{"query": "Populations — logistic growth", "count": 20, "results": [{"id": "card_ecology_populations", "title": "Populations — logistic growth", "shelf": "codex", "surface": "secular", "snippet": "A population grows fastest at half its carrying capacity (sealed); logistic growth and predator-prey cycles are the arithmetic of number in the wild.", "authority_tier": "engine_derived", "source": "Narrow Highway — ecology", "readable": false, "generated": false}, {"id": "card_theory_sabermetrics_sports_analytics", "title": "Sabermetrics / sports analytics (Pythagorean expectation, Elo)", "shelf": "theories", "surface": "secular", "snippet": "Sabermetrics / sports analytics (Pythagorean expectation, Elo) — an engine domain that can touch it: sports_analytics. Calibration: seals — expected win ratios and Elo updates compute. Bill James's Py", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_population_ecology", "title": "Population ecology (Lotka–Volterra, carrying capacity)", "shelf": "theories", "surface": "secular", "snippet": "Population ecology (Lotka–Volterra, carrying capacity) — an engine domain that can touch it: ecology. Calibration: seals — logistic and predator-prey models compute. Unlimited resources give exponenti", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_bridge_master_sigmoid", "title": "Master equation: y = 1 / (1 + e^(-x))", "shelf": "bridges", "surface": "secular", "snippet": "the S-curve — a soft switch between two states; a near-miss with logistic growth that IS a calculation. ONE equation, 7 domains, connected by a change of variable: economics [x = k(t - t0): the logist", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_calc_fermi_dirac", "title": "Fermi-Dirac occupation", "shelf": "calculations", "surface": "secular", "snippet": "Fermi-Dirac occupation — thermodynamics. Formula: n = 1/(1 + e^((E-mu)/kT)). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — the fraction of quantum states filled — the sigmoid, from Pauli exclusion.", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_titration_curve", "title": "Titration curve (Henderson-Hasselbalch)", "shelf": "calculations", "surface": "secular", "snippet": "Titration curve (Henderson-Hasselbalch) — chemistry. Formula: f = 1/(1 + 10^(pKa - pH)). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — the fraction deprotonated is a sigmoid in pH — a two-state occ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_replicator_payoff", "title": "Replicator dynamics (game)", "shelf": "calculations", "surface": "secular", "snippet": "Replicator dynamics (game) — game_theory. Formula: dx/dt = x (payoff - avg). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — a strategy spreads when its payoff beats the average — the same equation, ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_v_ecology_logistic_growth", "title": "Logistic growth", "shelf": "calculations", "surface": "secular", "snippet": "Logistic growth — ecology. Formula: N = K/(1+((K-N0)/N0) e^-rt). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — engine verifier ecology.logistic_growth — the deterministic check the concordance runs", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_calc_carrying_capacity", "title": "Logistic population (carrying capacity)", "shelf": "calculations", "surface": "secular", "snippet": "Logistic population (carrying capacity) — ecology. Formula: dP/dt = rP(1 - P/K). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — growth that slows as it nears the limit. Same form, different domain: ", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_src_openstax_concepts_biology_19_2_population_growth_and_regulation_f8657159", "title": "19.2 Population Growth and Regulation — Concepts of Biology", "shelf": "biology", "surface": "secular", "snippet": "19.2\n\nPopulation Growth and Regulation\n\nLearning ObjectivesBy the end of this section, you will be able to:\n\nExplain the characteristics of and differences between exponential and logistic growth patt", "authority_tier": "reference", "source": "OpenStax: Concepts of Biology (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_calculus_volume_2_4_4_the_logistic_equation_78cff0a4", "title": "4.4 The Logistic Equation — Calculus Volume 2", "shelf": "mathematics", "surface": "secular", "snippet": "4.4\n\nThe Logistic Equation\n\nLearning Objectives\n\n4.4.1\nDescribe the concept of environmental carrying capacity in the logistic model of population growth.