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Ecological Modelling 247 (2012) 83– 94

Contents lists available at SciVerse ScienceDirect

Ecological Modelling

jo ur n al homep ag e: www.elsev ier .com/ locate /eco lmodel

lobal sensitivity analysis of a marine ecosystem dynamic model of theanggou Bay

ei Zhenga,b, Honghua Shia,c,∗, Guohong Fanga,c, Long Hud, Shitao Penga,b, Mingyuan Zhua,b

The First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, ChinaKey Lab of Science and Engineering for Marine Ecology and Environment, State Oceanic Administration, Qingdao 266061, ChinaKey Lab of Marine Science and Numerical Modeling, State Oceanic Administration, Qingdao 266061, ChinaCollege of Mathematical Sciences, Fudan University, Shanghai 200433, China

r t i c l e i n f o

rticle history:eceived 21 September 2011eceived in revised form 5 August 2012

a b s t r a c t

A global sensitivity analysis (GSA) is performed on a marine ecosystem dynamic model by using Morris’method with testing 10,000 times in this paper. This statistical sampling method allows to display non-linear characteristics of interactions among various parameters and rank the most relevant parameters

ccepted 6 August 2012vailable online 20 September 2012

eywords:lobal sensitivity analysisorris’ statistical sampling method

of the model. With this method, we proposed the model’s parameter optimization strategy and appliedthis to the Sanggou Bay. The results showed that the method not only can safeguard the objectivity ofparameter set initial values, but also can avoid stochasticity of analytical results. Compared with the localone, the GSA is found to have a series of advantages, which supports necessity of developing it.

© 2012 Elsevier B.V. All rights reserved.

arine ecosystem dynamic model

. Introduction

Researches on marine ecosystem models are aimed at exploringhe varying mechanism of the ecosystem, simulating or predict-ng its changes, as well as providing scientific and decision-makingationales in order to maintain its health and recover its injuredarts (Wang, 1998). The uncertainty of a model is one of the upper-ost factors to limit its applications, while the uncertainty of its

pplication results originates mainly from changes in its param-ters, provided the ecological processes and model structure arexact.

The marine ecosystem dynamic model (MEDM), as an importantool to predict the health and impact of coastal marine ecosystem,as been extensively applied to study the functions and mecha-ism of marine ecosystems (Chen et al., 1999; Liu et al., 2007). Thencertainties of the model’s parameters will lead to the uncertain-ies of its operation results. If the uncertainties are very large, the

odel’s results cannot be used to reliable decision-making basis.n other words, in order to increase its prediction precision, it isecessary to decrease uncertainties of its various parameters. How-

ver, many ecosystem models have several tens of, even hundredsf parameters. So, it is very difficult to increase the precision ofach parameter at the same time. In addition, as the sea is a highly

∗ Corresponding author at: The First Institute of Oceanography, State Oceanicdministration, Qingdao 266061, China. Tel.: +86 532 88968672.

E-mail addresses: erashi@sina.com, coast2006@163.com (H. Shi).

304-3800/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.ecolmodel.2012.08.003

complicated system, each ecological process will be subject to var-ious uncertain factors.

Therefore, the uncertainties of some parameters in an ecosys-tem are unavoidable and their effects on model’s operational resultsneed to be evaluated with sensitivity analysis by increasing the pre-cisions of those parameters that have greater effects on operationalresults; while for the parameters with little effects, it is enough tochoose their empirical values.

The sensitivity analysis has been considered as one of the keysteps in the model establishment and an important base for param-eter optimization. However, different sensitivity analysis aims atdifferent studying objects and models. To the best of our knowl-edge, the sensitivity analysis is mainly classified into two kinds,i.e. local sensitivity analysis (LSA) and global sensitivity analysis(GSA). The former-LSA is the one that tests the effect of variation insingle parameter on model output result, while other parametersonly take their central values (Saltelli, 2000; Gao et al., 1997), how-ever, the latter-GSA is to check total effects caused by variationsin many parameters on model outputs, and analyze interactionbetween or among those parameters (Campolongo et al., 2000;Saltelli, 2000; Cai et al., 2008). The LSA’s advantage is its high cal-culating efficiency, but with following shortcomings (1) it takesonly one parameter variation into consideration, therefore, it isnot enough to reveal the impact of an input parameter on model

outputs by using LSA or one-factor-at-a-time (OAT) of an eco-environmental model; (2) different values of other parameters willaffect the sensitivity of the main parameter under consideration.The differences between the GSA to the LSA, however, are (1) each

