def __init__(self, lat, lon, rangebb, model, listparams=[], defaultparams=True, assumimm="P&L"):
self.Lat = lat
self.Lon = lon
self.RangeBB = rangebb
self.Model = model
self.ListParams = listparams
self.defaultParams = defaultparams
self.AssumImm = assumimm
self.MLD = ExtractEnvFor(self.Lat, self.Lon, self.RangeBB, 'mld')
self.N0X = ExtractEnvFor(self.Lat, self.Lon, self.RangeBB, 'n0x')
self.SST = ExtractEnvFor(self.Lat, self.Lon, self.RangeBB, 'sst')
self.PAR = ExtractEnvFor(self.Lat, self.Lon, self.RangeBB, 'par')
self.Params = self.setup_params()
self.initcond = self.initialconditions()
self.timedays = np.arange(0., self.Params['timeyears']*365., 1.0)
self.outvariables = self.modelrun()
def environemntalforcing(lat, lon, rbb):
tos = time.time()
print '\nComputation Time of Environmental Forcing'
# Extract environmental forcing using the class ExtractEnvFor in the module envforcing
MLD = ExtractEnvFor(lat, lon, rbb, 'mld')
tos1 = time.time()
print ' MLD=%4.3f seconds' % (tos1-tos)
PAR = ExtractEnvFor(lat, lon, rbb, 'par')
tos2 = time.time()
print ' PAR=%4.3f seconds' % (tos2-tos1)
SST = ExtractEnvFor(lat, lon, rbb, 'sst')
tos3 = time.time()
print ' SST=%4.3f seconds' % (tos3-tos2)
N0X = ExtractEnvFor(lat, lon, rbb, 'n0x')
tos4 = time.time()
print ' NOX=%4.3f seconds' % (tos4-tos3)
nt = np.arange(0., 366., 1.0)
# Figure NO.1
f1, (ax1, ax2, ax3, ax4) = plt.subplots(4, 1, sharex='col', sharey='row')
# MLD
ax1.plot(nt, MLD.dailyinterp(nt), c='black', lw=3)
ax1.set_ylabel('MLD (m)', multialignment='center', fontsize=12)
ax1.set_xlim(0, 365)
ax1.invert_yaxis()
# PAR
ax2.plot(nt, PAR.dailyinterp(nt), c='black', lw=3)
ax2.set_ylabel('PAR \n' '(Ein / m^2 / d^1)', multialignment='center', fontsize=12)
# SST
ax3.plot(nt, SST.dailyinterp(nt, k=5), c='black', lw=3)
ax3.set_ylabel('SST \n' '(degrees C)', multialignment='center', fontsize=12)
# N0
ax4.plot(nt, N0X.dailyinterp(nt, k=5), c='black', lw=3)
ax4.set_ylabel('N0 \n' '(mmol N / m^3)', multialignment='center', fontsize=12)
ax4.set_xlabel('Time (days)', fontsize=14)
class SM:
"""
Class with methods to quantify the size dynamics of phytoplankton
communities in the ocean using a 0-dimensional NPZD model structure.
Parameters
----------
lat: is the Latitude with a range from -90 to 90 degrees (North positive).
lon: is the Longitude with a range from -180 to 180 degrees (East positive).
rangebb: is the spatial range of the bounding box in degrees.
model: String specifying the size model variant to calculate, either 'Imm' for
a model with immigration, 'TraitDif' for a model with trait diffusion
mechanism, 'FixVar' for a model with a fix amount of variance or
'UnsustVar' for a model that does not sustain variance and "FullModel"
for a size model with a full spectrum of morphotypes.
listparams: A list of tuples ('parameter_name',value) with the new values for
for the specified parameter.
defaultparams: a boolean to specify the use of default parameters or to
modify the parameters suggested in ListParams.
:assumimm: Assumptions on the immigrating community (P_imm). In assumption "P&L"
the total biomass of P_imm is proportional to the biomass of the local
community (P) and the mean size of P_imm is equal to the size of P. In
assumption "L" only the size is equal between the two communities. In
assumption "P" only the total biomass is proportional between the
communities. Last if "None" P and L are different and independent for each
other.
Returns
-------
outvariables: is a ndarray that contains the results of all state variables.
In addition, in the ndarray we included dummy variables that quantify
the biomass fluxes between the state variables. Please refer to each model variant
to identify each variable.
"""
def __init__(self, lat, lon, rangebb, model, listparams=[], defaultparams=True, assumimm="P&L"):
self.Lat = lat
self.Lon = lon
self.RangeBB = rangebb
self.Model = model
self.ListParams = listparams
self.defaultParams = defaultparams
self.AssumImm = assumimm
self.MLD = ExtractEnvFor(self.Lat, self.Lon, self.RangeBB, 'mld')
self.N0X = ExtractEnvFor(self.Lat, self.Lon, self.RangeBB, 'n0x')
self.SST = ExtractEnvFor(self.Lat, self.Lon, self.RangeBB, 'sst')
self.PAR = ExtractEnvFor(self.Lat, self.Lon, self.RangeBB, 'par')
self.Params = self.setup_params()
self.initcond = self.initialconditions()
self.timedays = np.arange(0., self.Params['timeyears']*365., 1.0)
self.outvariables = self.modelrun()
def initialconditions(self):
"""
Method to assign the initial conditions depending on the type of model
structure, i.e. full or aggregate model.
