1. Modelling & model criteria Bas Kooijman Dept theoretical biology Vrije Universiteit Amsterdam Bas@bio.vu.nl http://www.bio.vu.nl/thb master course WTC methods Amsterdam, 2005/10/31

2. 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)

3. 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

4. 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

5. Causation Cause and effect sequences can work in chains A  B  C But are problematic in networks A B C Framework of dynamic systems allow for holistic approach

6. 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

7. 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 pH 1 – pH 2 ? 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!

8. 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 = y 0 (W/W 0 ) b, with y 0 = a W 0 b Alternative model: y = a L 2 + b L 3, where L  W 1/3 Freundlich’s model: C = k c 1/n C: density of compound in soil k: proportionality constant c: concentration in liquid n: parameter in (1.4, 5) Alternative form: C = C 0 (c/c 0 ) 1/n, with C 0 = kc 0 1/n Alternative model: C = 2C 0 c(c 0 +c) -1 (Langmuir’s model) Problem: No natural reference values W 0, c 0 Values of y 0, C 0 depend on the arbitrary choice

9. Model without dimension problem 1.2.3 Arrhenius model: ln k = a – T 0 /T k: some rate T: absolute temperature a: parameter T 0 : Arrhenius temperature Alternative form: k = k 0 exp{1 – T 0 /T}, with k 0 = exp{a – 1} Difference with allometric model: no reference value required to solve dimension problem

10. Egg development time 1.2.3 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

11. 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

12. Large scatter complicates parameter estimation complicates falsification Avoid large scatter by Standardization of factors that contribute to measurements Stratified sampling