1. Simple coupled physical-biogeochemical models of marine ecosystems
mathematical ^
Formulating quantitative mathematical models of conceptual ecosystems
Today will present how we can formulate mathematical models that encapsulate the processes we have seen in previous lectures that affect interactions between biological, geochemical and physical components of an ecosystem.
MS320: John Wilkin

2. Why use mathematical models?
Conceptual models often characterize an ecosystem as a set of “boxes” linked by processes
Processes e.g. photosynthesis, growth, grazing, and mortality link elements of the …
State (“the boxes”) e.g. nutrient concentration, phytoplankton abundance, biomass, dissolved gases, of an ecosystem
In the lab, field, or mesocosm, we can observe some of the complexity of an ecosystem and quantify these processes
With quantitative rules for linking the boxes, we can attempt to simulate the changes over time of the ecosystem state
Aside from the intrinsic geeky pleasure of applying math and physics to biology…
Whether a model behaves sensibly becomes a test of our ability to understand how the various components depend on each other, how accurately we know those interdependencies, and how well we can observe, model or understand the physical environment.

3. What can we learn?
Suppose a model can simulate the spring bloom chlorophyll concentration observed by satellite using: observed light, a climatology of winter nutrients, ocean temperature and mixed layer depth …
Then the model rates of uptake of nutrients during the bloom and loss of particulates below the euphotic zone give us quantitative information on net primary production and carbon export – quantities we cannot easily observe directly

4. Reality Model Individual plants and animals
Many influences from nutrients and trace elements
Continuous functions of space and time
Varying behavior, choice, chance
Unknown or incompletely understood interactions
Lump similar individuals into groups
express in terms of biomass and C:N ratio
Small number of state variables (one or two limiting nutrients)
Discrete spatial points and time intervals
Average behavior based on ad hoc assumptions
Must parameterize unknowns
We must lump discrete individuals into variables that are functions of space and time in order to write equations for them.
(Aside: IBM Individual Based Models attempt to characterize the behavior of individual animals and interactions among themselves – but most of the other limitations in formulating models remain)
Ideally the state variables and process rates should be well-defined, observable and within a certain range of accuracy – generally not the case.
Different from physics, where model equations are derived from basic laws. Here, many empirical assumptions and outright guesses must be made based on limited observations or laboratory calculations that may not be valid in the real ecosystem.

5. The steps in constructing a model
Identify the scientific problem (e.g. seasonal cycle of nutrients and plankton in mid-latitudes; short-term blooms associated with coastal upwelling events; human-induced eutrophication and water quality; global climate change)
Determine relevant variables and processes that need to be considered
Develop mathematical formulation
Numerical implementation, provide forcing, parameters, etc.
There are many different biological models that have been developed for different applications. Choices follow applications, interests, and available understanding or data.

6. State variables and Processes
“NPZD”: model named for and characterized by its state variables
State variables are concentrations (in a common “currency”) that depend on space and time
Processes link the state variable boxes
The arrows that link boxes denote processes that move the e.g. nitrogen or carbon mass from one state variable to another

7. Processes Biological: Growth Death Photosynthesis Grazing
Bacterial regeneration of nutrients
Physical:
Mixing
Transport (by currents from tides, winds …)
Light
Air-sea interaction (winds, heat fluxes, precipitation)

8. State variables and Processes
Can use Redfield ratio to give e.g. carbon biomass from nitrogen equivalent
Carbon-chlorophyll ratio
Where is the physics?
The arrows that link boxes denote processes that move the e.g. nitrogen or carbon mass from one state variable to another

9. Examples of conceptual ecosystems that have been modeled
A model of a food web might be relatively complex
Several nutrients
Different size/species classes of phytoplankton
Different size/species classes of zooplankton
Detritus (multiple size classes)
Predation (predators and their behavior)
Multiple trophic levels
Pigments and bio-optical properties
Photo-adaptation, self-shading
3 spatial dimensions in the physical environment, diurnal cycle of atmospheric forcing, tides

10. Silicic acid – important limiting nutrient in N. Pacific
particulate silicon
Silicic acid – important limiting nutrient in N. Pacific
gelatinous zooplankton, euphausids, krill
copepods
ciliates
Fig. 1 – Schematic view of the NEMURO lower trophic level ecosystem model. Solid black arrows indicate nitrogen flows and dashed blue arrows indicate silicon. Dotted black arrows represent the exchange or sinking of the materials between the modeled box below the mixed layer depth.
Kishi, M., M. Kashiwai, and others, (2007), NEMURO - a lower trophic level model for the North Pacific marine ecosystem, Ecological Modelling, 202(1-2),

11. Soetaert K, Middelburg JJ, Herman PMJ, Buis K. 2000
Soetaert K, Middelburg JJ, Herman PMJ, Buis K On the coupling of benthic and pelagic bio-geochemical models. Earth-Sci. Rev. 51:

12. Schematic of ROMS “Fennel” ecosystem model
Phytoplankton concentration absorbs light
Att(x,z) = AttSW + AttChl*Chlorophyll(x,z,t)

13. Examples of conceptual ecosystems that have been modeled
In simpler models, elements of the state and processes can be combined if time and space scales justify this
e.g. bacterial regeneration can be treated as a flux from zooplankton mortality directly to nutrients
A very simple model might be just: N – P – Z
Nutrients
Phytoplankton
Zooplankton … all expressed in terms of equivalent nitrogen concentration
Then the bacterial loop becomes simply a process (‘arrow between boxes) that represents the immediate conversion of dead phytoplankton and zooplankton to nutrients

14. ROMS fennel.h (carbon off, oxygen off, chl not shown)

15. Mathematical formulation
Mass conservation
Mass M (kilograms) of e.g. carbon or nitrogen in the system
Concentration Cn (kg m-3) of state variable n is mass per unit volume V
Source for one state variable will be a sink for another
Concentration is a straightforward concept for nutrients, harder if you imagine distinct plankton cells. The assumption is that a very large number of cells in a volume, say 1 cubic meter, will be transported and mixed by the ocean currents as if the equivalent amount of nitrogen contained in the cells were simply dissolved in the water.