"""
if self.Model == 'FullModel':
return np.concatenate(([[self.Params['N0']], [self.Params['Z0']], [self.Params['D0']],
[self.Params['P0']]*self.Params['NoMtype']]), 0)
else:
return np.concatenate(([[self.Params['N0']], [self.Params['Z0']], [self.Params['D0']], [self.Params['P0']],
[self.Params['L_mean0']], [self.Params['L_var0']], [0]*29]), 0)
def slns(self, ms, sv):
"""
Method to log-transform the mean trait, based on the log-normal relationship,
suggested by Bruggeman (2009).
Parameters
----------
ms: Mean trait or mean size
sv: Trait variance or size variance
"""
return np.log(ms / np.sqrt(sv / ms ** 2 + 1.))
def vlnv(self, ms, sv):
"""
Method to log-transform the trait variance, based on the log-normal
relationship,suggested by Bruggeman (2009).
Parameters
----------
ms: Mean trait or mean size
sv: Trait variance or size variance
"""
return np.log(sv / ms ** 2 + 1.)
def setup_params(self):
"""
Method to assign either the default values to all parameters
or to specify the new values according to the values provided
on ListParams.
"""
lnvar0 = self.vlnv(25.1, 500)
lnmean0 = self.slns(25, 500)
n0 = np.mean(self.N0X.outForcing[:12])
default_parameters = {
'kappa': 0.1, # Diffusive mixing across thermocline (m*d^-1)
'kw': 0.1, # Light attenuation constant (m^-1)
'OptI': 30., # Optimum irradiance (einstein*m^-2*d^-1)
'muP': 1.5, # Phytoplankton max growth rate (d^-1)
'moP': 0.05, # Phytoplankton mortality rate (d^-1)
'muZ': 1.35, # Zooplankton maximum grazing rate (d^-1)
'moZ': 0.3, # Zooplankton mortality
'Kp': 0.1, # P half-saturation constant (mmol*m^-3)
'deltaZ': 0.31, # P assimilation coefficient (-)
'deltaD': 0.1, # Mineralization rate (d^-1)
'deltaI': 0.008, # Immigration rate (mmol*m^-3 * d^-1)
'nu': 0.008, # Trait diffusivity parameter (-)
'alphaG': -0.75, # Slope of allometric grazer preference ([micro m ESD]^-1)
'alphaU': 0.81, # Slope of Kn allometric function (mmol * m^-3 * [micro m ESD]^-1) (Modified from Litchman et. al 2007)
'betaU': 0.14275, # Intercept of Kn allometric function (mmol * m^-3) (Modified from Litchman et. al 2007)
'alphav': 1.17, # Slope of Sinking allometric function (meters * d^-1 * [micro m ESD]^-1) (Modified from Kiorboe 1993)
'betav': 0.01989, # Intercept of Sinking allometric function (meters * d^-1) (Modified from Kiorboe 1993)
'L_var0': lnvar0, # Size variance of the immigrating community (Ln [micro m ESD]^2)
'L_mean0': lnmean0, # Mean size of the immigrating community (Ln micro m ESD)
'N0': n0, # Initial Nutrient concentration (mmol*m^-3)
'Z0': 0.1, # Initial Zooplankton concentration (mmol*m^-3)
'D0': 0.01, # Initial Detritus concentration (mmol*m^-3)
'P0': 0.1, # Initial Phytoplankton concentration (mmol*m^-3)
'timeyears': 10.0, # Maximum running time of the model (years)
'NoMtype': 10, # Number of morphotypes for the full model
'sizemin': 0.2, # Minimum size of the phytoplankton community (micro m ESD)
'sizemax': 50. # Maximum size of the phytoplankton community (micro m ESD)
}
if self.defaultParams:
return default_parameters
else:
for parnames, values in self.ListParams:
default_parameters[parnames] = values
return default_parameters
def sizemodel_imm(self, x, t):
"""
This size based model variant is based on Acevedo-Trejos et al. (2015) in Sci. Rep.
