Andrew Edwards

1. Mathematical modelling of the plankton ecosystem Andrew Edwards Department of Biology Dalhousie University Biological Oceanography Lecture OCEA-4140 edwards@mathstat.dal.ca 18th Feb 2004

2. Aim of lecture • Explain difficulties of marine ecological modelling • Show how a model is constructed • Discuss two important models from the literature, to give you an indication of their implementation and utility.

3. Outline of lecture • What is a model? • Why model the marine ecosystem? • Physics envy - or not • Constructing a model • Two models in detail • Tuning models to data - data assimilation • Other modelling approaches • Summary

4. What is a model? Some representation of reality. Doesn’t necessarily have to be mathematical (though it will be today). Is NOT going to precisely simulate reality. All models are wrong because they leave something out. But experiments and fieldwork do not consider everything either....

5. Caswell (1988) counted no. of experimental factors considered in posters on Terrestrial Ecology, Pine Forests and Nutrient Cycling at a conference: no. posters no. experimental factors

6. But surely no forest ecologist would argue that nutrient cycling is completely determined by two or three factors. no. posters no. experimental factors

7. But surely no forest ecologist would argue that nutrient cycling is completely determined by two or three factors. Nor would they assume that no other factors were important. Similar to modelling - too many factors in a model may make understanding intractable. But vary too many factors in an experiment and interpretation becomes difficult. In oceanography, hard to concurrently measure everything that’s of interest.

8. Chlorophyll Sea-surface temperature BIO SeaWiFS Archive

9. Why model the marine ecosystem? Same general motivation as why we study the marine ecosystem (to better understand it, global carbon cycle, etc.). Modelling helps us toquantifyprocesses and fluxes. Can indicate gaps in knowledge; e.g. which processes need to be measured more often or in more detail. Explore scenarios (What if ....?)

10. Physical oceanography In physical oceanography we can start with a small parcel of water, and derive the equations of motion:

11. Physical oceanography We thus have basic equations of motion, which tell us (in theory) how to model fluid motion:

12. Physical oceanography Unfortunately, these physical equations cannot be solved analytically (algebraically), but can be integrated numerically on a computer. Difficulties can arise in the numerical implementation, but at least we have a high degree of confidence in the equations.

13. But in ecology (in general) we do not have the equivalent of these equations or Newton’s laws of motion. Ecologists often call this ‘physics envy’. Thus a major problem is knowing precisely how to start. ?

14. However, it can be argued that we do have basic rules from which we can start, and these are somewhat analagous to Newton’s laws.

15. Consider a flask (or ocean) containing water plus a small concentration of phytoplankton. Let the conditions be ideal for growth (enough nutrients and light), then the population will grow exponentially. time

16. Population increase then given by: Thus population increases exponentially.

17. A population will experience unabated exponential growth in the absence of any limiting factors. Analagous to Newton’s first law of motion: “an object will continue in its state of momentum in the absence of any other forces.” Turchin (2001) True that these rules are analagous to Newton’s laws, but it then gets difficult when we increase complexity.

18. Setting up a model Need to explain: purpose physical setting ecological structure units equations parameter values

19. Models vary greatly in structure: • temporal – week-long or decadal time series

20. 0.8 Synechococcus 0.6 Cell volume (μm3) 0.4 0.2 1000 Radiation (Wm-2) 500 0 29 Jul 31 Jul 2 Aug 4 Aug Date in 2001 H. Sosik, WHOI

21. Bedford Basin Monitoring Program, BIO

22. Models vary greatly in structure: • temporal - spring bloom or decadal time series • spatial - local, regional, global

23. Oschlies & Garçon (1999)

24. Models vary greatly in structure: • temporal - spring bloom or decadal time series • spatial - local, regional, global • biological - simple vs detailed structure

25. Models vary greatly in structure: • biological - simple vs detailed structure e.g. one zooplankton compartment encompassing all species and size classes, or multiple compartments giving the population size of each stage (nauplii, C1, C2, ...., adult)

26. Models vary greatly in structure: • physical - homogeneous mixed layer vs detailed vertical structure model to be discussed in class assumes a mixed layer within which the biological components have no vertical dependence (more later), whereas...

27. Oschlies & Garçon (1999) 37 vertical layers with depths (m): 11 23 35 .... 5,000 5,250 5,500

28. Models vary greatly in structure: • chemical - single currency vs multi-element e.g. represent all biological entities in terms of nitrogen, or within the model explicitly track nitrogen, carbon, iron, silica, phosphorous, ...

29. Models vary greatly in structure: temporal physical spatial chemical biological All these factors result in great variation in complexity of models, and hence in the mathematical formulae used to represent processes.

30. What to do with the model? - any analysis possible? - numerical implementation - comparison with data - tuning to fit data better - data assimilation - can examine sensitivity to: parameter values functional forms in equations ecological structure physical forcing So what? Conclude.

31. Art of modelling is in selecting appropriate level of resolution pertinent to the question at hand. e.g. if interested in life stages of a copepod, then the NPZ model about to be discussed will not be of much help.

32. Often a modeller’s background/upbringing has an influence: Biology background - predilection and training for paying attention to detail. Mathematics/physics background - prefer abstraction and like to keep it simple. Getz (1998)

33. A common assumption Homogeneous mixed-layer (i.e. biology uniform with depth within this layer).

34. N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N

35. A common assumption Homogeneous mixed-layer (i.e. biology uniform with depth within this layer). One of the limitations - cannot simulate a deep chlorophyll maximum. Although now we are seeing a greater use of coupled physical-biological models (e.g. Oschlies & Garçon, 1999).

36. Constructing a model Say we want to model nutrient concentration, phytoplankton population and zooplankton population in a region of the open ocean for which we consider the previous diagram to be a reasonable representation.

37. First, specify components: nutrients

38. phytoplankton

39. zooplankton

40. input (diffusive mixing)

41. uptake

42. sinking respiration

43. grazing

44. fecal pellets excretion “sloppy feeding”

45. higher predation i.e. losses to predators that are not being explicitly modelled

46. excretion by higher predators

47. need formulae to represent processes

50. model Case study 1: Evans and Parslow (1985) Previous models had simulated the details of a single bloom. P time winter spring summer fall