Mathematical Biology Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Mathematical Biology Ontology within the Kireth Research Semantic Library. It formalises mathematical biology as a machine-readable ontology for representing living systems with differential equations, stochastic processes, networks, optimisation, dynamical systems and computational methods. It sits beneath the Biological Mathematics parent layer and provides a core semantic framework for population dynamics, epidemiology, evolution, genetics, systems biology, biochemical kinetics, ecology, physiology, spatial biology and stochastic biological modelling.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = MathematicalBiology
 ENTITY_CLASS = FormalMathematicalBiologicalSystemsOntology
 PRIMARY_CONTEXT = BiologicalMathematics
 DEFINITION = "Mathematical biology is the formal application of mathematical models, equations, stochastic processes and computational methods to biological systems and processes."
 SYMBOLIC_FORMS = [
  "dx/dt=f(x,p)",
  "dN/dt=rN(1-N/K)",
  "dX_t=a(X_t,t)dt+b(X_t,t)dW_t",
  "partial u/partial t=D*nabla^2u+f(u)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for biological state variables, parameters, growth, interaction, disease, evolution, biochemical dynamics, spatial organisation, stochasticity and model validation."
}

SCOPE {
 INCLUDE = [
  MathematicalBiology,
  BiologicalMathematics,
  BiologicalSystem,
  BiologicalModel,
  StateVariable,
  BiologicalParameter,
  InitialCondition,
  BoundaryCondition,
  PopulationDynamics,
  PopulationGrowth,
  ExponentialGrowth,
  LogisticGrowth,
  CarryingCapacity,
  BirthRate,
  DeathRate,
  Competition,
  Cooperation,
  Predation,
  PredatorPreyModel,
  LotkaVolterraModel,
  Epidemiology,
  CompartmentalModel,
  SIRModel,
  SEIRModel,
  SISModel,
  BasicReproductionNumber,
  EffectiveReproductionNumber,
  EpidemicThreshold,
  EndemicEquilibrium,
  EvolutionaryDynamics,
  NaturalSelection,
  Fitness,
  ReplicatorEquation,
  EvolutionarilyStableStrategy,
  PopulationGenetics,
  AlleleFrequency,
  HardyWeinbergEquilibrium,
  GeneticDrift,
  Mutation,
  Migration,
  SystemsBiology,
  BiochemicalKinetics,
  EnzymeKinetics,
  MichaelisMentenKinetics,
  HillEquation,
  GeneRegulatoryNetwork,
  ProteinInteractionNetwork,
  MetabolicNetwork,
  NeuralDynamics,
  NeuronModel,
  PhysiologicalModel,
  Pharmacokinetics,
  Pharmacodynamics,
  Ecology,
  CommunityDynamics,
  FoodWeb,
  SpatialEcology,
  ReactionDiffusion,
  Chemotaxis,
  PatternFormation,
  TuringPattern,
  StochasticBiology,
  BirthDeathProcess,
  BranchingProcess,
  MarkovProcess,
  MasterEquation,
  NetworkBiology,
  ComputationalBiology,
  ParameterEstimation,
  SensitivityAnalysis,
  Identifiability,
  StabilityAnalysis,
  BifurcationAnalysis,
  ModelValidation
 ]
 EXCLUDE_AS_PRIMARY = [
  DescriptiveBiologyWithoutMathematicalRepresentation,
  PureBioinformaticsWithoutBiologicalMathematicalModel,
  GeneralStatisticsWithoutBiologicalMechanisticContext,
  PureChemistryWithoutBiologicalSystemContext
 ]
 DISAMBIGUATION_RULE = "When biological processes are formalised using equations, probabilities, dynamical systems, networks or optimisation, interpret the context as mathematical biology."
}