Parameters
----------
x: array with initial conditions for the state variables
t: time
"""
# Initialization of state variables and dummy variables to store the biomass fluxes
N = x[0]
Z = x[1]
D = x[2]
P = x[3]
L = x[4]
V = x[5]
dxdt = np.zeros(35)
# Edible phytoplankton
Ped = P*(np.exp(L)**self.Params['alphaG']+0.5*V*self.Params['alphaG']**2*np.exp(L)**self.Params['alphaG'])
# Gains of phytoplankton biomass
NutrientUptake = N/(N+self.Params['betaU']*np.exp(L)**self.Params['alphaU'])
LightHarvesting = 1./(self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0)) * (-np.exp(1. - self.PAR.dailyinterp(t) / self.Params['OptI']) - (-np.exp((1. - (self.PAR.dailyinterp(t) * np.exp(-self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0))) / self.Params['OptI']))))
TemperatureDepGrowth = np.exp(0.063 * self.SST.dailyinterp(t, k=5))
Gains = self.Params['muP'] * NutrientUptake * LightHarvesting * TemperatureDepGrowth
# Losses of phytoplankton biomass
Grazing = self.Params['muZ']*Z*np.exp(L)**self.Params['alphaG']/(Ped+self.Params['Kp'])
Sinking = (self.Params['betav']*np.exp(L)**self.Params['alphav'])/self.MLD.dailyinterp(t, k=3, s=0)
OtherPMortalities = self.Params['moP']
Mixing = (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
Losses = Grazing + Sinking + OtherPMortalities + Mixing
# Other Processes
ZooGrowth = self.Params['deltaZ'] * Grazing * P
ZooMortality = self.Params['moZ'] * Z**2
ZooMixing = Z * self.MLD.firstderivspl(t, k=3, s=0) / self.MLD.dailyinterp(t, k=3, s=0)
UnassimilatedProduction = (1.-self.Params['deltaZ']) * Grazing * P
Mineralization = self.Params['deltaD']*D
DetritusMixing = D * (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
NMixing = Mixing * (self.N0X.dailyinterp(t, k=5) - N)
# Derivatives for the growth components of phytoplankton with respect to the trait
tt0 = self.Params['muP']*LightHarvesting*TemperatureDepGrowth
tt2 = self.Params['muZ']*Z
d1r0 = -N*tt0*self.Params['betaU']*self.Params['alphaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d1r2 = tt2*self.Params['alphaG']*np.exp(L)**self.Params['alphaG']/(Ped + self.Params['Kp'])
d1r3 = self.Params['betav']*self.Params['alphav']*np.exp(L)**self.Params['alphav']/self.MLD.dailyinterp(t, k=3, s=0)
d2r0 = N*tt0*self.Params['betaU']*self.Params['alphaU']**2*(2*self.Params['betaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU']) - 1)*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d2r2 = tt2*self.Params['alphaG']**2*np.exp(L)**self.Params['alphaG']/(Ped + self.Params['Kp'])
d2r3 = self.Params['betav']*self.Params['alphav']**2*np.exp(L)**self.Params['alphav']/self.MLD.dailyinterp(t, k=3, s=0)
d1 = d1r0-d1r2-d1r3
d2 = d2r0-d2r2-d2r3
# Corrections of higher order moments
E = 0.5*V*d2
EN = 0.5*V*d2r0
EZ = self.Params['deltaZ'] * 0.5*V*d2r2 * P
EZD = (1.-self.Params['deltaZ']) * 0.5*V*d2r2 * P
# Skewness and kurtosis according to normal distribution
m3 = 0.
m4 = 3.
# Assumptions on the immigrating community
if self.AssumImm == 'P&L':
# Default assumptions as used by Acevedo-Trejos et al. (2015) in Sci.Rep.
self.Params['L_mean0'] = L
Immigration = self.Params['deltaI'] * P
elif self.AssumImm == 'L':
self.Params['L_mean0'] = L
Immigration = self.Params['deltaI']
elif self.AssumImm == 'P':
Immigration = self.Params['deltaI'] * P
elif self.AssumImm == 'None':
Immigration = self.Params['deltaI']
# state variables
dxdt[0] = -P * (Gains + EN) + Mineralization + NMixing # Nutrients
dxdt[1] = ZooGrowth + EZ - ZooMortality - ZooMixing # Zooplankton
dxdt[2] = UnassimilatedProduction + P * OtherPMortalities + EZD + ZooMortality - Mineralization - DetritusMixing # Detritus
dxdt[3] = P * (Gains - Losses + E) + Immigration # Phytoplankton
dxdt[4] = V*d1 + m3*0.5*d2 + Immigration/P*(self.Params['L_mean0']-L) # Mean Size
dxdt[5] = 0.5*(m4-1.)*V**2*d2 + Immigration/P*((self.Params['L_var0']-V) + (self.Params['L_mean0']-L)**2) # Size Variance
# Biomass Fluxes
dxdt[6] = P * Gains # gross growth
dxdt[7] = NutrientUptake # Nutrient Uptake
dxdt[8] = LightHarvesting # Light Harvesting
dxdt[9] = TemperatureDepGrowth # Phytoplankton temperature dependency
dxdt[10] = P * Grazing # Grazing
dxdt[11] = P * Sinking # Phytoplankton Sinking
dxdt[12] = P * OtherPMortalities # Other P mortalities
dxdt[13] = P * Mixing # Phytoplankton Mixing
dxdt[14] = ZooGrowth # Zooplankton Growth
dxdt[15] = ZooMortality # Zooplankton Predator Mortality
dxdt[16] = ZooMixing # Zooplankton Mixing
dxdt[17] = UnassimilatedProduction # Unassimilated Production
dxdt[18] = Mineralization # Mineralization
dxdt[19] = DetritusMixing # Detritus Mixing
dxdt[20] = NMixing # Nutrients Mixing
dxdt[21] = Immigration # Immigration
dxdt[22] = P*E # P Higher order correction
dxdt[23] = P*EN # N Higher order correction
dxdt[24] = EZ # Grazing Higher order correction
dxdt[25] = EZD # Unassimilated Production Higher order correction
dxdt[26] = V*d1 # Changes in mean size
dxdt[27] = Immigration/P*(self.Params['L_mean0']-L) # Immigrating mean size
dxdt[28] = 0.5*(m4-1.)*V**2*d2 # Changes in size variance
dxdt[29] = Immigration/P*((self.Params['L_var0']-V) + (self.Params['L_mean0']-L)**2) # Immigrating size variance
dxdt[30] = self.Params['betaU']*np.exp(L)**self.Params['alphaU'] # N Half Saturation
dxdt[31] = d2 # second derivative with respect to trait
dxdt[32] = d2r0 # second derivative of Nutrient Uptake with respect to the trait
dxdt[33] = d2r2 # second derivative of Grazing with respect to the trait
dxdt[34] = d2r3 # second derivative of Sinking with respect to the trait
return dxdt
def sizemodel_traitdif(self, x, t):
"""
This size based model variant is based on Acevedo-Trejos et al. (2015) in Sci. Rep.