8 al Modelling 247 (2012) 83– 94

piiit

bihoitn

m2spetmiqtsHstpt

gwibiptmMp

2

s(stb2a⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−ˇ1UkQ (T) minDIN

KkN +DIN,

DIPKkP +DIP

K + ˛1

˛20.003+0.001 cos 2�

T −180365( ) ( ( ))

4 W. Zheng et al. / Ecologic

arameter varies in the finite even infinite range; (2) the changesn the model’s results caused by the changes in some parameters global, that is, the changes in its results come from the commonnteraction of all the parameters. But due to the model’s complexity,he calculating cost by the global sensitivity analysis is very high.

As the ecosystem model develops, its sensitivity analysisecomes more and more important. Due to the simplicity, the LSA

s widely used. Recently, qualitative GSA begins to occur greatly,owever most of which use the multiple regression that was basedn Latin hypercube sampling (Xu et al., 2004). The quantitative GSAs the hot spot of the current sensitivity analyses, but its applica-ions in ecosystem models especially in marine ecosystem are stillot mature.

Many scholars have presented multiple sensitivity analysisethods and got some primary applications (Morris, 1991; Xu et al.,

004; Cossarini and Solidoro, 2008). Morris suggested his factorampling method which can be used to study the effect of a modelarameter on its system outputs in the global range, i.e. as a param-ter varies in a pretty large range, how and at what a degree it affectshe system output (Morris, 1991). The sampling principle of Morris’

ethod was designed very cleverly, which is capable of compar-ng the sensitivities of global parameters by size and obtaining theualitative description of interactions among those parameters athe cost of a smaller calculation. Even for those parameters withmall sensitivity, this method was also effective (Jia et al., 2008).owever, due to its random, errors occur in the process of random

ampling and randomization, so it needs to be repeated many timeso take the average as the parameter’s sensitivity. Morris’ methodrovides a GSA frame, it, after further modification, can be appliedo analyze the sensitivities of MEDM’s parameters.

In this paper, we firstly proposed the MEDM of NPKZD in Sang-ou Bay which is in the Shandong Province of China. Secondly,ith the help of GSA, we studied all variable parameter sensitiv-

ties in the model and sorted their sensitivities by size. Thirdly,ased on the above analyses and combining the effects of variations

n parameters on multiple state variables, we put forward to thearameter optimization strategy of the MEDM. Lastly, we applied ito the marine ecosystem based on Sanggou Bay. We remark that our

ethod in this paper can be applied to optimize parameters of theEDM, and we will consider these applications in our forthcoming

ublications.

. Marine ecosystem dynamic model

Considering phytoplankton (P), zooplankton (Z), kelp (K), dis-olved inorganic nitrogen (DIN), dissolved inorganic phosphorousDIP), organic debris (D) as state variables, and irradiance on theurface of sea (I), and temperature (T) as forced functions, we can gethe schematic diagram of the NPKZD model (see Fig. 1). Meanwhile,ased on these and corresponding previous publications (Shi et al.,011), the dynamic model of pelagic ecosystem can be presenteds follows:

dP

dt= Um min

(DIN

KN + DIN,

DIPKP + DIP

)f (I)g(T)P − Gmh(T)Z(1 − e−�P ) − MpP

dZ

dt= �Gmh(T)Z(1 − e−�P ) − MZ Z

dK

dt= UkQ (T) min

(DIN

KkN + DIN,

DIPKkP + DIP

)K − MK K

dDINdt

=˛1

(−Um min

(DIN

KN +DIN,

DIPKP +DIP

)f (I)g(T)P+�Gmh(T)Z(1−e−�P )+eD

)( ( ) )

dDIPdt

=˛2 −Um minDIN

KN +DIN,

DIPKP +DIP

f (I)g(T)P+�Gmh(T)Z(1−e−�p)+eD − ˇ2

dD

dt= (1 − � − �)Gmh(T)Z(1 − e−�P ) − eD + MP P + MZ Z + MK K

( ) ( ( )) (1)

