Transcript DEB theory
Modelling & model criteria Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam [email protected] http://www.bio.vu.nl/thb master course WTC methods Amsterdam, 2005/10/31 Modelling 1 1.2.1 • model: scientific statement in mathematical language “all models are wrong, some are useful” • aims: structuring thought; the single most useful property of models: “a model is not more than you put into it” how do factors interact? (machanisms/consequences) design of experiments, interpretation of results inter-, extra-polation (prediction) decision/management (risk analysis) Modelling 2 1.2.1 • language errors: mathematical, dimensions, conservation laws • properties: generic (with respect to application) realistic (precision) simple (math. analysis, aid in thinking) plasticity in parameters (support, testability) • ideals: assumptions for mechanisms (coherence, consistency) distinction action variables/meausered quantities core/auxiliary theory Modelling criteria • Consistency dimensions, conservation laws, realism (consistency with data) • Coherence consistency with neighbouring fields of interest, levels of organisation • Efficiency comparable level of detail, all vars and pars are effective numerical behaviour • Testability amount of support, hidden variables Causation Cause and effect sequences can work in chains ABC But are problematic in networks A B C Framework of dynamic systems allow for holistic approach Dynamic systems 1.2.2 Defined by simultaneous behaviour of input, state variable, output Supply systems: input + state variables output Demand systems: input state variables + output Real systems: mixtures between supply & demand systems Constraints: mass, energy balance equations State variables: span a state space behaviour: usually set of ode’s with parameters Trajectory: map of behaviour state vars in state space Parameters: constant, functions of time, functions of modifying variables compound parameters: functions of parameters Dimension rules 1.2.3 • quantities left and right of = must have equal dimensions • + and – only defined for quantities with same dimension • ratio’s of variables with similar dimensions are only dimensionless if addition of these variables has a meaning within the model context • never apply transcendental functions to quantities with a dimension log, exp, sin, … What about pH, and pH1 – pH2? • don’t replace parameters by their values in model representations y(x) = a x + b, with a = 0.2 M-1, b = 5 y(x) = 0.2 x + 5 What dimensions have y and x? Distinguish dimensions and units! Models with dimension problems 1.2.3 • Allometric model: y = a W b y: some quantity a: proportionality constant W: body weight b: allometric parameter in (2/3, 1) Usual form ln y = ln a + b ln W Alternative form: y = y0 (W/W0 )b, with y0 = a W0b Alternative model: y = a L2 + b L3, where L W1/3 • Freundlich’s model: C = k c1/n C: density of compound in soil k: proportionality constant c: concentration in liquid n: parameter in (1.4, 5) Alternative form: C = C0 (c/c0 )1/n, with C0 = kc01/n Alternative model: C = 2C0 c(c0+c)-1 (Langmuir’s model) Problem: No natural reference values W0 , c0 Values of y0 , C0 depend on the arbitrary choice Model without dimension problem Arrhenius model: ln k = a – T0 /T k: some rate T: absolute temperature a: parameter T0: Arrhenius temperature Alternative form: k = k0 exp{1 – T0 /T}, with k0 = exp{a – 1} Difference with allometric model: no reference value required to solve dimension problem 1.2.3 Egg development time 1.2.3 D exp(3.3956 0.2193ln(T ) 0.3414(ln(T ))2 ) D exp(a b ln(T ) c(ln(T ))2 ) dim(a ) ln t ln t dim(b) ln K ln t dim(c) (ln K ) 2 D egg developmen t time T temperatur e in Kelvin Bottrell, H. H., Duncan, A., Gliwicz, Z. M. , Grygierek, E., Herzig, A., Hillbricht-Ilkowska, A., Kurasawa, H. Larsson, P., Weglenska, T. 1976 A review of some problems in zooplankton production studies. Norw. J. Zool. 24: 419-456 Complex models • hardly contribute to insight • hardly allow parameter estimation • hardly allow falsification Avoid complexity by • delineating modules • linking modules in simple ways • estimate parameters of modules only Large scatter • complicates parameter estimation • complicates falsification Avoid large scatter by • Standardization of factors that contribute to measurements • Stratified sampling