but with a trait diffusion mechanism to sustain size variance as
suggested by Merico et al. (2014) in Frontiers in Ecology and Evolution.
Parameters
----------
x: array with initial conditions for the state variables
t: time
"""
# Initialization of state variables and dummy variables to store the biomass fluxes
N = x[0]
Z = x[1]
D = x[2]
P = x[3]
L = x[4]
V = x[5]
dxdt = np.zeros(35)
# Edible phytoplankton
Ped = P*(np.exp(L)**self.Params['alphaG']+0.5*V*self.Params['alphaG']**2*np.exp(L)**self.Params['alphaG'])
# Gains of phytoplankton biomass
NutrientUptake = N/(N+self.Params['betaU']*np.exp(L)**self.Params['alphaU'])
LightHarvesting = 1./(self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0)) * (-np.exp(1. - self.PAR.dailyinterp(t) / self.Params['OptI']) - (-np.exp((1. - (self.PAR.dailyinterp(t) * np.exp(-self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0))) / self.Params['OptI']))))
TemperatureDepGrowth = np.exp(0.063 * self.SST.dailyinterp(t, k=5))
Gains = self.Params['muP'] * NutrientUptake * LightHarvesting * TemperatureDepGrowth
# Losses of phytoplankton biomass
Grazing = self.Params['muZ']*Z*np.exp(L)**self.Params['alphaG']/(Ped+self.Params['Kp'])
Sinking = (self.Params['betav']*np.exp(L)**self.Params['alphav'])/self.MLD.dailyinterp(t, k=3, s=0)
OtherPMortalities = self.Params['moP']
Mixing = (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
Losses = Grazing + Sinking + OtherPMortalities + Mixing
# Other Processes
ZooGrowth = self.Params['deltaZ'] * Grazing * P
ZooMortality = self.Params['moZ'] * Z**2
ZooMixing = Z * self.MLD.firstderivspl(t, k=3, s=0) / self.MLD.dailyinterp(t, k=3, s=0)
UnassimilatedProduction = (1.-self.Params['deltaZ']) * Grazing * P
Mineralization = self.Params['deltaD']*D
DetritusMixing = D * (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
NMixing = Mixing * (self.N0X.dailyinterp(t, k=5) - N)
# Derivatives for the growth components of phytoplankton with respect to the trait
tt0 = self.Params['muP']*LightHarvesting*TemperatureDepGrowth
tt2 = self.Params['muZ']*Z
d1r0 = -N*tt0*self.Params['betaU']*self.Params['alphaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d1r2 = tt2*self.Params['alphaG']*np.exp(L)**self.Params['alphaG']/(Ped + self.Params['Kp'])
d1r3 = self.Params['betav']*self.Params['alphav']*np.exp(L)**self.Params['alphav']/self.MLD.dailyinterp(t, k=3, s=0)
d2r0 = N*tt0*self.Params['betaU']*self.Params['alphaU']**2*(2*self.Params['betaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU']) - 1)*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d2r2 = tt2*self.Params['alphaG']**2*np.exp(L)**self.Params['alphaG']/(Ped + self.Params['Kp'])
d2r3 = self.Params['betav']*self.Params['alphav']**2*np.exp(L)**self.Params['alphav']/self.MLD.dailyinterp(t, k=3, s=0)
d3r0 = N*self.Params['betaU']*self.Params['alphaU']**3*tt0*(-6*self.Params['betaU']**2*np.exp(L)**(2*self.Params['alphaU'])/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2 + 6*self.Params['betaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU']) - 1)*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d4r0 = N*self.Params['betaU']*self.Params['alphaU']**4*tt0*(24*self.Params['betaU']**3*np.exp(L)**(3*self.Params['alphaU'])/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**3 - 36*self.Params['betaU']**2*np.exp(L)**(2*self.Params['alphaU'])/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2 + 14*self.Params['betaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU']) - 1)*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d1 = d1r0-d1r2-d1r3
d2 = d2r0-d2r2-d2r3
# Corrections of higher order moments
E = 0.5*V*d2
EN = 0.5*V*d2r0
EZ = self.Params['deltaZ'] * 0.5*V*d2r2 * P
EZD = (1.-self.Params['deltaZ']) * 0.5*V*d2r2 * P
# Skewness and kurtosis according to normal distribution
m3 = 0.
m4 = 3.
