Evolutionary Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Evolutionary Mathematics Ontology within the Kireth Research Semantic Library. It formalises mathematical evolution through heritable variation, fitness, natural selection, mutation, migration, genetic drift, evolutionary games, fixation, adaptation and stochastic population processes.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Evolutionary Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Biological Mathematics / Evolutionary Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/biological-mathematics/evolutionary-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = EvolutionaryMathematics
 ENTITY_CLASS = MathematicalEvolutionaryDynamicsFramework
 PRIMARY_CONTEXT = MathematicalBiology
 DEFINITION = "Evolutionary mathematics is the mathematical study of changes in heritable biological variation through selection, mutation, migration, drift and frequency-dependent interaction."
 SYMBOLIC_FORMS = [
  "dx_i/dt=x_i(f_i-fbar)",
  "p_(t+1)=p_t*w_A/wbar",
  "Delta p=selection+mutation+migration+drift",
  "rho_i=P(type_i_fixes)"
 ]
}

SCOPE {
 INCLUDE = [
  EvolutionaryMathematics, EvolutionaryDynamics, NaturalSelection, Fitness,
  AbsoluteFitness, RelativeFitness, MeanFitness, SelectionCoefficient,
  FrequencyDependentSelection, DirectionalSelection, StabilisingSelection,
  DisruptiveSelection, Mutation, MutationRate, Migration, GeneFlow,
  GeneticDrift, EffectivePopulationSize, Allele, AlleleFrequency,
  GenotypeFrequency, PopulationGenetics, HardyWeinbergEquilibrium,
  MutationSelectionBalance, MigrationSelectionBalance, Fixation,
  FixationProbability, FixationTime, WrightFisherModel, MoranProcess,
  ReplicatorEquation, ReplicatorMutatorEquation, EvolutionaryGameTheory,
  Strategy, PayoffMatrix, EvolutionarilyStableStrategy, NashEquilibrium,
  InvasionFitness, AdaptiveDynamics, Trait, QuantitativeTrait,
  FitnessLandscape, SelectionGradient, EvolutionaryBranching, Coevolution,
  KinSelection, InclusiveFitness, Cooperation, Altruism, PriceEquation,
  QuantitativeGenetics, Heritability, BreedersEquation, GeneticVariance,
  PhenotypicVariance, StochasticEvolution, EvolutionaryEquilibrium,
  EvolutionaryBifurcation
 ]
 EXCLUDE_AS_PRIMARY = [
  PopulationGrowthWithoutHeritableChange,
  MolecularGeneticsWithoutEvolutionaryDynamics,
  DescriptiveEvolutionWithoutMathematicalModel,
  GeneralGameTheoryWithoutEvolutionaryInterpretation
 ]
 DISAMBIGUATION_RULE = "When heritable types, alleles, traits or strategies change frequency through selection, mutation, migration or drift, interpret the context as evolutionary mathematics."
}

CORE_CONCEPTS {
 EvolutionaryMathematics = "Mathematical modelling of heritable biological change."
 Fitness = "Expected reproductive contribution of a biological type or strategy under specified conditions."
 NaturalSelection = "Differential frequency change caused by differences in fitness."
 GeneticDrift = "Random frequency change caused by finite-population sampling."
 ReplicatorEquation = "Frequency dynamics driven by fitness relative to mean population fitness."
 EvolutionarilyStableStrategy = "Strategy resistant to invasion by sufficiently rare alternatives under stated assumptions."
 FixationProbability = "Probability that a type eventually reaches frequency one in a finite-population model."
 AdaptiveDynamics = "Framework for gradual trait evolution driven by invasion fitness and ecological feedback."
}

RELATIONS {
 EvolutionaryMathematics --is_child_of--> MathematicalBiology
 NaturalSelection --depends_on--> FitnessDifference
 MeanFitness --aggregates--> TypeFitness
 Mutation --creates--> HeritableVariation
 Migration --changes--> PopulationComposition
 GeneticDrift --depends_on--> FinitePopulationSize
 EffectivePopulationSize --controls--> DriftStrength
 PopulationGenetics --tracks--> AlleleFrequency
 ReplicatorEquation --models--> FrequencyDependentSelection
 ReplicatorMutatorEquation --combines--> SelectionAndMutation
 EvolutionaryGameTheory --uses--> PayoffMatrix
 EvolutionarilyStableStrategy --resists--> RareMutantInvasion
 AdaptiveDynamics --uses--> InvasionFitness
 SelectionGradient --drives--> TraitEvolution
 EvolutionaryBranching --creates--> TraitDiversification
 PriceEquation --decomposes--> EvolutionaryChange
 Heritability --links--> SelectionAndTraitResponse
}