CORE_CONCEPTS {
 MathematicalBiology = "Formal mathematical modelling of biological systems."
 BiologicalModel = "Mathematical abstraction of a biological mechanism or process."
 BiologicalStateVariable = "Quantity describing a biological system state."
 BiologicalParameter = "Model quantity controlling biological rates, interactions or structure."
 PopulationDynamics = "Mathematical dynamics of populations through time."
 Epidemiology = "Mathematical modelling of infectious or population-level disease processes."
 EvolutionaryDynamics = "Mathematical modelling of selection and changes in fitness-related traits."
 SystemsBiology = "Mathematical modelling of interacting biological components as systems."
 SpatialBiology = "Mathematical representation of biological variation and movement across space."
 StochasticBiology = "Mathematical modelling of random biological events and trajectories."
}

RELATIONS {
 MathematicalBiology --is_child_of--> BiologicalMathematics
 MathematicalBiology --models--> BiologicalSystem
 BiologicalModel --contains--> StateVariable
 BiologicalModel --contains--> BiologicalParameter
 BiologicalModel --uses--> InitialCondition
 PopulationDynamics --uses--> DifferentialEquations
 LogisticGrowth --uses--> CarryingCapacity
 PredatorPreyModel --is_type_of--> PopulationDynamics
 Epidemiology --uses--> CompartmentalModel
 SIRModel --is_type_of--> CompartmentalModel
 BasicReproductionNumber --characterises--> EpidemicThreshold
 EvolutionaryDynamics --depends_on--> Fitness
 ReplicatorEquation --models--> EvolutionaryDynamics
 PopulationGenetics --tracks--> AlleleFrequency
 SystemsBiology --uses--> BiologicalNetworks
 MichaelisMentenKinetics --models--> EnzymeKinetics
 HillEquation --models--> CooperativeBinding
 ReactionDiffusion --models--> SpatialBiology
 Chemotaxis --models--> DirectedCellularMovement
 StochasticBiology --uses--> StochasticProcesses
 BirthDeathProcess --models--> PopulationEventDynamics
 ParameterEstimation --calibrates--> BiologicalModel
 SensitivityAnalysis --assesses--> BiologicalParameter
 Identifiability --constrains--> ParameterEstimation
 StabilityAnalysis --assesses--> BiologicalEquilibrium
 BifurcationAnalysis --detects--> QualitativeDynamicChange
}

OPERATORS {
 EXPONENTIAL_GROWTH(N,r) := dN/dt=r*N
 LOGISTIC_GROWTH(N,r,K) := dN/dt=r*N*(1-N/K)
 LOTKA_VOLTERRA_PREY(x,alpha,beta,y) := dx/dt=alpha*x-beta*x*y
 LOTKA_VOLTERRA_PREDATOR(y,delta,gamma,x) := dy/dt=delta*x*y-gamma*y
 SIR_S(S,I,beta,N) := dS/dt=-beta*S*I/N
 SIR_I(S,I,beta,gamma,N) := dI/dt=beta*S*I/N-gamma*I
 SIR_R(I,gamma) := dR/dt=gamma*I
 BASIC_REPRODUCTION_NUMBER(beta,gamma) := R0=beta/gamma under_basic_SIR_assumptions
 REPLICATOR_EQUATION(x_i,f_i,fbar) := dx_i/dt=x_i*(f_i-fbar)
 HARDY_WEINBERG(p,q) := frequencies=[p^2,2*p*q,q^2]
 MICHAELIS_MENTEN(S,Vmax,Km) := v=Vmax*S/(Km+S)
 HILL_RESPONSE(x,K,n) := y=x^n/(K^n+x^n)
 REACTION_DIFFUSION(u,D,f) := partial u/partial t=D*nabla^2u+f(u)
 STOCHASTIC_BIRTH_DEATH(lambda,mu,N) := transitions=[N->N+1 at lambda*N, N->N-1 at mu*N]
}