# state variables
dxdt[0] = -P * (Gains + EN) + Mineralization + NMixing # Nutrients
dxdt[1] = ZooGrowth + EZ - ZooMortality - ZooMixing # Zooplankton
dxdt[2] = UnassimilatedProduction + P * OtherPMortalities + EZD + ZooMortality - Mineralization - DetritusMixing # Detritus
dxdt[3] = P * (Gains - Losses + E + 0.5*self.Params['nu']*(V*d4r0-3.*d2r0)) # Phytoplankton
dxdt[4] = V*d1 + m3*0.5*d2 + self.Params['nu']*(V*d3r0-3.*d1r0) # Mean Size
dxdt[5] = 0.5*(m4-1.)*V**2*d2 + self.Params['nu']*(V**2*d4r0-5.*V*d2r0+2.*Gains) # Size Variance
# Biomass Fluxes
dxdt[6] = P * Gains # gross growth
dxdt[7] = NutrientUptake # Nutrient Uptake
dxdt[8] = LightHarvesting # Light Harvesting
dxdt[9] = TemperatureDepGrowth # Phytoplankton Temperature Dependency
dxdt[10] = P * Grazing # Grazing
dxdt[11] = P * Sinking # Phytoplankton Sinking
dxdt[12] = P * OtherPMortalities # Other P mortalities
dxdt[13] = P * Mixing # Phytoplankton Mixing
dxdt[14] = ZooGrowth # Zooplankton Growth
dxdt[15] = ZooMortality # Zooplankton Predator Mortality
dxdt[16] = ZooMixing # Zooplankton Mixing
dxdt[17] = UnassimilatedProduction # Unassimilated Production
dxdt[18] = Mineralization # Mineralization
dxdt[19] = DetritusMixing # Detritus Mixing
dxdt[20] = NMixing # Nutrients Mixing
dxdt[21] = 0.5*self.Params['nu']*(V*d4r0-3.*d2r0) # Trait Diffusion into P
dxdt[22] = P*E # P Higher order correction
dxdt[23] = P*EN # N Higher order correction
dxdt[24] = EZ # Grazing Higher order correction
dxdt[25] = EZD # Unassimilated Production Higher order correction
dxdt[26] = V*d1 # Changes in mean size
dxdt[27] = self.Params['nu']*(V*d3r0-3.*d1r0) # Trait Diffusion into S
dxdt[28] = 0.5*(m4-1.)*V**2*d2 # Changes in size variance
dxdt[29] = self.Params['nu']*(V**2*d4r0-5.*V*d2r0+2.*Gains) # Trait Diffusion into V
dxdt[30] = self.Params['betaU']*np.exp(L)**self.Params['alphaU'] # N Half Saturation
dxdt[31] = d2 # second derivative with respect to trait
dxdt[32] = d2r0 # second derivative of Nutrient Uptake with respect to the trait
dxdt[33] = d2r2 # second derivative of Grazing with respect to the trait
dxdt[34] = d2r3 # second derivative of Sinking with respect to the trait
return dxdt
def sizemodel_fixvar(self, x, t):
"""
This size based model variant is based on Acevedo-Trejos et al. (2015) in Sci. Rep.
but with a fix size variance.
Parameters
----------
x: array with initial conditions for the state variables
t: time
"""
# Initialization of state variables and dummy variables to store the biomass fluxes
N = x[0]
Z = x[1]
D = x[2]
P = x[3]
L = x[4]
V = x[5]
dxdt = np.zeros(35)
# Edible phytoplankton
Ped =P*(np.exp(L)**self.Params['alphaG']+0.5*V*self.Params['alphaG']**2*np.exp(L)**self.Params['alphaG'])
# Gains of phytoplankton biomass
NutrientUptake = N/(N+self.Params['betaU']*np.exp(L)**self.Params['alphaU'])
LightHarvesting = 1./(self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0)) * (-np.exp(1. - self.PAR.dailyinterp(t) / self.Params['OptI']) - (-np.exp((1. - (self.PAR.dailyinterp(t) * np.exp(-self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0))) / self.Params['OptI']))))
TemperatureDepGrowth = np.exp(0.063 * self.SST.dailyinterp(t, k=5))
Gains = self.Params['muP'] * NutrientUptake * LightHarvesting * TemperatureDepGrowth
# Losses of phytoplankton biomass
Grazing = self.Params['muZ']*Z*np.exp(L)**self.Params['alphaG']/(Ped+self.Params['Kp'])
Sinking = (self.Params['betav']*np.exp(L)**self.Params['alphav'])/self.MLD.dailyinterp(t, k=3, s=0)
OtherPMortalities = self.Params['moP']
Mixing = (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
Losses = Grazing + Sinking + OtherPMortalities + Mixing