Fig. 1. Schematic diagram of the NPKZD model.

where g(T) and h(T) (with T in ◦C) are the temperature dependencefunction for growth and grazing respectively. According to ‘Q10’ rule(Fransz and Verhagen, 1985), they can be parameterized as:

g(T) = Q1, h(T) = Q2

and Q (T) represents the temperature limited function which canbe given as follows (Duarte et al., 2003):

Q (T) = 2.0(1 + ˇ)xt

x2t + 2.0ˇxt + 1.0

, xt = T − Tleth

Topt − Tleth

f (I) = Ie[1−(I/I0)]/I0 is the light limited function which is given fol-lowing the photosynthetic response to irradiance (Steele, 1962) andthe sea surface irradiance is simulated according to observed datamonthly as follows:

IS = 200.38 − 116.47 cos(

2�t − 60365

)I is photosynthetically available radiation (PAR) with optimal lightintensity I0. I is considered to be a function of the water attenuationcoefficient kext and the surface irradiance Is:

I = Ise−kextz

where z is the depth below the sea surface. In the coastal water, kext

is regarded as a linear function of water transparency (S, in meters)as shown in (Fei, 1984)

kext = 1.51S

UkQ (T) minDIN

KkN +DIN,

DIPKkP +DIP

K + 0.003+0.001 cos 2�T −180

365

W. Zheng et al. / Ecological Mo

lt

I

dv

3

M

3

npptr

Z

o

E

1

crt

fcTdiFt

Fig. 2. Schematic of Morris’ statistical sampling.

In general, phytoplankton photosynthesis occurs in the euphoticayer. Let H be the depth of euphotic layer, and then we can calculatehe PAR as

= 1H

∫ H

0

Ise−kexthdh = Is

Hkext(1 − e−kextH) = IsS

1.51H(1 − e− 1.51H

S )

It is necessary to note that the initial time of this model (t = 0)enotes the initial stage of kelp growing. All the description andalues of the parameters of this model given in Table 1.

. Method of sensitivity analysis

Based on Morris’ method, we can establish our GSA frame ofEDM.

.1. Sampling principle

Assuming the system model is y = y(z1, z2, . . ., zm), where m is theumber of parameters. We first mapped the varying range of eacharameter onto the interval [0,1], discretized them, and made everyarameter value only from

{0, 1

p−1 , 2p−1 , ..., 1 − 1

p−1

}, where p is

he number of sampling points of a parameter, every parameterandomly value at p sampling points. Thus, we had the vector

� = [z1, z2, . . . , zm] (2)

As the i-th parameter zi of the vector �Z varies �, the basic factorf �Z , EE, is expressed as:

Ei(Z0) = y(z0

1, z02, . . . , z0

i+ �, . . . , z0

m) − y(Z0)

�(3)

where � is the preset varying increment of the parameter, � =/(p − 1).

Then, we randomly selected another non-i parameter k, made ithange �, through the model operations twice, got another EE, andepeated this process till all the parameters change once. Thus allhe EE depicted a curve in the parametric space, as shown in Fig. 2.

As an example of a 3-D parametric space, z10, z2

0, z30, and z4

0 areour points in it, respectively, and form a curve. Through numericalculations, we obtained three EEs: EE1 (Z1), EE2 (Z2), and EE3 (Z3).hrough repeating the process r times, each starting from the ran-

om position (Z1, Z2, . . ., Zj, . . ., Zr), and the sequence of change

n parameters also is random. At last, we obtained the distributioni(i = 1, 2,. . ., m)of EE that was calculated r times for each parame-er. The sample expected value (mean) of the distribution Fi, �, and

delling 247 (2012) 83– 94 85

the standard deviation, �, are used to evaluate the sensitivity of theparameter. � is a measure of the total effect on some parameter.The greater � is, the bigger the effect of the parameter on the sys-tem output. � is the degree of interaction among parameters, thegreater � is, the stronger the interaction among the parameter andother parameters.