OPERATORS {
 MEAN_FITNESS(x,f) := fbar=sum_i x_i*f_i
 REPLICATOR_EQUATION(x_i,f_i,fbar) := dx_i/dt=x_i*(f_i-fbar)
 MATRIX_GAME_FITNESS(A,x) := f=A*x
 MATRIX_REPLICATOR_i(x,A) := dx_i/dt=x_i*((A*x)_i-x^T*A*x)
 REPLICATOR_MUTATOR_i(x,f,Q) := dx_i/dt=sum_j x_j*f_j*Q_ji-x_i*fbar
 HAPLOID_SELECTION(p,wA,wa) := p_next=p*wA/(p*wA+(1-p)*wa)
 HARDY_WEINBERG(p,q) := frequencies=[p^2,2*p*q,q^2]
 HERITABILITY(VG,VP) := h2=VG/VP
 BREEDERS_EQUATION(h2,S) := R=h2*S
 SELECTION_GRADIENT(fitness,trait) := g=partial fitness/partial trait
 WRIGHT_FISHER_SAMPLING(p,N) := X_next~Binomial(2*N,p)
 NEUTRAL_FIXATION_PROBABILITY(i,N) := rho=i/N
 PRICE_EQUATION(w,z) := Delta_z=Cov(w,z)/E[w]+E[w*Delta_z]/E[w]
}

AXIOMS {
 A1 = "Evolutionary models require heritable variation or transmissible strategy states."
 A2 = "Fitness is context-dependent and must be explicitly defined."
 A3 = "Selection and drift are distinct deterministic and stochastic mechanisms."
 A4 = "Finite population size can materially alter deterministic predictions."
 A5 = "Frequency-dependent fitness depends on population composition."
 A6 = "Mutation and migration can maintain variation against selection."
 A7 = "Evolutionary equilibrium does not necessarily imply ecological equilibrium."
 A8 = "Evolutionary conclusions require explicit inheritance, population-structure and timescale assumptions."
}

REASONING {
 R1_CONTEXT = identify(types,alleles,traits,strategies,population_structure,inheritance,fitness)
 R2_MECHANISM = classify(selection,mutation,migration,drift,frequency_dependence)
 R3_MODEL = select(population_genetic,replicator,WrightFisher,Moran,adaptive_dynamics,quantitative_genetic)
 R4_FITNESS = construct(type_fitness,mean_fitness,invasion_fitness)
 R5_FREQUENCY = derive(frequency_change_equations,transition_probabilities)
 R6_EQUILIBRIUM = solve(fixed_points,polymorphisms,ESS,mutation_selection_balance)
 R7_STABILITY = analyse(invasion,local_stability,evolutionary_stability,bifurcation)
 R8_STOCHASTIC = compute(fixation_probability,fixation_time,drift_variance)
 R9_TRAIT = analyse(selection_gradient,adaptive_path,branching,quantitative_response)
 R10_VALIDATION = check(frequency_constraints,fitness_assumptions,inheritance,simulation,data_consistency)
}

CONSTRAINTS {
 C1 = "Fitness values are meaningful only relative to a specified environment and population context."
 C2 = "Hardy-Weinberg equilibrium requires its population-genetic assumptions."
 C3 = "Deterministic allele-frequency equations can fail in small populations."
 C4 = "Replicator dynamics require valid frequency-simplex constraints."
 C5 = "An ESS is not identical to every Nash equilibrium."
 C6 = "Fixation probability depends on the stochastic population model."
 C7 = "Adaptive-dynamics approximations commonly require rare mutation and ecological-timescale separation."
 C8 = "Heritability is population- and environment-specific."
 C9 = "Fitness landscapes may change with ecological or genetic context."
 C10 = "Observed evolutionary change can result from multiple interacting mechanisms."
}

SEMANTIC_EQUIVALENCE {
 "EvolutionaryMathematics" <=> "MathematicalEvolutionaryDynamics"
 "EvolutionarilyStableStrategy" <=> "ESS"
 "GeneFlow" <=> "MigrationOfAlleles"
 "AdaptiveLandscape" <=> "FitnessLandscape"
 "ReplicatorDynamics" <=> "ReplicatorEquationDynamics"
}