AXIOMS {
 A1 = "Mathematical biological variables must have explicit biological interpretations."
 A2 = "Biological model parameters require declared units and meanings."
 A3 = "Model validity depends on biological scale and modelling assumptions."
 A4 = "Deterministic and stochastic models represent different levels of biological variability."
 A5 = "Mechanistic models should preserve relevant biological constraints."
 A6 = "Spatial models require explicit movement, transport or diffusion assumptions."
 A7 = "Parameter identifiability and parameter sensitivity are distinct properties."
 A8 = "Mathematical biology models require biological and numerical validation."
}

REASONING {
 R1_CONTEXT = identify(biological_system,scale,state_variables,parameters,data,question)
 R2_MODEL_SELECTION = classify(population,epidemiology,evolution,genetics,biochemical,physiological,ecological,network,spatial,stochastic)
 R3_FORMULATION = construct(equations,transition_rules,initial_conditions,boundary_conditions)
 R4_EQUILIBRIUM = solve(fixed_points,steady_states)
 R5_STABILITY = analyse(local_stability,eigenvalues,thresholds)
 R6_DYNAMIC_BEHAVIOUR = analyse(growth,oscillation,bifurcation,pattern,extinction,persistence)
 R7_STOCHASTIC_BEHAVIOUR = compute(distributions,transition_probabilities,first_passage,extinction,fixation)
 R8_PARAMETER_REASONING = estimate(parameters,confidence,identifiability)
 R9_SENSITIVITY_REASONING = quantify(parameter_and_initial_state influence)
 R10_VALIDATION = compare(predictions,data,known_biological_constraints,numerical_convergence)
}

CONSTRAINTS {
 C1 = "Continuous models may be inappropriate when biological populations or molecule counts are very small."
 C2 = "Mass-action assumptions require suitable mixing and scale."
 C3 = "Basic reproduction numbers are model-dependent."
 C4 = "Equilibrium stability is not equivalent to global dynamic behaviour."
 C5 = "Deterministic models can hide demographic or molecular noise."
 C6 = "Model fit alone does not prove a biological mechanism."
 C7 = "Highly parameterised models can be structurally or practically unidentifiable."
 C8 = "Spatial pattern formation depends on domain, diffusion and boundary conditions."
 C9 = "Network topology alone may not determine biological dynamics."
 C10 = "Numerical discretisation must preserve biologically meaningful behaviour."
}

SEMANTIC_EQUIVALENCE {
 "MathematicalBiology" <=> "BiologicalMathematicsCore"
 "PopulationDynamics" <=> "MathematicalPopulationDynamics"
 "StochasticBiology" <=> "StochasticMathematicalBiology"
 "NetworkBiology" <=> "BiologicalNetworkMathematics"
 "BiochemicalKinetics" <=> "BiologicalReactionKinetics"
}

VALIDATION {
 V1 = "Biological scale is explicit."
 V2 = "State variables and units are explicit."
 V3 = "Parameters and mechanisms are documented."
 V4 = "Initial and boundary conditions are explicit where needed."
 V5 = "Equilibria and thresholds are verified where applicable."
 V6 = "Parameter identifiability is assessed for fitted models."
 V7 = "Sensitivity and uncertainty are evaluated."
 V8 = "Stochastic sampling error is assessed where relevant."
 V9 = "Spatial and temporal numerical convergence are checked."
 V10 = "Model outputs are compared with empirical or mechanistic biological evidence."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MATHEMATICAL_BIOLOGY_DEFINITION = 0.99
 MODEL_SELECTION_REASONING = 0.99
 POPULATION_REASONING = 0.99
 EPIDEMIOLOGY_REASONING = 0.98
 EVOLUTIONARY_REASONING = 0.98
 BIOCHEMICAL_REASONING = 0.98
 SPATIAL_REASONING = 0.98
 STOCHASTIC_REASONING = 0.98
 PARAMETER_REASONING = 0.97
 POLICY = "Reduce confidence when biological scale, mechanisms, state variables, parameter meanings, initial conditions, spatial structure, data quality or validation evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MathematicalBiologyReasoning,
  BiologicalModelReasoning,
  PopulationDynamicsReasoning,
  EpidemiologyReasoning,
  EvolutionaryDynamicsReasoning,
  PopulationGeneticsReasoning,
  SystemsBiologyReasoning,
  BiochemicalKineticsReasoning,
  NeuralDynamicsReasoning,
  PhysiologicalReasoning,
  EcologicalReasoning,
  SpatialBiologyReasoning,
  ReactionDiffusionReasoning,
  StochasticBiologyReasoning,
  NetworkBiologyReasoning,
  ParameterEstimationReasoning,
  IdentifiabilityReasoning,
  StabilityReasoning,
  BifurcationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "mathematical biology" -> MathematicalBiology
  "biological mathematics" -> BiologicalMathematics
  "population dynamics" -> PopulationDynamics
  "mathematical epidemiology" -> Epidemiology
  "systems biology" -> SystemsBiology
  "reaction diffusion" -> ReactionDiffusion
  "stochastic biology" -> StochasticBiology
  "network biology" -> NetworkBiology
 }
}