16. Mathematical formulation
e.g. inputs of nutrients from rivers or sediments
e.g. burial in sediments
e.g. nutrient uptake by phytoplankton
The key to model building is finding appropriate formulations for transfers, and not omitting important state variables

17. Some calculus Slope of a continuous function of x is
Baron Gottfried Wilhelm von Leibniz

18. For example: State variables: Nutrient and Phytoplankton Process: Photosynthetic production of organic matter
Large N
Small N
Photosynthetic production of org. matter requires light and nutrients, but lets just look at the nutrients for now
Michaelis_Menten uptake kinetics was derived for enzyme catalyzed reactions (ask Oscar).
This equation has the following qualitative features we like:
Growth is independent of nutrient concentrations if nutrient concentrations are high (when nothing is limiting, phytoplankton can only grow so fast at rate nu_max)
If nutrient concentrations are low they limit the growth rate to be proportional to the available nutrients
At small N << k, f(N) = N/k and growth rate is nu_max N/k
Michaelis and Menten (1913)
vmax is maximum growth rate (units are time-1) kn is “half-saturation” concentration; at N=kn f(kn)=0.5

19. State variables: Nutrient and Phytoplankton Process: Photosynthetic production of organic matter
The nitrogen consumed by the phytoplankton for growth must be lost from the Nutrients state variable
The photosynthesis process links the N and P boxes. The transfer of nitrogen between the boxes must balance for this process. So the N equation has the same term but with an opposite sign on the right-hand-side. The growth of phytoplankton (in equivalent nitrogen) is at the expense of drawing down the dissolved inorganic nitrogen in the surrounding waters.
The total inventory of nitrogen is conserved

20. Suppose there are ample nutrients so N is not limiting: then f(N) = 1
Growth of P will be exponential
Suppose f(N)=1
What does the solution to dP/dt = mu P look like? … exponential growth
Growth of P increases the right-hand-side of the N equation, driving down the available N until f(N) becomes small and then limits the P growth.

21. N will decrease linearly with time as it is consumed to grow P
Suppose the plankton concentration held constant, and nutrients again are not limiting: f(N) = 1
N will decrease linearly with time as it is consumed to grow P
Suppose we could hold the plankton concentration constant (old ones are dying as fast as new ones are growing, OR, they eat just enough to stay the same size), and f(N) = N/k, then the N equation is
dN/dt = -vmax N/k P = -a N where a = constant = vmax P /k
What does the solution to dN/dt = -a N look like? … exponential decay
The two state variables are linked by the photosynthesis process. Eventually N will diminish and dP/dt -> 0.

22. N will exponentially decay to zero until it is exhausted
Suppose the plankton concentration held constant, but nutrients become limiting: then f(N) = N/kn
N will exponentially decay to zero until it is exhausted
Suppose we could hold the plankton concentration constant, and f(N) = N/k, then the N equation is
dN/dt = -vmax N/k P = -a N where a = constant = vmax P /k
What does the solution to dN/dt = -a N look like? … exponential decay
The two state variables are linked by the photosynthesis process. Eventually N will diminish and will tend toward dP/dt -> 0 which means P stops changing, the P can grow no further because there is nothing to eat. So then what happens?

23. Can the right-hand-side of the P equation be negative?
Can the right-hand-side of the N equation be positive?
… So we need other processes to complete our model.

24. Rates of grazing by zooplankton and mortality of phytoplankton
what must be the units of G and epsilon?

26. Coupling to physical processes
Advection-diffusion-equation:
physics
turbulent
mixing
Biological dynamics
advection
C is the concentration of any biological state variable

27. I0 winter spring summer fall
Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness)

28. Simple 1-dimensional vertical model of mixed layer and N-P ecosystem
Windows program and inputs files are at:
Run the program called Phyto_1d.exe using the default input files
Sharples, J., Investigating the seasonal vertical structure of phytoplankton in shelf seas, Marine Models Online, vol 1, 1999, 3-38.
Control run with default input files shows spring and fall bloom
Physicsb.dat includes S2 tide and produces spring-neap cycle
Physicsc.dat has stronger PAR attenuation and eliminates mid-depth chl-max
Phyto1d.dat has greater respiration rate ands delays bloom until photosynthesis rate is greater (to balance respiration)

29. Fig. 1. Schematic of the model grid, and the physical processes
Fig. 1. Schematic of the model grid, and the physical processes. Velocities and scalars are associated with the centres of a grid cell, and vertical turbulent fluxes with the lower boundary of a grid cell.
Fig. 2. Schematic diagram of the biological scalars and processes at each grid cell.

30. grazing mortality
phytoplankton growth
vertical sinking at velocity ws
vertical turbulent mixing of phytoplankton

32. Physicsc.dat: stronger PAR attenuation eliminates mid-depth chl-max
Change settings ….
Physicsc.dat: stronger PAR attenuation eliminates mid-depth chl-max
Phyto1d.dat: greater respiration rate delays bloom until photosynthesis rate is greater
Physicsc.dat has stronger PAR attenuation and eliminates mid-depth chl-max
Phyto1d.dat has greater respiration rate and delays bloom until photosynthesis rate is greater (to balance respiration)

33. I0 winter spring summer fall bloom
Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness)
bloom

34. I0 winter spring summer fall bloom secondary bloom
Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness)
bloom
secondary bloom

35. I0 winter spring summer fall bloom secondary bloom
Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness)
bloom
secondary bloom