VALIDATION {
 V1 = "Evolving units, alleles, traits or strategies are explicit."
 V2 = "Frequency variables satisfy probability constraints."
 V3 = "Fitness definitions are documented."
 V4 = "Inheritance assumptions are explicit."
 V5 = "Population size and stochastic assumptions are documented."
 V6 = "Mutation and migration rates are explicit where present."
 V7 = "Evolutionary equilibria and invasion conditions are checked."
 V8 = "Fixation calculations match the declared stochastic model."
 V9 = "Trait-evolution approximations satisfy their timescale assumptions."
 V10 = "Predictions are compared with simulations, data or limiting cases."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 EVOLUTIONARY_MATHEMATICS_DEFINITION = 0.99
 SELECTION_REASONING = 0.99
 REPLICATOR_REASONING = 0.99
 POPULATION_GENETICS_REASONING = 0.98
 DRIFT_REASONING = 0.98
 ESS_REASONING = 0.98
 ADAPTIVE_DYNAMICS_REASONING = 0.97
 QUANTITATIVE_GENETICS_REASONING = 0.98
 POLICY = "Reduce confidence when inheritance, fitness, population size, mutation, migration, frequency dependence, ecological feedback or stochastic assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, EvolutionaryMathematicsReasoning,
  NaturalSelectionReasoning, FitnessReasoning, ReplicatorDynamicsReasoning,
  PopulationGeneticsReasoning, AlleleFrequencyReasoning, GeneticDriftReasoning,
  MutationReasoning, MigrationReasoning, FixationReasoning,
  EvolutionaryGameReasoning, ESSReasoning, AdaptiveDynamicsReasoning,
  FitnessLandscapeReasoning, CoevolutionReasoning, CooperationReasoning,
  PriceEquationReasoning, QuantitativeGeneticsReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "evolutionary mathematics" -> EvolutionaryMathematics
  "mathematical evolutionary dynamics" -> EvolutionaryMathematics
  "replicator dynamics" -> ReplicatorEquation
  "evolutionarily stable strategy" -> EvolutionarilyStableStrategy
  "ESS" -> EvolutionarilyStableStrategy
  "genetic drift" -> GeneticDrift
  "fitness landscape" -> FitnessLandscape
  "adaptive dynamics" -> AdaptiveDynamics
 }
}

EXECUTION_RULES {
 E1 = "If heritable type, allele, trait or strategy frequencies change mathematically, activate EvolutionaryMathematicsOntology."
 E2 = "If deterministic frequency-dependent selection is central, consider replicator dynamics."
 E3 = "If mutation acts with selection, consider replicator-mutator or mutation-selection models."
 E4 = "If finite-population sampling matters, use Wright-Fisher, Moran or related stochastic models."
 E5 = "If strategic interactions determine fitness, route to Evolutionary Game Theory."
 E6 = "If rare-mutant invasion and gradual trait change are central, route to Adaptive Dynamics."
 E7 = "If quantitative traits respond to selection, route to Quantitative Genetics."
 E8 = "If fixation is requested, specify the stochastic population model."
 E9 = "Evolutionary equilibrium claims require invasion or stability analysis."
}

