Simple coupled physical-biogeochemical models of marine ecosystems

1. 1 Simple coupled physical-biogeochemical models of marine ecosystems 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.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.

2. 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. 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. 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

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5. 5 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. 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.

6. 6 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. There are many different biological models that have been developed for different applications. Choices follow applications, interests, and available understanding or data.

7. 7 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 The arrows that link boxes denote processes that move the e.g. nitrogen or carbon mass from one state variable to another

8. 8 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)

9. 9 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 The arrows that link boxes denote processes that move the e.g. nitrogen or carbon mass from one state variable to another

10. 10 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

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18. 18 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 nutrientsThen the bacterial loop becomes simply a process (�arrow between boxes) that represents the immediate conversion of dead phytoplankton and zooplankton to nutrients

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21. 21 Mathematical formulation Mass conservation Mass M (kilograms) of e.g. carbon or nitrogen in the system Concentration Cn (kilograms m-3) of state variable n: (mass per unit volume V) 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. 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.

22. 22 Mathematical formulation

23. 23 Some calculus

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25. 25 State variables: Nutrient and PhytoplanktonProcess: Photosynthetic production of organic matter 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 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

26. Average PP saturates at high PAR

27. 27 Sometimes you don�t get ks and Vmax or only with large uncertainties You can get different values in different regions Figure top left in Matlab: >> k=1.4; N=0:0.1:80; plot(N,N./(k+N),�-�,1.4,0.5,�rx�)Sometimes you don�t get ks and Vmax or only with large uncertainties You can get different values in different regions Figure top left in Matlab: >> k=1.4; N=0:0.1:80; plot(N,N./(k+N),�-�,1.4,0.5,�rx�)

28. 28 Uptake expressions There are other possible parameterizations of the nutrient-growth function, which can be justified on various grounds. A mathematical model has to use something that can be represented as a function. One test of the validity of the parameterization is the sensitivity of the model results (maybe it doesn�t matter, maybe it matters only at extremes (high/low) of nutrient concentration). Kn is probably a function of the plankton species, and possibly their age or size etc. Note for the exponential formula the limiting function assumes the value 0.6321 for kN; not strictly half-saturation concentration anymoreThere are other possible parameterizations of the nutrient-growth function, which can be justified on various grounds. A mathematical model has to use something that can be represented as a function. One test of the validity of the parameterization is the sensitivity of the model results (maybe it doesn�t matter, maybe it matters only at extremes (high/low) of nutrient concentration). Kn is probably a function of the plankton species, and possibly their age or size etc. Note for the exponential formula the limiting function assumes the value 0.6321 for kN; not strictly half-saturation concentration anymore

29. 29 State variables: Nutrient and PhytoplanktonProcess: Photosynthetic production of organic matter 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 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.

30. 30 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.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.

31. 31 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. 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.

32. 32 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? 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?

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34. 34 Rates of grazing by zooplankton and mortality of phytoplankton what must be the units of G and epsilon? Rates of grazing by zooplankton and mortality of phytoplankton what must be the units of G and epsilon?

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36. 36 There are many possible parameterizations for processes: e.g. Zooplankton grazing Again, no a priori principle to the mathematical form of the expression Mayzaud and Poulet argue that there can be food-level acclimation: herbivores modify their gut enzymes in response to phytoplankton concentration and be better adapted to succeed in grazing a large phytoplankton bloom Some grazers cannot graze large cells and the grazing rate falls for large particle sizesAgain, no a priori principle to the mathematical form of the expression Mayzaud and Poulet argue that there can be food-level acclimation: herbivores modify their gut enzymes in response to phytoplankton concentration and be better adapted to succeed in grazing a large phytoplankton bloom Some grazers cannot graze large cells and the grazing rate falls for large particle sizes

37. 37 Light

38. 38 Coupling to physical processes Advection-diffusion-equation:

39. 39 Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness) Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness)

40. 40 Simple 1-dimensional vertical model of mixed layer and N-P-Z type ecosystem Windows program and inputs files are at: http://marine.rutgers.edu/dmcs/ms320/Phyto1d/ Run the program called Phyto_1d.exe using the default input files Sharples, J., Investigating theseasonal 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)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)

41. 41 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) 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)

42. 42 Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness) Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness)

43. 43 Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness) Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness)

44. 44 Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness) Timing of the spring bloom related to light availability and thermal stratification (mixed layer thickness)

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