Comparing simple ecosystem models in state space

1. Comparing simple ecosystem models in state space Nicky Grigg, CLW Fabio Boschetti, CMAR

2. Typical features of aquatic ecosystem models • Dynamic • Track the flow of nutrients through sediment and water column processes • Nonlinear interactions, feedbacks, hysteretic responses

3. Typical uses of aquatic ecosystem models • Seek inconsistencies between system understanding and observations • Inform monitoring strategies • Identify system vulnerabilities • Investigate ecosystem responses to changed forcing

4. How to fit a ‘wrong’ model? • All models are ‘wrong’. • A good model captures system characteristics of interest. • Dynamics process models: dynamical behaviour is important? • Aggregate, statistical quantities in model validation and sensitivity analyses throw away information about the system dynamics . • Given a ‘wrong’ model for a system, what criteria can we use to characterise and compare the dynamics of the two systems?

5. Toy example: stochastically forced food chain qZn Zooplankton mortality: Linear mortality (n = 1): only one basin of attraction Quadratic mortality (n = 2): alternative basins of attraction, and large flips between basins possible. Source: Edwards and Brindley (1999)

6. Given: observations from quadratic mortality system Aim: model the dynamics with a linear mortality model

7. Parameter search result:least squares fit

8. Is this a ‘good fit’?

9. Least squares fit State space fit Comparisons in reconstructed state space

10. ? Fitting in state space: how?

11. Density estimation

12.  Joint Probability Density =

13. Evaluate joint pdf for set of points

14. f(x,b1) p y p(y|x,c1) f(x,b2) p(y|x,c2) p(x|c1) p(c1) p(c2) p(x|c2) m1 m2 x Estimating probability density:Cluster-weighted modelling Reference: Gershenfeld (1999)

15. Why? • Model-data comparisons • Parameter sensitivity analysis • Sensitivity to choice of model structure • Identifying appropriate ‘simpler’ or lower-dimensional models (including how to ‘lump’ foodwebs and other networks)

16. Extrapolating with ensembles of ‘acceptable’ models • Typical justification for using dynamic process models – their ability to extrapolate and make predictions outside the calibration conditions. • Toy example: find an ensemble of ‘acceptable’ models given observations from one basin of attraction. Test the fitted models’ ability to extrapolate given novel forcing.

17. High nutrient load Low nutrient load NPZ model withdifferent nutrient loads

18. ‘Best’ fit to low-nutrient case using the 2D parameter search and the ‘wrong’ model Best time domain fit Best state-space fit

19. Extrapolating high-nutrient response using the ‘best’ fits Extrapolating with model fitted in state space Extrapolating with model fitted in time domain

20. Food Webs Source: Neo Martinez, http://online.sfsu.edu/~webhead/

21. 1 2 5 3 6 4 7 Food web questions Can we ‘lump’ a food web to retain dynamic characteristics? Can we find relationships between network properties and dynamic characteristics? Relative importance of network properties vs functional form of the links between species?

22. Effect of changing one link strength on two fitness measures

23. Pros and cons? • A single measure which captures many characteristics of the time series simultaneously – aggregate statistical measures, frequency information and geometry of state space trajectories. • Suitable for non-stationary, stochastically forced and chaotic systems. • Assumes there is a ‘useful’ geometrical structure in reconstructed state space. • Faced with numerical searches through high-dimensional space.

24. Conclusions • Can characterise non-stationary time series from forced dynamical systems in a way that allows quantitative comparison between systems • Currently applying these techniques to NPZ and low-dimensional foodweb models • Need to move beyond toy systems and look at more realistic ecosystem models and higher-dimensional foodwebs