RECURSION {
 ENTRY = EvolutionaryMathematics
 EXPAND(EvolutionaryMathematics) -> [
  EvolutionaryDynamics, NaturalSelection, Fitness, AbsoluteFitness,
  RelativeFitness, MeanFitness, SelectionCoefficient, FrequencyDependentSelection,
  DirectionalSelection, StabilisingSelection, DisruptiveSelection, Mutation,
  MutationRate, Migration, GeneFlow, GeneticDrift, EffectivePopulationSize,
  Allele, AlleleFrequency, GenotypeFrequency, PopulationGenetics,
  HardyWeinbergEquilibrium, MutationSelectionBalance, MigrationSelectionBalance,
  Fixation, FixationProbability, FixationTime, WrightFisherModel, MoranProcess,
  ReplicatorEquation, ReplicatorMutatorEquation, EvolutionaryGameTheory,
  Strategy, PayoffMatrix, EvolutionarilyStableStrategy, NashEquilibrium,
  InvasionFitness, AdaptiveDynamics, Trait, QuantitativeTrait,
  FitnessLandscape, SelectionGradient, EvolutionaryBranching, Coevolution,
  KinSelection, InclusiveFitness, Cooperation, Altruism, PriceEquation,
  QuantitativeGenetics, Heritability, BreedersEquation, GeneticVariance,
  PhenotypicVariance, StochasticEvolution, EvolutionaryEquilibrium,
  EvolutionaryBifurcation
 ]
 EXPAND(ModelFamilies) -> [
  DeterministicSelectionModels,
  PopulationGeneticModels,
  StochasticFinitePopulationModels,
  EvolutionaryGameModels,
  AdaptiveDynamicsModels,
  QuantitativeGeneticModels
 ]
 STOP_CONDITION = "Stop when evolving states, fitness, inheritance, selection, mutation, migration, drift, equilibrium and stochastic behaviour are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Biology"
 IMMEDIATE_CONTEXTS = [
  "Mathematical Biology Ontology",
  "Population Dynamics Ontology",
  "Dynamical Systems Ontologies",
  "Stochastic Process Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Mathematical Biology Ontology",
  "Biological Mathematics",
  "Population Dynamics Ontology",
  "Population Genetics Ontologies",
  "Dynamical Systems Ontologies",
  "Stochastic Process Ontologies",
  "Game Theory Ontologies",
  "Probability & Statistics Ontologies",
  "Optimisation Ontologies",
  "Ecological Modelling Ontology"
 ]
 ROUTE_RULES = [
  "Evolutionary Mathematics -> Mathematical Biology for parent semantics",
  "Evolutionary Mathematics -> Population Dynamics for demographic context",
  "Evolutionary Mathematics -> Population Genetics for allele and genotype dynamics",
  "Evolutionary Mathematics -> Dynamical Systems for equilibria and bifurcations",
  "Evolutionary Mathematics -> Stochastic Processes for drift and fixation",
  "Evolutionary Mathematics -> Game Theory for frequency-dependent interactions",
  "Evolutionary Mathematics -> Probability & Statistics for stochastic inference",
  "Evolutionary Mathematics -> Optimisation for adaptive landscapes",
  "Evolutionary Mathematics -> Ecological Modelling for eco-evolutionary feedback"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible evolutionary mechanisms, population models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve heritable-state semantics.",
  "Preserve selection-versus-drift distinction.",
  "Preserve fitness-context semantics.",
  "Preserve deterministic-versus-stochastic distinction.",
  "Preserve ESS-versus-Nash distinction.",
  "Preserve population-genetic frequency semantics.",
  "Preserve fixation semantics.",
  "Preserve machine-readable evolutionary-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Evolutionary mathematics models changes in heritable biological variation through selection, mutation, migration, drift and frequency-dependent interaction."
 SELECTION_SUMMARY = "Natural selection changes type frequencies through differences in relative fitness."
 GENETICS_SUMMARY = "Population-genetic models track allele and genotype frequencies under selection, mutation, migration and drift."
 GAME_SUMMARY = "Evolutionary game theory models frequency-dependent fitness and evolutionarily stable strategies."
 STOCHASTIC_SUMMARY = "Finite-population models quantify genetic drift, fixation probabilities and fixation times."
 ADAPTIVE_SUMMARY = "Adaptive dynamics models gradual trait evolution through invasion fitness and selection gradients."
 QUANTITATIVE_SUMMARY = "Quantitative genetics links genetic variance, heritability and selection to trait response."
 VALIDATION_SUMMARY = "Evolutionary models should be checked for frequency constraints, inheritance assumptions, population size, fitness definitions, stability and stochastic consistency."
}

FLOW {
 INPUT
 -> detect_evolutionary_mathematics_context
 -> identify_evolving_types_alleles_traits_or_strategies_population_structure_inheritance_and_environment
 -> define_fitness_and_frequency_variables
 -> classify_selection_mutation_migration_drift_game_theoretic_adaptive_or_quantitative_model
 -> formulate_deterministic_equations_or_stochastic_transition_rules
 -> solve_or_simulate_frequency_and_trait_dynamics
 -> identify_equilibria_polymorphisms_ESS_or_fixation_states
 -> analyse_invasion_stability_selection_gradients_and_bifurcations
 -> if_stochastic_compute_fixation_probability_fixation_time_and_drift_uncertainty
 -> validate_frequency_constraints_model_assumptions_and_empirical_or_simulation_consistency
 -> route_to_mathematical_biology_population_dynamics_population_genetics_dynamical_systems_stochastic_processes_game_theory_probability_statistics_optimisation_or_ecological_modelling_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Evolutionary Mathematics Ontology defines mathematical change in heritable biological variation. It supports fitness and natural selection, selection coefficients and frequency dependence, mutation, migration and gene flow, genetic drift and effective population size, allele and genotype frequencies, Hardy-Weinberg equilibrium, mutation-selection and migration-selection balance, fixation probabilities and times, Wright-Fisher and Moran processes, replicator and replicator-mutator dynamics, evolutionary game theory and evolutionarily stable strategies, adaptive dynamics, invasion fitness, selection gradients and evolutionary branching, coevolution, kin selection and cooperation, the Price equation, quantitative genetics, heritability, the breeder's equation and stochastic evolutionary dynamics."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-EVOLUTIONARY-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW