1. Modelling in Ecology Predictions in ecology rely on models.

2. 1.What is a model? 2.Matrix algebra 3.Linear regression models 4.More on regression 5.Variance analysis 6.Model selection techniques 7.Classification techniques 8.Eigenvector techniques 9.More on eigenvectors 10.Species distribution modelling Our program

3. What is a model? A biological model is a formal representation of any biological process. Models serve to 1.Simplify a process 2.Make a process analytically tractable 3.Identify basic patterns 4.Identify basic variables (drivers) 5.Make qualitative predictions 6.Make quantitative predictions 7.Derive testable hypotheses 8.Provide guidelines for conservation and decision making There are many different types of models: Brain models, Cellular automata, Food web models, Species distribution models, infectious disease models, demographic models, ecosystem models … In general, there are two types of models: 1.Analytical models 2.Descriptive models 3.Simulation models

4. A simple analytical model A species – area relationship is modelled by two different analytical functions. These trend lines predict central tendencies (averages) around which the observed data scatter. The model predicts alpha, beta, and gamma diversities

5. A descriptive (qualitative) model of slug carcass colonisation Hyperparasitoids Idiotypa nigriceps Basalys parva Aspilota A Aspilota B Aspilota C Aspilota C Orthostigma sp Megaselia ruficornis Megaselia pulicaria Arion ater Necrophilus spp. Carabus spp. Aspilota A Aspilota E Kleridotoma psiloides Pentapleura sp. Gymnophora arcuata Limosina sp. Conicera schnittmani Fannia immutica Psychoda sp Time Primary parasitoids Necrophagous flies

6. Modelling starts with a graphical representation The classical Silver Springs semi-quantitative model of ecosystem functioning by H. T. Odum (1971)

7. Industry Emmisions according to the credits Local authorities permit emmisssions Carbon credits Lower emmissions Trading credits with other firms Higher emmissions Payment Trading for other permisions The carbon credit system

8. Modelling is essentially a trade-off (compromise) between 1. Generality 2. Realism 3. Precision 1.A good model does not only refer to a special case but allows for some generalisation. 2.A model must be realistic with regard to its components and drivers. 3.Predictive models must be sufficiently precise. A too precise model is rarely general. A too realistic model is rarely of general application (too case specific). A too general model is rarely precise. Generality Precision Realism Trade- off

9. What is interesting: the prediction or the deviation? This quantitative model has low predictive power. It is not able to precisely predict species richness for a given area. The model might serve as a standard with which deviations (residuals) are compared. We are interested in patterns of deviation along the gradient for which the model is defined.

10. Steps in model formulation Question Define the elements (drivers) of the models Provide a flowchart Identify the necessary parameters to quantify the drivers Parameterisation Model validation Derivation of questions from ecological theory Theory Validate the model with independent data sets. Assess the degree of imprecision. Assess predictive power Do not overparameterise the model

11. Null models A null model is a pattern generating model that is based on randomization of ecological data or random sampling from a known or imagined distribution. The null model is designed with respect to some ecological or evolutionary process of interest ’. (Gotelli and Graves 1996) Classical Person-Neyman hypothesis testing confronts a hypothesis with its counterpart, that is most often a random assumption. Does a IQ of 129 kg deviate from the average IQ of Europeans? We use a Z-test. A Z-test confronts the observation with a distribution ( normal distribution) that is linked to the Z-value. The null assumption refers to a random draw from a normal distribution If Z > 1.96 we accept the hypothesis at the two-sided 5% error level.

12. Now we want to test whether couples are similar in IQ We have a precise starting hypothesis H 0. There is no precisely defined null hypothesis with an associated null (random) distribution. We have to define a null model that simulates random draws of couples from the whole population. Null models often define simulations to obtain a desired random distribution with which the observed pattern is compared. We draw 1000 women and 1000 men at random from the observed distributions and calculate the average IQ difference and the associated standard deviation.

13. NoManWomenDifferenceMeanStdDev Sorted difference 1108.0663100.89337.1730271818.6911914.334540.056137 2101.5697.719833.84015654 0.122529 3113.053591.1154121.9381392 0.135907 4112.513854.1981458.3156358 0.172403 5108.824133.060824.2368334 0.208191 696.5690981.6981314.8709596 0.234321 792.5625885.869786.69279548 0.237816 889.25959104.823315.563699 0.240457 9116.8267104.347212.4794160.247577 1091.3924794.357752.96528410.261283 11115.4509104.73210.71893720.387984 12117.400590.2950527.10549050.434614 13100.6308100.60550.025359640.490975 14135.1346112.722822.41189140.492127 1594.76996102.79358.023542060.634511 1693.1209128.912735.7918140.643094 1791.30526109.59318.2877524P0.671985 18139.0272111.627227.39993060.018Observed0.689628 1964.082695.9336431.85103370.703756 A normal random number : =200*(LOS()+LOS()+LOS()+LOS()+LOS()+LOS()+LOS()+LOS()+LOS()+LOS()+LOS()+LOS())/12 A simple null model