# Other Processes
ZooGrowth = self.Params['deltaZ'] * Grazing * P
ZooMortality = self.Params['moZ'] * Z**2
ZooMixing = Z * self.MLD.firstderivspl(t, k=3, s=0) / self.MLD.dailyinterp(t, k=3, s=0)
UnassimilatedProduction = (1.-self.Params['deltaZ']) * Grazing * P
Mineralization = self.Params['deltaD']*D
DetritusMixing = D * (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
NMixing = Mixing * (self.N0X.dailyinterp(t, k=5) - N)
# Derivatives for the growth components of phytoplankton with respect to the trait
tt0 = self.Params['muP']*LightHarvesting*TemperatureDepGrowth
tt2 = self.Params['muZ']*Z
d1r0 = -N*tt0*self.Params['betaU']*self.Params['alphaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d1r2 = tt2*self.Params['alphaG']*np.exp(L)**self.Params['alphaG']/(Ped + self.Params['Kp'])
d1r3 = self.Params['betav']*self.Params['alphav']*np.exp(L)**self.Params['alphav']/self.MLD.dailyinterp(t, k=3, s=0)
d2r0 = N*tt0*self.Params['betaU']*self.Params['alphaU']**2*(2*self.Params['betaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU']) - 1)*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d2r2 = tt2*self.Params['alphaG']**2*np.exp(L)**self.Params['alphaG']/(Ped + self.Params['Kp'])
d2r3 = self.Params['betav']*self.Params['alphav']**2*np.exp(L)**self.Params['alphav']/self.MLD.dailyinterp(t, k=3, s=0)
d1 = d1r0-d1r2-d1r3
d2 = d2r0-d2r2-d2r3
# Corrections of higher order moments
E = 0 # 0.5*V*d2
EN = 0 # 0.5*V*d2r0
EZ = 0 # self.Params['deltaZ'] * 0.5*V*d2r2 * P
EZD = 0 # (1.-self.Params['deltaZ']) * 0.5*V*d2r2 * P
# state variables
dxdt[0] = -P * (Gains + EN) + Mineralization + NMixing # Nutrients
dxdt[1] = ZooGrowth + EZ - ZooMortality - ZooMixing # Zooplankton
dxdt[2] = UnassimilatedProduction + P * OtherPMortalities + EZD + ZooMortality - Mineralization - DetritusMixing # Detritus
dxdt[3] = P * (Gains - Losses + E) # Phytoplankton
dxdt[4] = V*d1 # Mean Size
dxdt[5] = 0.0 # Size Variance
# Biomass Fluxes
dxdt[6] = P * Gains # gross growth
dxdt[7] = NutrientUptake # Nutrient Uptake
dxdt[8] = LightHarvesting # Light Harvesting
dxdt[9] = TemperatureDepGrowth # Phytoplankton Temperature Dependency
dxdt[10] = P * Grazing # Grazing
dxdt[11] = P * Sinking # Phytoplankton Sinking
dxdt[12] = P * OtherPMortalities # Other P mortalities
dxdt[13] = P * Mixing # Phytoplankton Mixing
dxdt[14] = ZooGrowth # Zooplankton Growth
dxdt[15] = ZooMortality # Zooplankton Predator Mortality
dxdt[16] = ZooMixing # Zooplankton Mixing
dxdt[17] = UnassimilatedProduction # Unassimilated Production
dxdt[18] = Mineralization # Mineralization
dxdt[19] = DetritusMixing # Detritus Mixing
dxdt[20] = NMixing # Nutrients Mixing
dxdt[21] = 0. # No addition into P
dxdt[22] = P*E # P Higher order correction
dxdt[23] = P*EN # N Higher order correction
dxdt[24] = EZ # Grazing Higher order correction
dxdt[25] = EZD # Unassimilated Production Higher order correction
dxdt[26] = V*d1 # Changes in mean size
dxdt[27] = 0. # No addition into S
dxdt[28] = 0. # Changes in size variance
dxdt[29] = 0. # No addition into V
dxdt[30] = self.Params['betaU']*np.exp(L)**self.Params['alphaU'] # N Half Saturation
dxdt[31] = 0. # second derivative with respect to trait
dxdt[32] = 0. # second derivative of Nutrient Uptake with respect to the trait
dxdt[33] = 0. # second derivative of Grazing with respect to the trait
dxdt[34] = 0. # second derivative of Sinking with respect to the trait
return dxdt
def sizemodel_unvar(self, x, t):
"""
This size based model variant is based on Acevedo-Trejos et al. (2015) in Sci. Rep.
but without any mechanism to sustain variance.