3.2. GSA procedure

Assuming the matrix D* is a m-dimensional diagonal matrix,each diagonal element takes equiprobably its value +1 or −1,the matrix B is a strict down triangle matrix, its elements are 1,B ∈ R(m+1)m, i.e.,

B =

⎡⎢⎢⎢⎣

0 0 0 0 0

1 0 0 . . . 0...

......

. . ....

1 1 1 1 1

⎤⎥⎥⎥⎦ (4)

Let Jm+1,m be ((m + 1)m) dimensional matrix, its all elementsare 1, then the matrix J* = (2B − Jm+1,m)D* + Jm+1,m also is ((m + 1)m)dimensional matrix;

(2) Assuming that Z* is a base vector with the input parameterZ, every parameter of Z* randomly takes its value from{

0,1

p − 1,

2p − 1

, ..., 1 − 1p − 1

};

(3) Suppose that P* is (m × m) dimensional random transpositionmatrix, its every column and row have and only have one value, 1,and the others are zero. Then the random matrix B* of the samplingmatrix B is:

B∗ =[

Jm+1,1Z∗ +(

�

2

)J∗]

P∗ (5)

Because D*, Z*, and P* take values randomly and independently,it guarantees the matrix B* takes its value stochastically, and thereis only one different parameter value in every two adjacent rowsof B*.

Suppose that only the j-th column elements in two adjacentrows of the matrix are different, then

B(j) =[

z1 · · · zj1 · · · zm

z1 · · · zj2 · · · zm

](6)

where Zj1 − Zj2 = �.Selecting B(j) as the input parameter of the model, we could

find the sensitivity of the j-th parameter, SA, according to Eq. (7)

SAj =(y(z1, . . . zj1 , . . . , zm) − y(z1, . . . zj2 , . . . , zm))/y(z1, . . . zj2 , . . . , zm)

�(7)

Taking consecutively the adjacent row elements of all m groupsas the input parameters of the model, we obtained all the sensitiv-ities of m parameters.

Thus it is obvious that with only one random sampling, the sen-sitivities of all m parameters can be found.

(4) Select the number of sampling r, repeat Steps (1)–(3) r timesto obtain r number of SA values of each parameter;

(5) Find the mean � and standard deviation � of SAs of eachinput parameter zi;

(6) According to the calculated results, determine the GSA of theinput parameter.

4. Simulation of Sanggou Bay ecosystem

4.1. Study area

Sanggou Bay is located in the Yellow Sea near Rongcheng City ofShangdong Province in China (37◦01′–37◦09′N, 122◦24′–122◦35′E).

86 W. Zheng et al. / Ecological Modelling 247 (2012) 83– 94

Table 1Parameter of NPKZD model.

Parameter Value Unit Description Reference

Um 1.8 d−1 Phytoplankton maximum nutrient uptake rate Wang et al., 2006kN 8 �mol/L Phytoplankton half-saturation constant for nitrogen uptake Wang et al., 2006kP 0.6 �mol/L Phytoplankton half-saturation constant for phosphorous uptake Wang et al., 2006KkN 8 �mol/L Kelp half-saturation constant for nitrogen uptake Wang et al., 2006KkP 0.6 �mol/L Kelp half-saturation constant for phosphorous uptake Wang et al., 2006Uk 0.04 d−1 Kelp maximum nutrient uptake rate Mao et al., 1993Gm 0.6 d−1 Zooplankton maximum grazing rate This paper� 0.1 umol/L Ivlev constant for grating Wang et al., 2006� 0.2 d−1 Zooplankton growth coefficient Gao et al., 1998 0.12 d−1 Zooplankton excretion coefficient Wang et al., 2006I0 180 W/m2 Optimum light intensity Wu, 2009Q1 2.08 Temperature dependence coefficient for growth Fransz and Verhagen, 1985Q2 3.1 Temperature dependence coefficient for grazing Fransz and Verhagen, 1985MP 0.1 d−1 Phytoplankton death rate McCreary et al.,1996MZ 0.2 d−1 Zooplankton death rate Franks et al., 1986MK 5 × 10−4 d−1 Kelp death rate This papere 0.05 d−1 Detrital remineralization rate McCreary et al., 1996ˇ 3 Temperature adjustive parameter for kelp growth Andersen and Nival, 1989Topt 13 ◦C Optimum temperature for kelp growth Petrell et al., 1993Tleth 25 ◦C Death temperature for kelp growth Petrell et al., 1993˛1 16/106 molN/(molC) Phytoplankton N/C Redfield, 1958

