{"paper":{"title":"Populations with interaction and environmental dependence: from few, (almost) independent, members into deterministic evolution of high densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"F.C. Klebaner, P. Chigansky, P. Jagers","submitted_at":"2018-06-09T16:56:39Z","abstract_excerpt":"Many populations, e.g. of cells, bacteria, viruses, or replicating DNA molecules, start small, from a few individuals, and grow large into a noticeable fraction of the environmental carrying capacity $K$. Typically, the elements of the initiating, sparse set will not be hampering each other and their number will grow from $Z_0=z_0$ in a branching process or Malthusian like, roughly exponential fashion, $Z_t \\sim a^tW$, where $Z_t$ is the size at discrete time $t\\to\\infty$, $a>1$ is the offspring mean per individual (at the low starting density of elements, and large $K$), and $W$ a sum of $z_0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.03506","kind":"arxiv","version":1},"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"}