c Simple physical-biogeochemical ticoupled a m he t
c Simple physical-biogeochemical ticoupled a m he t a m models of almarine ecosystems ^ Formulating quantitative mathematical models of conceptual ecosystems 1 MS320: John Wilkin
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 variables (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 2 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 modeled 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 3 Reality 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 Model Lump similar individuals into groups Might express some biomass as 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 4 The steps in constructing a model 1)
2) 3) 4) 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. 5 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 6 Processes Biological: Growth Death / mortality Photosynthesis Grazing Bacterial regeneration of nutrients Excretion of dissolved matter
Egestion of undigested food Physical: Mixing Transport (by currents from tides, winds ) Light Air-sea interaction (winds, heat fluxes, precipitation) 7 State variables and Processes Can use Redfield ratio to give e.g. carbon biomass from nitrogen equivalent Carbon-chlorophyll ratio might be assumed Where is the physics? 8 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 and light; tides 9
A very simple model NPZ Nutrients Phytoplankton Zooplankton all expressed in terms of equivalent nitrogen concentration Several elements of the state and multiple processes are combined e.g. the action of bacterial regeneration is treated as a flux from zooplankton mortality directly to nutrients with no explicit bacteria in the state 10 To run this yourself download the java program NPZ Visualizer from the
class web site ROMS fennel.h (carbon off, oxygen off, chl not shown) Banas, N. S., E. J. Lessard, R. M. Kudela, P. MacCready, T. D. Peterson, B. M. Hickey, and E. Frame (2009), Planktonic growth and grazing in the Columbia River plume region: A biophysical model study, J. Geophys. Res., 114, C00B06, doi:10.1029/2008JC004993. 11 Schematic of ROMS Fennel ecosystem model Phytoplankton concentration absorbs light Att(x,z) = AttSW + AttChl*Chlorophyll(x,z,t) dI =Att(z) * I (z)
dz Mathematical formulation d V Cn =sourcesn - sinksn + transfern, j dt j 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 13 Some calculus Slope of a continuous function of x is df slope = f = dx Baron Gottfried Wilhelm von Leibniz 1646-1716 14 For example:
State variables: Nutrient and Phytoplankton Process: Photosynthetic production of organic matter d P =vmax f ( N ) P dt N f (N ) = kN + N Michaelis and Menten (1913) Large N f (N ) 1 dP dt =vmax P Small N f ( N ) N / kn
dP dt = vmax N / kn P vmax is maximum growth rate (units are time-1) kn is half-saturation concentration; at N=kn f(kn)=0.5 15 State variables: Nutrient and Phytoplankton Process: Photosynthetic production of organic matter d P =vmax f ( N ) P dt d N =- vmax f ( N ) P dt d (P + N) =0 dt
The nitrogen consumed by the phytoplankton for growth must be lost from the Nutrients state variable The total inventory of nitrogen is conserved 16 Suppose there are ample nutrients so N is not limiting: then f(N) = 1 dP =vmax P dt Growth of P will be exponential P = Ae
vmax t 17 Suppose the plankton concentration held constant, and nutrients again are not limiting: f(N) = 1 dN =- vmax P dt N will decrease linearly with time as it is consumed to grow P 18 Suppose the plankton concentration held constant, but nutrients become limiting: then f(N) = N/kn
vmax P dN =N dt kn N will exponentially decay to zero until it is exhausted N = Ae - vmax P t kn 19 d P =vmax f (N)P
( N)P - ... dt d N =- vmax f (N)P ( N)P + ... dt 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. 20 21 N ~~~~~~~~~~~ No DN dN
m(No - N) m k Dz dz * Rate of vertical mixing from below depends on vertical gradient in nutrients and turbulent mixing coefficient 22 Coupling to physical processes Advection-diffusion-equation: sic y h
p s C + v C - DDC = gain (C ) - loss (C ) t turbulent Biological dynamics advection mixing C is the concentration of any biological state variable 23 I0 winter
spring summer fall 24 Simple 1-dimensional vertical model of mixed layer and N-P 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 the seasonal vertical structure of phytoplankton in shelf seas,
Marine Models Online, vol 1, 1999, 3-38. 25 wind stress drives mixing at surface z sea surface ~~~~~~~~~~~~~~~~ PAR n PAR n-1 PAR n-2 tidal currents drive mixing at the bottom
Biological equations are solved in each grid element depending on the local N, P and Z and available light. Vertical turbulent mixes causes N, P and Z to exchange between the grid element, and P and Z can sink. 26 wind stress drives mixing at surface z sea surface ~~~~~~~~~~~~~~~~ PAR
n ~~~~~~~~~~~~~~~~ O O PAR n-1 O O PAR n-2 tidal currents drive
mixing at the bottom OO Chl DIN 27 The rate of change of phytoplankton biomass is described by the equation P P P = k + m P - GP - ws t z z z
ve gra ve ph r z yto ve rtica ing mi tica loc l s pla mo ph xing l tur ity ink n rta kto yto of bu w ing
lity l en ng pla s at t row nk ton th The first term on the right describes the vertical turbulent transport of biomass. The second term represents growth of phytoplankton, with the specific instantaneous growth rate (s-1). Growth can be either light-limited or nutrientlimited, so the growth rate is taken as the lesser of: kQ m =mm 1 Q
m =qchl a I - r B (19a) describes nutrient-determined growth. max is the max growth rate, kQ the subsistence cell nutrient (19a) quota. (19b) describes light-determined growth, driven by mean photosynthetically-available (19b) radiation (PAR) in a depth element (I W/m2) of the model. qchl is cell chlorophyll:carbon ratio, is the max quantum yield, and rB is the respiration rate. Run Change Change settings settings . . Physicsc.dat: Physicsc.dat: stronger stronger PAR
PAR attenuation attenuation eliminates eliminates mid-depth mid-depth chl-max chl-max Phyto1d.dat: Phyto1d.dat: greater greater respiration respiration rate rate delays delays bloom bloom until until photosynthesis photosynthesis rate rate is is greater greater
29 I0 winter spring bloom summer fall secondary bloom 30 I0
winter spring bloom summer fall secondary bloom 31
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