae1cb3bda89170cad0de101c0ae12227f1f6bd10214f","last_reissued_at":"2026-05-18T00:58:04.392234Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:04.392234Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On impulsive reaction-diffusion models in higher dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["q-bio.PE"],"primary_cat":"math.AP","authors_text":"Hao Wang, Mark Lewis, Mostafa Fazly","submitted_at":"2015-11-02T23:34:54Z","abstract_excerpt":"Assume that $N_m(x)$ denotes the density of the population at a point $x$ at the beginning of the reproductive season in the $m$th year. We study the following impulsive reaction-diffusion model for any $m\\in \\mathbb Z^+$ \\begin{eqnarray*}\\label{}\n  \\ \\ \\ \\ \\ \\left\\{ \\begin{array}{lcl}\n  u^{(m)}_t = div(A\\nabla u^{(m)}-a u^{(m)}) + f(u^{(m)}) \\quad \\text{for} \\ \\ (x,t)\\in\\Omega\\times (0,1]\n  u^{(m)}(x,0)=g(N_m(x)) \\quad \\text{for} \\ \\ x\\in \\Omega\n  N_{m+1}(x):=u^{(m)}(x,1) \\quad \\text{for} \\ \\ x\\in \\Omega \\end{array}\\right.\n  \\end{eqnarray*} for functions $f,g$, a drift $a$ and a diffusion mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00743","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1511.00743","created_at":"2026-05-18T00:58:04.392306+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.00743v2","created_at":"2026-05-18T00:58:04.392306+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.00743","created_at":"2026-05-18T00:58:04.392306+00:00"},{"alias_kind":"pith_short_12","alias_value":"TN7NPFXKUCRU","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TN7NPFXKUCRUNX47","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TN7NPFXK","created_at":"2026-05-18T12:29:42.218222+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF","json":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF.json","graph_json":"https://pith.science/api/pith-number/TN7NPFXKUCRUNX47VYOLHPNISF/graph.json","events_json":"https://pith.science/api/pith-number/TN7NPFXKUCRUNX47VYOLHPNISF/events.json","paper":"https://pith.science/paper/TN7NPFXK"},"agent_actions":{"view_html":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF","download_json":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF.json","view_paper":"https://pith.science/paper/TN7NPFXK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.00743&json=true","fetch_graph":"https://pith.science/api/pith-number/TN7NPFXKUCRUNX47VYOLHPNISF/graph.json","fetch_events":"https://pith.science/api/pith-number/TN7NPFXKUCRUNX47VYOLHPNISF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF/action/storage_attestation","attest_author":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF/action/author_attestation","sign_citation":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF/action/citation_signature","submit_replication":"https://pith.science/pith/TN7NPFXKUCRUNX47VYOLHPNISF/action/replication_record"}},"created_at":"2026-05-18T00:58:04.392306+00:00","updated_at":"2026-05-18T00:58:04.392306+00:00"}