EXECUTION_RULES {
 E1 = "If a biological system is represented mathematically, activate MathematicalBiologyOntology."
 E2 = "If population abundance or interaction changes through time, route to PopulationDynamics."
 E3 = "If transmission between health states is central, route to Epidemiology."
 E4 = "If selection or inheritance is central, route to EvolutionaryDynamics or PopulationGenetics."
 E5 = "If reaction rates or molecular interactions are central, route to BiochemicalKinetics or SystemsBiology."
 E6 = "If spatial transport or movement is central, route to SpatialBiology, ReactionDiffusion or Chemotaxis."
 E7 = "If randomness materially alters outcomes, route to StochasticBiology."
 E8 = "If biological components form interaction graphs, route to NetworkBiology."
 E9 = "If parameters are calibrated, require identifiability, uncertainty and validation checks."
}

RECURSION {
 ENTRY = MathematicalBiology
 EXPAND(MathematicalBiology) -> [
  BiologicalSystem, BiologicalModel, StateVariable, BiologicalParameter,
  InitialCondition, BoundaryCondition, PopulationDynamics, PopulationGrowth,
  ExponentialGrowth, LogisticGrowth, CarryingCapacity, Competition, Cooperation,
  Predation, PredatorPreyModel, LotkaVolterraModel, Epidemiology,
  CompartmentalModel, SIRModel, SEIRModel, SISModel, BasicReproductionNumber,
  EffectiveReproductionNumber, EpidemicThreshold, EndemicEquilibrium,
  EvolutionaryDynamics, NaturalSelection, Fitness, ReplicatorEquation,
  EvolutionarilyStableStrategy, PopulationGenetics, AlleleFrequency,
  HardyWeinbergEquilibrium, GeneticDrift, Mutation, Migration,
  SystemsBiology, BiochemicalKinetics, EnzymeKinetics,
  MichaelisMentenKinetics, HillEquation, GeneRegulatoryNetwork,
  ProteinInteractionNetwork, MetabolicNetwork, NeuralDynamics, NeuronModel,
  PhysiologicalModel, Pharmacokinetics, Pharmacodynamics, Ecology,
  CommunityDynamics, FoodWeb, SpatialEcology, ReactionDiffusion, Chemotaxis,
  PatternFormation, TuringPattern, StochasticBiology, BirthDeathProcess,
  BranchingProcess, MarkovProcess, MasterEquation, NetworkBiology,
  ComputationalBiology, ParameterEstimation, SensitivityAnalysis,
  Identifiability, StabilityAnalysis, BifurcationAnalysis, ModelValidation
 ]
 EXPAND(ModelFamilies) -> [
  PopulationModels,
  EpidemiologicalModels,
  EvolutionaryModels,
  GeneticModels,
  BiochemicalModels,
  PhysiologicalModels,
  EcologicalModels,
  SpatialModels,
  StochasticModels,
  NetworkModels
 ]
 STOP_CONDITION = "Stop when biological scale, model class, mechanisms, equations, parameters, dynamics, uncertainty and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Biological Mathematics"
 IMMEDIATE_CONTEXTS = [
  "Biological Mathematics",
  "Differential Equation Ontologies",
  "Dynamical Systems Ontologies",
  "Probability & Statistics Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Biological Mathematics",
  "Applied Mathematics Ontologies",
  "Differential Equation Ontologies",
  "Dynamical Systems Ontologies",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Graph Theory Ontologies",
  "Optimisation Ontologies",
  "Computational Mathematics Ontologies",
  "Scientific Ontologies",
  "Biological Ontologies"
 ]
 ROUTE_RULES = [
  "Mathematical Biology -> Biological Mathematics for parent semantics",
  "Mathematical Biology -> Differential Equations for continuous dynamics",
  "Mathematical Biology -> Dynamical Systems for stability and bifurcation",
  "Mathematical Biology -> Probability & Statistics for inference and uncertainty",