Parameters
----------
x: array with initial conditions for the state variables
t: time
"""
# Initialization of state variables and dummy variables to store the biomass fluxes
N = x[0]
Z = x[1]
D = x[2]
P = x[3]
L = x[4]
V = x[5]
dxdt = np.zeros(35)
# Edible phytoplankton
Ped = P*(np.exp(L)**self.Params['alphaG']+0.5*V*self.Params['alphaG']**2*np.exp(L)**self.Params['alphaG'])
# Gains of phytoplankton biomass
NutrientUptake = N/(N+self.Params['betaU']*np.exp(L)**self.Params['alphaU'])
LightHarvesting = 1./(self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0)) * (-np.exp(1. - self.PAR.dailyinterp(t) / self.Params['OptI']) - (-np.exp((1. - (self.PAR.dailyinterp(t) * np.exp(-self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0))) / self.Params['OptI']))))
TemperatureDepGrowth = np.exp(0.063 * self.SST.dailyinterp(t, k=5))
Gains = self.Params['muP'] * NutrientUptake * LightHarvesting * TemperatureDepGrowth
# Losses of phytoplankton biomass
Grazing = self.Params['muZ']*Z*np.exp(L)**self.Params['alphaG']/(Ped+self.Params['Kp'])
Sinking = (self.Params['betav']*np.exp(L)**self.Params['alphav'])/self.MLD.dailyinterp(t, k=3, s=0)
OtherPMortalities = self.Params['moP']
Mixing = (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
Losses = Grazing + Sinking + OtherPMortalities + Mixing
# Other Processes
ZooGrowth = self.Params['deltaZ']*Grazing * P
ZooMortality = self.Params['moZ'] * Z**2
ZooMixing = Z * self.MLD.firstderivspl(t, k=3, s=0) / self.MLD.dailyinterp(t, k=3, s=0)
UnassimilatedProduction = (1.-self.Params['deltaZ']) * Grazing * P
Mineralization = self.Params['deltaD']*D
DetritusMixing = D * (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
NMixing = Mixing * (self.N0X.dailyinterp(t, k=5) - N)
# Derivatives for the growth components of phytoplankton with respect to the trait
tt0 = self.Params['muP']*LightHarvesting*TemperatureDepGrowth
tt2 = self.Params['muZ']*Z
d1r0 = -N*tt0*self.Params['betaU']*self.Params['alphaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d1r2 = tt2*self.Params['alphaG']*np.exp(L)**self.Params['alphaG']/(Ped + self.Params['Kp'])
d1r3 = self.Params['betav']*self.Params['alphav']*np.exp(L)**self.Params['alphav']/self.MLD.dailyinterp(t, k=3, s=0)
d2r0 = N*tt0*self.Params['betaU']*self.Params['alphaU']**2*(2*self.Params['betaU']*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU']) - 1)*np.exp(L)**self.Params['alphaU']/(N + self.Params['betaU']*np.exp(L)**self.Params['alphaU'])**2
d2r2 = tt2*self.Params['alphaG']**2*np.exp(L)**self.Params['alphaG']/(Ped + self.Params['Kp'])
d2r3 = self.Params['betav']*self.Params['alphav']**2*np.exp(L)**self.Params['alphav']/self.MLD.dailyinterp(t, k=3, s=0)
d1 = d1r0-d1r2-d1r3
d2 = d2r0-d2r2-d2r3
# Corrections of higher order moments
E = 0.5*V*d2
EN = 0.5*V*d2r0
EZ = self.Params['deltaZ'] * 0.5*V*d2r2 * P
EZD = (1.-self.Params['deltaZ']) * 0.5*V*d2r2 * P
# Skewness and kurtosis according to normal distribution
m3 = 0.
m4 = 3.
# state variables
dxdt[0] = -P * (Gains + EN) + Mineralization + NMixing # Nutrients
dxdt[1] = ZooGrowth + EZ - ZooMortality - ZooMixing # Zooplankton
dxdt[2] = UnassimilatedProduction + P * OtherPMortalities + EZD + ZooMortality - Mineralization - DetritusMixing # Detritus
dxdt[3] = P*(Gains - Losses + E) # Phytoplankton
dxdt[4] = V*d1 + m3*0.5*d2 # Mean Size
dxdt[5] = 0.5*(m4-1.)*V**2*d2 # Size Variance
# Biomass Fluxes
dxdt[6] = P * Gains # gross growth
dxdt[7] = NutrientUptake # Nutrient Uptake
dxdt[8] = LightHarvesting # Light Harvesting
dxdt[9] = TemperatureDepGrowth # Phytoplankton Temperature Dependency
dxdt[10] = P * Grazing # Grazing
dxdt[11] = P * Sinking # Phytoplankton Sinking
dxdt[12] = P * OtherPMortalities # Other P mortalities
dxdt[13] = P * Mixing # Phytoplankton Mixing
dxdt[14] = ZooGrowth # Zooplankton Growth
dxdt[15] = ZooMortality # Zooplankton Predator Mortality
dxdt[16] = ZooMixing # Zooplankton Mixing
dxdt[17] = UnassimilatedProduction # Unassimilated Production
dxdt[18] = Mineralization # Mineralization
dxdt[19] = DetritusMixing # Detritus Mixing
dxdt[20] = NMixing # Nutrients Mixing
dxdt[21] = 0. # No addition into P
dxdt[22] = P*E # P Higher order correction
dxdt[23] = P*EN # N Higher order correction
dxdt[24] = EZ # Grazing Higher order correction
dxdt[25] = EZD # Pellets Production Higher order correction
dxdt[26] = V*d1 # Changes in mean size
dxdt[27] = 0. # No addition into S
dxdt[28] = 0.5*(m4-1.)*V**2*d2 # Changes in size variance
dxdt[29] = 0. # No addition into V
dxdt[30] = self.Params['betaU']*np.exp(L)**self.Params['alphaU'] # N Half Saturation
dxdt[31] = d2 # second derivative with respect to trait
dxdt[32] = d2r0 # second derivative of Nutrient Uptake with respect to the trait
dxdt[33] = d2r2 # second derivative of Grazing with respect to the trait
dxdt[34] = d2r3 # second derivative of Sinking with respect to the trait
return dxdt
def fullmodel(self, x, t):
"""
This size based model variant calculates structural changes of a community
composed by n number of morphologically different phytoplankton.