/C

IitbtaaoatlTk

t

4

phc

˛2 1/106 molP/(molC) Phytoplankton Pˇ1 1/6.5 molN/(molC) Kelp N/C

ˇ2 1/215 molP/(molC) Kelp P/C

t is a coastal embayment with 143.2 km2 water area and 20 km2

ntertidal areas. The average water depth of this Bay is 7–8 m, withhe maximum 15 m. There are no large rivers running into this bay,ut some small rivers such as the Sanggan River, the Yetao River,he Gu River, and the Xiaoluo River. The annual average air temper-ture is 10.90 ◦C. The average monthly precipitation is 68.02 mm,nd it rains mostly in July and August. There are 88 days with windver grade six one year. Mariculture is the main anthropogenicctivity in Sanggou bay, which started in the 1960s. The main cul-ivated species is kelp, scallop and oyster. The mariculture area isarger than 50% of the total area and kelp cultivation is dominant.herefore, it is important to study the dynamic character of theelp.

We have investigated 9 sites in Sanggou Bay in order to selecthe data, the diagram of the study area is given by Fig. 3.

.2. Result of model simulation

In order to analyze the effect of kelp-culturing density on thelanktonic ecosystem, we divided the kelp-nurturing area into theigh-, middle-, and low-density zones according to its density, andarried out numerical simulations for Sanggou Bay ecosystem. The

Fig. 3. Observation sites(9 sites in Sanggou Bay).

Redfield, 1958Huang et al., 2005Huang et al., 2005

measured time was fixed within 0–200 days (the kelp-nurturingperiod in the experimental area was from the early November toMay of the next year). It can be seen from Fig. 4(a)–(c) that no mat-ter how high or low kelp-nurturing density is, kelp always winsan advantage over phytoplankton in the competition. It is clearthat the higher the kelp-nurturing density (kelp’s initial value) is,the slower the phytoplankton grow. However, zooplankton, whichmainly feed phytoplankton, grows or lessens along with the des-tiny of phytoplankton, so does organic debris. We can also seethat there is a small peak of phytoplankton biomass in spring,which is in agreement with observation (see Fig. 5). The peakbecomes smaller with the kelp-nurturing density increasing. Inthe late period of kelp growth, photosynthesis promotes kelp tofeed more nutrients, which makes DIN decrease gradually andthen tend to stable. The reason may be both the phytoplanktonbiomass and the kelp-nurturing density stay at higher values, wast-ing a large quantity of DIP, which makes it be the major limitingfactor.

Simulations results that trends of phytoplankton, DIN, DIP agreewith those shown from the observational data (see Fig. 4(a) andFig. 5), which, especially, can also simulate the algal blooms of phy-toplankton in Spring. Although we assume that there does existsmall errors between simulated and observed data, which indi-cates, however, the importance for optimizing parameters of theMEDM, whose basis is the sensitivity analysis, and we will considerall of these in our forthcoming publications.

5. GSA results

In order to avoid huge diversity of state variables in the begin-ning, the simulation starts from March to May for GSA with themid kelp-nurturing density, and with land source in consideration(see Fig. 4). Generally speaking, smaller parameters range will bemore efficient in GSA provided the premise of satisfying precisionrequirement. Moreover, the oversized parameter range would beprone to making some parameter values go beyond the reasonablerange. Therefore, in the calculation of GSA, all the model param-

eters (24 ones) were changed simultaneously and the range ofvariation in parameter was selected as ±10% of the default param-eter,in which all the parameters were uniformly divided into 10intervals (P = 11), that is, the varying range of the parameter each

W. Zheng et al. / Ecological Modelling 247 (2012) 83– 94 87

Fig. 4. The trend of N–P–Z–K–D ecosystem dynamic model simulation.

tiwd

feeding rates were more sensitive compared with other parame-ters, which were different from what obtained by doing LSA (Shi

ime was 2% of the default value. The parameter and its correspond-ng sensitivity were logged and collected under 10,000 simulations,

hich will be used to calculate the average sensitivity and standard

eviation.