\n\n4.4.2\nDraw a direction field for a logistic ", "authority_tier": "reference", "source": "OpenStax: Calculus Volume 2 (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_concepts_biology_chapter_summary_cc08f57e", "title": "Chapter Summary — Concepts of Biology", "shelf": "biology", "surface": "secular", "snippet": "Chapter Summary\n\n19.1\n\nPopulation Demographics and Dynamics\n\nPopulations are individuals of a species that live in a particular habitat. Ecologists measure characteristics of populations: size, densit", "authority_tier": "reference", "source": "OpenStax: Concepts of Biology (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_biology_2e_45_3_environmental_limits_to_population__9b5c5687", "title": "45.3 Environmental Limits to Population Growth — Biology 2e", "shelf": "biology", "surface": "secular", "snippet": "45.3\n\nEnvironmental Limits to Population Growth\n\nLearning Objectives\nBy the end of this section, you will be able to do the following:\n\nExplain the characteristics of and differences between exponenti", "authority_tier": "reference", "source": "OpenStax: Biology 2e (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_biology_45_3_environmental_limits_to_population__79ebaf1a", "title": "45.3 Environmental Limits to Population Growth — Biology", "shelf": "biology", "surface": "secular", "snippet": "45.3\n\nEnvironmental Limits to Population Growth\n\nLearning ObjectivesBy the end of this section, you will be able to:\n\nExplain the characteristics of and differences between exponential and logistic gr", "authority_tier": "reference", "source": "OpenStax: Biology (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_biology_ap_courses_36_3_environmental_limits_to_population__ea3884dc", "title": "36.3 Environmental Limits to Population Growth — Biology for AP® Courses", "shelf": "biology", "surface": "secular", "snippet": "36.3\n\nEnvironmental Limits to Population Growth\n\nLearning Objectives\n\nIn this section, you will explore the following questions:\n\nWhat are the characteristics of and differences between exponential an", "authority_tier": "reference", "source": "OpenStax: Biology for AP® Courses (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_biology_45_4_population_dynamics_and_regulation_bd8d993b", "title": "45.4 Population Dynamics and Regulation — Biology", "shelf": "biology", "surface": "secular", "snippet": "45.4\n\nPopulation Dynamics and Regulation\n\nLearning ObjectivesBy the end of this section, you will be able to:\n\nGive examples of how the carrying capacity of a habitat may change\n\nCompare and contrast ", "authority_tier": "reference", "source": "OpenStax: Biology (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_openstax_biology_2e_45_4_population_dynamics_and_regulation_2ca2c793", "title": "45.4 Population Dynamics and Regulation — Biology 2e", "shelf": "biology", "surface": "secular", "snippet": "45.4\n\nPopulation Dynamics and Regulation\n\nLearning Objectives\nBy the end of this section, you will be able to do the following:\n\nGive examples of how the carrying capacity of a habitat may change\n\nCom", "authority_tier": "reference", "source": "OpenStax: Biology 2e (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_calc_logistic_activation", "title": "Logistic activation / regression", "shelf": "calculations", "surface": "secular", "snippet": "Logistic activation / regression — computer_science. Formula: sigma(x) = 1/(1 + e^(-x)). Canonical FORM: logistic (dP/dt = rP(1 - P/K)) — a neuron's soft switch and logistic regression's link — the sa", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_floor_logistics", "title": "Logistics — moving things at least cost", "shelf": "codex", "surface": "secular", "snippet": "Getting goods from here to there at least cost: shortest paths, scheduling, inventory, the travelling salesman. The optimization arithmetic is already sealed on the logistics stick; this floor places ", "authority_tier": "engine_derived", "source": "Narrow Highway — logistics", "readable": false, "generated": false}, {"id": "card_bridge_dual_logistic__exponential", "title": "Duality: logistic  ↔  exponential  (growth with a ceiling)", "shelf": "bridges", "surface": "secular", "snippet": "A deep bridge between two forms — growth with a ceiling. logistic growth IS exponential growth with a ceiling — for a small population it is indistinguishable from the exponential", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_ecology_populations"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}