Andrew Edwards Mathematical modelling of the OCEA-4140 Department of Biology Dalhousie University [email protected] plankton ecosystem Biological.

Andrew Edwards

Mathematical modelling of the

OCEA-4140

Department of BiologyDalhousie University

[email protected]

plankton ecosystem

Biological Oceanography Lecture

18th Feb 2004

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.

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

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....

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

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

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

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.

Nor would they assume that no other factors were important.

Chlorophyll Sea-surface temperature

BIO SeaWiFS Archive

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 to quantify processes 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 ....?)

Physical oceanography

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

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

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.

Physical oceanography

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.

?

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.

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

Population increase then given by:

Thus population increases exponentially.

True that these rules are analagous to Newton’s laws, but it then gets difficult when we increase complexity.

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)

Setting up a model

Need to explain: purpose physical setting ecological structure units equations parameter values

Models vary greatly in structure:

• temporal – week-long or decadal time series

Date in 2001 29 Jul 31 Jul 2 Aug 4 Aug

Cell volume

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H. Sosik, WHOI

Bedford Basin Monitoring Program, BIO

Models vary greatly in structure:

• temporal - spring bloom or decadal time series

• spatial - local, regional, global

Oschlies &Garçon (1999)

Models vary greatly in structure:

• temporal - spring bloom or decadal time series

• spatial - local, regional, global

• biological - simple vs detailed structure

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)

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...

Oschlies &Garçon (1999)

37 vertical layers with depths (m):

11 23 35 ....

5,000 5,250 5,500

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, ...

Models vary greatly in structure:temporal physicalspatial chemicalbiological

All these factors result in great variation in complexity of models, and hence in the mathematical formulae used to represent processes.

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.

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.

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)

Homogeneous mixed-layer (i.e. biology uniform with depth within this layer).

A common assumption

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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).

A common assumption

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.

Constructing a model

nutrients

First, specify components:

phytoplankton

zooplankton

input (diffusive mixing)

uptake

respirationsinking

grazing

excretion

fecal pellets

“sloppy feeding”

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

excretion by higher predators

need formulae to represent processes

model

Case study 1: Evans and Parslow (1985)

Previous models had simulated the details of a single bloom.

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timewinter spring summer fall

Case study 1: Evans and Parslow (1985)

EP85 modelled spring bloom as part of a repeating annual cycle, rather than as an isolated single event.

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timeyear 1 year 2 year 3

Case study 1: Evans and Parslow (1985)

If a bloom happens every year then it must be part of a repeating annual cycle, rather than than a simple cause and effect.

Annual model hopefully settles down to a repeating annual cycle, independent of initial conditions (sketch on board).

Physical forcing - irradiance, water temperature, mixed-layer depth. Often treated as known, and prescribed to be the same year after year.

Case study 1: Evans and Parslow (1985)

p328: bloom in spring, but ecosystem recreates over the fall and winter the conditions necessary for the bloom in spring.

Response of phytoplankton to light is modelled in detail (as given in the appendix, we won’t go into today).

The Model

p329: “We modelled the exchanges of matter (expressed as nitrogen, although the choice of unit is not important)”

What do they mean by this last part?

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The Equations

Volunteer from the audience to construct the food-web diagram on the board.

Table 1: Parameter values, obtained from various sources. Many of these values then became ‘standard’ in modelling. [Mention example].

Figure 1: Bottom diagram shows forcing: mixed-layer depth photosynthetic rate (α - not the one in the table!)

Mixed layer shallows - spring bloom.

BUT is shallowing of mixed layer completely necessary to obtain a spring bloom?

Fig 2 has mixed layer fixed at 80m. Still gives bloom.

Fig 3 shows bloom is ‘larger’.

Fig 4 has mixed layer fixed at 25m. Gives no bloom!!!!!So does bloom depend on deep rather than shallow mixed layers?

So consider simpler model: remove N equation simplify growth rate term (notation alert).

Fig 5 has mixed layer fixed at 80m - compare to Fig 2. Still gives bloom.

Fig 6 has mixed layer fixed at 25m. No bloom, as for Fig 4.

So still retaining (in simple model) the factors causing the bloom.

Some analysis. If α(t) held constant, P and H head to steady state.

Over year we get quasi-equilibrium cycle - Fig 7.

Conclude: rapid specific changes in growth rate cause spring bloom.

Case study 2:Fasham, Ducklow and McKelvie (1990) = FDM.

A far more detailed model. Uses many ideas from EP85.

We’ll discuss food web, won’t go through equations.

Popular model, has been used by many other people. Why?

Popular because:

Clearly explained: 15 pages (+appendix) just to explain equations and parameter values!

Somewhat overwhelming at first, but essential if others are to use model and thoroughly understand it.

Fasham gave away FORTRAN code, saving others the onerous task of having to code it themselves.

Case study 2:Fasham, Ducklow and McKelvie (1990)

NnPNd

Z DB

Nr

Data assimilation

FDM tuned one or two parameters.

Data assimilation techniques being used recently.

These prescribe cost function, which is then minimised by varying parameters.

Expect use of Bayesian statistical methods in the future – MCMC (Markov Chain Monte Carlo)

Lagrangian models

All models talked about today are Eulerian. They consider concentrations of biological entities.

Lagrangian models consider individuals (or rather collections of individuals), and track their location, history, nutrient status, etc., recorded in a ‘parish register’. Very computationally intensive. Main

proponent is John Woods in UK.

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ReferencesEvans and Parslow (1985) A model of annual plankton cycles, Biol. Oceanogr. 3:327-347.Fasham, Ducklow and McKelvie (1990) A nitrogen- based model of plankton dynamics in the oceanic mixed layer, J. Mar. Res., 48:591-639.Recent review papers:Franks (2002) NPZ models of plankton dynamics: their construction, coupling to physics, and application, J. Oceanogr. 58:379-387.Gentleman (2002) A chronology of plankton dynamics in silico: how computer models have been used to study marine ecosystems, Hydrobiologia 480:69-85.