The main analytical results are as follows.

Fig. 5. Observation data of Sanggou Bay ((from November of 2003 to May of 2004)).

5.1. Sensitivity distribution

In order to avoid the excessive dispersion of sensitivity distri-bution, we divided the calculated data into 50 groups and everygroup simulate 200 times. The sensitivity distributions of variousstate variables relevant to the parameters were shown in Fig. 6(a)and we also calculate the absolute sensitivity distributions of var-ious state variables versus 24 parameters of the model, the figuresare given by Fig. 6(b).

It can be seen from Fig. 6(a) that many sensitivity distri-butions contained simultaneously positives and negatives, thatmeans under different parameters, the same state variable mightbe positively, or negatively correlated to some model parameter,which also shows the complexity of using the MEDM to ana-lyze sensitivity. Hence, it might not be reasonable if only usingthe mean to describe sensitivity of the state variables to theparameters.

At the same time, it was found that the differences between sen-sitivities of different state variables relative to the same parameterwere also various. For example, from Fig. 6(a)(2), the sensitivityof the phytoplankton half saturation constant for DIN uptake washigher relative to those of phytoplankton, DIN, and DIP, while lowerrelative to those of zooplankton, kelp, and organic debris. For otherparameters, conclusions were similar. In addition, the differencesin quantity of the sensitivities of various state variables relative todifferent parameters were also big, even the order of magnitudedifferences exist. The similar results can be obtained from Fig. 6(b)for absolute sensitivity.

Therefore, it is necessary to consider the contributions of vari-ous parameters to the model’s stability through the whole range.Further analyses on the average sensitivities and absolute averagesensitivity of the model’s various state variables relative to changesin parameter were shown in Fig. 7(a) and (b), respectively.

It was clear from Fig. 7(a) that phytoplankton’s maximumgrowth rate, the optimal illumination, and the growth temperaturedependent on coefficient, zooplankton’s mortality and maximum

et al., 2011), which showed that the maximum feeding rate Gm, thegrowth coefficient �, and the mortality Mz were ranked as the top

8 al Mo

seGe

F2

8 W. Zheng et al. / Ecologic

ensitive parameters, that indicated zooplankton’s growth param-

ters had great effect on the model’s reliability. Combining withSA, we can see the importance of phytoplankton’s growth param-ters, such as the maximum growth rate Um, the optimal irridiance

ig. 6. (a)Sensitivity distributions of various state variables vs. 24 parameters of the mod4 parameters of the model, respectively.

delling 247 (2012) 83– 94

I0, the growth temperature dependent coefficient Qg10. We can

also get similar results from Fig. 7(b) for absolute average sensi-tivity of the model’s various state variables relative to changes inparameter.

el, respectively. (b) Absolute sensitivity distributions of various state variables vs.

W. Zheng et al. / Ecological Modelling 247 (2012) 83– 94 89

(conti

5

t

Fig. 6.

.2. Gradation of parameter sensitivities

As the model referred to 24 parameters, we graded or rankedhem according to their sensitivities. Each state variable (such

nued ).

as phytoplankton, zooplankton, etc.) could be seen as a unit andeach parameter’s sensitivity was calculated respectively (meanand standard deviation). Among these 24 parameters, the 1stclass with the maximum sensitivity (the absolute value) was

90 W. Zheng et al. / Ecological Modelling 247 (2012) 83– 94

(conti

rgawhtpt

geer

Fig. 6.

anked as the first grade; the 2nd–4th classes as the secondrade, the 5–10th classes as the third grade, the 11–15th classess the fourth grade, the remains as the fifth one. The gradesere marked with different colors, the darker the box is, theigher its grade is. First class is its highest grade. The grada-ions divided by the mean and the standard deviation of thearameters’ sensitivities were shown in Fig. 8 and Fig. 9, respec-ively.

From Fig. 8, we could clearly discern those parameters that

reatly affect various state variables, and know how various param-ters affect different state variables. By average sensitivity, it isasy to see that the effects of phytoplankton’s maximum growthate Um on the state variables, except that on kelp, were the first

nued ).

or second grade, while zooplankton’s mortalities on DIN and DIPwere the most sensitive parameter. If we concerned individual statevariable, see the x-axis of Fig. 8, the most sensitive parameterswere ranked as the phytoplankton’s maximum growth rate Um andgrowth temperature dependent coefficient Q1, the zooplankton’sgrowth coefficient �, the phytoplankton’s half saturation constantabsorbing DIP, the kelp’s half saturation constant absorbing DIP,name a few. Similar analysis could be performed for other statevariables.