  "Mathematical Biology -> Stochastic Processes for random dynamics",
  "Mathematical Biology -> Graph Theory for biological networks",
  "Mathematical Biology -> Optimisation for calibration and control",
  "Mathematical Biology -> Computational Mathematics for numerical solution",
  "Mathematical Biology -> Biological Ontologies for domain entities and processes"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible biological model families or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve biological interpretation semantics.",
  "Preserve scale semantics.",
  "Preserve deterministic-versus-stochastic distinction.",
  "Preserve mechanistic-versus-phenomenological distinction.",
  "Preserve spatial-versus-nonspatial distinction.",
  "Preserve parameter-identifiability semantics.",
  "Preserve validation semantics.",
  "Preserve machine-readable mathematical-biology semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Mathematical biology formalises biological systems using equations, stochastic processes, networks and computational models."
 POPULATION_SUMMARY = "Population models describe growth, competition, cooperation and predation."
 EPIDEMIOLOGY_SUMMARY = "Compartmental disease models represent transmission, recovery and threshold behaviour."
 EVOLUTION_SUMMARY = "Evolutionary and genetic models represent fitness, selection, inheritance, mutation and drift."
 SYSTEMS_SUMMARY = "Systems biology represents interacting genes, proteins, metabolites and biochemical reactions."
 SPATIAL_SUMMARY = "Reaction-diffusion and chemotaxis models represent biological processes in space."
 STOCHASTIC_SUMMARY = "Stochastic models represent demographic, molecular and environmental randomness."
 VALIDATION_SUMMARY = "Mathematical biology models should be checked for biological meaning, identifiability, sensitivity, uncertainty and empirical consistency."
}

FLOW {
 INPUT
 -> detect_mathematical_biology_context
 -> identify_biological_system_scale_state_variables_parameters_data_and_question
 -> classify_model_family
 -> formulate_equations_transition_rules_initial_conditions_and_boundary_conditions
 -> solve_or_simulate_model
 -> analyse_equilibria_stability_thresholds_bifurcations_patterns_or_extinction
 -> if_stochastic_compute_probabilistic_outcomes_and_uncertainty
 -> calibrate_parameters_and_assess_identifiability
 -> perform_sensitivity_and_uncertainty_analysis
 -> validate_biological_and_numerical_consistency
 -> route_to_biological_mathematics_differential_equations_dynamical_systems_probability_statistics_stochastic_processes_graph_theory_optimisation_computational_mathematics_or_biological_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Mathematical Biology Ontology provides the formal core of biological mathematics. It supports biological systems and state variables, population growth, competition and predator-prey dynamics, epidemiological compartment models and reproduction thresholds, evolutionary dynamics and population genetics, systems biology and biochemical kinetics, enzyme models and regulatory networks, neural and physiological modelling, pharmacokinetics and pharmacodynamics, ecology and food webs, spatial ecology, reaction-diffusion, chemotaxis and pattern formation, stochastic birth-death and branching processes, network biology, parameter estimation, sensitivity, identifiability, stability, bifurcation and model validation."
}

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

}

END_FLOW