Parameters
----------
x: array with initial conditions for the state variables
t: time
"""
N = x[0]
Z = x[1]
D = x[2]
Ps = x[3:]
dxdt = np.zeros(len(x))
sizerange = np.linspace(np.log(self.Params['sizemin']), np.log(self.Params['sizemax']), num=self.Params['NoMtype'])
# Edible phytoplankton
Ped = np.sum(Ps*np.exp(sizerange)**self.Params['alphaG'])
# Non-P Processes
Mixing = (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
ZooMortality = self.Params['moZ'] * Z**2
ZooMixing = Z * self.MLD.firstderivspl(t, k=3, s=0) / self.MLD.dailyinterp(t, k=3, s=0)
Mineralization = self.Params['deltaD']*D
DetritusMixing = D * (self.Params['kappa'] + max(self.MLD.firstderivspl(t, k=3, s=0), 0.)) / self.MLD.dailyinterp(t, k=3, s=0)
NMixing = Mixing * (self.N0X.dailyinterp(t, k=5) - N)
dxdt[0] = Mineralization + NMixing
dxdt[1] = - ZooMortality - ZooMixing
dxdt[2] = ZooMortality - Mineralization - DetritusMixing
for i in range(len(Ps)):
S = sizerange[i]
P = Ps[i]
# Gains of phytoplankton biomass
NutrientUptake = N/(N+self.Params['betaU']*np.exp(S)**self.Params['alphaU'])
LightHarvesting = 1. / (self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0)) * (-np.exp(1. - self.PAR.dailyinterp(t) / self.Params['OptI']) - (-np.exp((1. - (self.PAR.dailyinterp(t) * np.exp(-self.Params['kw'] * self.MLD.dailyinterp(t, k=3, s=0))) / self.Params['OptI']))))
TemperatureDepGrowth = np.exp(0.063 * self.SST.dailyinterp(t, k=5))
Gains = self.Params['muP'] * NutrientUptake * LightHarvesting * TemperatureDepGrowth
# Losses of phytoplankton biomass
Grazing = self.Params['muZ']*Z*np.exp(S)**self.Params['alphaG']/(Ped+self.Params['Kp'])
Sinking = (self.Params['betav']*np.exp(S)**self.Params['alphav'])/self.MLD.dailyinterp(t, k=3, s=0)
OtherPMortalities = self.Params['moP']
Losses = Grazing + Sinking + OtherPMortalities + Mixing
# Other Processes
ZooGrowth = self.Params['deltaZ']* Grazing * P
UnassimilatedProduction = (1.-self.Params['deltaZ']) * Grazing * P
dxdt[0] = dxdt[0] - P*Gains
dxdt[1] = dxdt[1] + ZooGrowth
dxdt[2] = dxdt[2] + UnassimilatedProduction + P*OtherPMortalities
dxdt[3+i] = P*(Gains - Losses)
return dxdt
def modelrun(self):
"""
Method to integrate a specific size-based model variant.
The integration it is done with the module "integrate.odeint",
from the library scipy.
"""
try:
if self.Model == "Imm":
outarray = odeint(self.sizemodel_imm, self.initcond, self.timedays)
return outarray
elif self.Model == "TraitDif":
outarray = odeint(self.sizemodel_traitdif, self.initcond, self.timedays)
return outarray
elif self.Model == "FixVar":
outarray = odeint(self.sizemodel_fixvar, self.initcond, self.timedays)
return outarray
elif self.Model == "UnsustVar":
outarray = odeint(self.sizemodel_unvar, self.initcond, self.timedays)
return outarray
elif self.Model == "FullModel":
outarray = odeint(self.fullmodel, self.initcond, self.timedays, rtol=1e-12, atol=1e-12)
return outarray
raise Exception("InputError:", "it is not a valid size model variant. Please specify one model variant "
"from: Imm, TraitDif, Fixvar, UnsustVar or FullModel.\n")
except Exception as expt:
print expt.args[0], self.Model, expt.args[1]
raise
def get_derivatives(self):
"""
Symbolic solutions for the derivatives of the phytoplankton growth terms with
respect to the trait using library sympy.
"""
# variables definition
tt0, N, a, S, b, Ped, c, d, tt2, e, f, M, lnS = sympy.symbols('tt0,N,a,S,b,Ped,c,d,tt2,e,f,M,lnS')
S = sympy.exp(lnS)
trait = lnS
# derivatives of growth terms with respect to the trait
NutrientUptake = tt0*N/(N+a*S**b)
print 'NutrientUptake d=', sympy.diff(NutrientUptake, trait)
print 'NutrientUptake d^2=', sympy.diff(NutrientUptake, trait, 2)
print 'NutrientUptake d^3=', sympy.diff(NutrientUptake, trait, 3)
print 'NutrientUptake d^4=', sympy.diff(NutrientUptake, trait, 4)
Grazing = S**c/(Ped+d)*tt2
print 'Grazing d=', sympy.diff(Grazing, trait)
print 'Grazing d^2=', sympy.diff(Grazing, trait, 2)
Sinking = e*S**f/M
print 'Sinking d=', sympy.diff(Sinking, trait)
print 'Sinking d^2=', sympy.diff(Sinking, trait, 2)
PED = S**c
print 'Edible P d^2=', sympy.diff(PED, trait, 2)