Table 2 listed those parameters which had major effects on statevariables, while Table 3 listed those parameters which had minoreffects. The tables could provide important basis for us to study themodel’s state parameters.

W. Zheng et al. / Ecological Modelling 247 (2012) 83– 94 91

Fig. 6. (continued ).

Table 2Parameters who had major effects on state variables.

Influencing degree Those who had important influence on three or more state variablesand the influence were ranked as the first or second grade.

Those who had major influences on two or more state variables andthe influences were ranked as the first or second grade.

Parameters Um , Kp Um , Kp , MZ , KkN , kN , Kkp , Gm , Io , e

Table 3Parameters who had minor effects on state variables.

Influencing degree Those who had some influences on four or more state variables and theinfluences were ranked as the fifth grade, and no the first grade one.

Those who had major influences on three or more state variables andthe influences were ranked as the fifth grade, and no the first or secondgrade.

Parameters Q2, Topt, ˇ1, ˛2 Q2, Topt, ˇ1, ˛2, ˇ, �, Uk

92 W. Zheng et al. / Ecological Modelling 247 (2012) 83– 94

(b) Ab

eststei(o

6

m

Fig. 7. Average sensitivity of various state variables vs. all model parameters.

We could also obtain similar conclusions by grading the param-ters according to the standard deviation of the parameters’ensitivity, which indicated that the expected value (mean) andhe variance of parameters’ sensitivities had strong correlations totate variables. The greater the average sensitivity was, the greaterhe fluctuation was, which mainly resulted from the accumulationffects caused by the model’s high nonlinearity. These results weren agreement with that was obtained by Cossarini and Solidoro2008). The conclusions were similar as we discussed the gradationf absolute parameter sensitivities.

. Discussions

The sensitivity analysis of the parameters is the key ele-ent to build the model (Gao et al., 1997; Xu et al., 2004).

solute average sensitivity of various state variables vs. all model parameters.

Increasing the model’s reliability mainly depends on the opti-mization of parameters with higher sensitivity (Cossarini andSolidoro, 2008). However, the MEDM’s high complexity makesthe sensitivities of its parameters uncertainty which is character-ized by (1) the difference between the sensitivities of differentstate variables related to the same parameter; (2) various ini-tial values; (3) the dependence on other parameters. Therefore,taking LSA method is difficult to fully display the true responserelation between MEDM’s state variables and the changes inparameter.

In conclusion, we proposed a MEDM based on the NPKZD ecosys-

tem of the Sanggou Bay by using Morris’ method, which performedglobal sensitivity analysis of the model with initial values and dif-ferent parameter sets obtained by randomly sampling. This methodalso allows to display nonlinear characteristics of interactions

W. Zheng et al. / Ecological Modelling 247 (2012) 83– 94 93

Fig. 8. Gradations according to the mean of sensitivity.

the st

aoehwsplai

Fig. 9. Gradations according to

mong various parameters and rank the most relevant parametersf the model. With this method, we proposed the model’s param-ter optimization strategy, that is, optimizing the parameters withigh effects, and choosing their empirical values for the parametersith little effects, and applied this to the Sanggou Bay. The results

howed that the method not only can safeguard the objectivity of

arameter set initial values, but also can avoid stochasticity of ana-

ytical results. Compared with the local one, the GSA is found to have series of advantages, which supports the necessary of developingt (Shi et al., 2011).

andard deviation of sensitivity.

Acknowledgements

This work is supported by the National Special Fund forBasic Science and Technology of China (No. 2012FY112500), theNational Natural Science Foundation of China (No. 41206111 &No. 41206112) and the Basic Research Fund for National Non-

profit Institute of China (No. FIO2011T06). We would like to thanktwo anonymous reviewers for their constructive comments to thispaper. We are very grateful to Dr. Zexun Wei for his help in prepar-ing this paper.

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