e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 547–552
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
A novel approach to combine response functions
in ecological process modelling
Claus Florian Stange a,b,∗
a
b
UFZ Helmholtz Centre for Environmental Research, Department of Soil Physics, Theodor-Lieser-Str. 4, 06120 Halle, Germany
Institute of Agricultural and Nutritional Sciences, Martin-Luther-University Halle Wittenberg, Germany
a r t i c l e
i n f o
a b s t r a c t
Article history:
A novel approach to combining response functions, e.g., temperature and soil moisture
Received 24 June 2005
dependency, is presented. This approach is in analogy of resistances connected in parallel
Received in revised form
and mathematically to the inverse function of the sum of reciprocal response functions. The
3 January 2007
approach presented is applicable for a wide range of response functions, and demonstrate
Accepted 15 January 2007
better performance as the multiplicative approach if the limiting factor dominates the pro-
Published on line 23 February 2007
cess rate more than the other factors. It was applied to a gross nitrification data set acquired
from beech litter samples in the laboratory using the barometric process separation (BaPS)
Keywords:
method. Compared with the minimum and the multiplicative approaches, the best fit was
Biogeochemical modelling
achieved with the novel approach, using the Residual Sum of Squares and r2 values as indi-
Gross nitrification
cators. Additionally, two examples from the literature were presented to demonstrate the
Barometric process separation
potential and benefits of the approach, which is a good alternative combining two or more
Soil temperature
response functions.
© 2007 Elsevier B.V. All rights reserved.
Soil moisture
Stress
Plant
1.
Introduction
Processes in ecosystems such as transformation processes or
transport often depend upon more then one factor; hence to
simulate processes in ecosystems by mathematical modelling,
an accurate combination of the specific response function is
necessary. Due to the complexity of interactions between environmental factors, primarily the influence of each factor was
investigated discretely and the multiplicative approach was
used to combine the response functions.
A handful of functions are available to describe the
dependency of processes from temperature and soil moisture
(Rodrigo et al., 1997; Antonopoulos, 1999). Nevertheless,
approaches to combine two or more response functions
are scarce. Current models use the multiplicative approach
(Rodrigo et al., 1997; Antonopoulos, 1999; Paul et al., 2003):
F = kmax f (x)f (y)
(1)
where F is the transformation, kmax the rate under optimal
environmental conditions, and f(x) and f(y) are the response
functions of environmental factors, normally scaled between
0 and 1. This approach does not consider any interaction
between moisture and temperature effects. A way to account
∗
Correspondence address: UFZ Helmholtz Centre for Environmental Research, Department of Soil Physics, Theodor-Lieser-Str. 4, 06120
Halle, Germany. Tel.: +49 345 5585 418; fax: +49 345 5585 559.
E-mail address: fl[email protected].
0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2007.01.005
548
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 547–552
for an interaction between temperature and moisture has
been considered in the minimum approach, based on
the Liebig law (Liebig, 1840), in which the limiting factor
dominates the process:
were dried (48 h at 65 ◦ C) and milled before analysis of total N
content (Nt ) and organic carbon (Corg ) by an element-analyser
(Elementar, Germany).
2.2.
F = kmax min(f (x), f (y))
Unfortunately, the discontinuous differentiability causes
problems in the numerical solution of the differential
equation and restricts possible applications.
In this paper an approach is presented which is based
on the description of parallel connected resistances and also
combines the idea of Liebig with the advantage of continuous differentiability. The approach is tested against three data
sets, first at a gross nitrification data set from beech litter.
Nitrification is an important process of the soil N-cycle, since
nitrification controls the allocation between the less mobile
NH4 + and the more mobile NO3 − (Abbasi and Adams, 1998).
Another two examples are presented: one is plant transpiration which depends upon salinity and water stress (Homaee et
al., 2002) and the other is malondialdehyde content in plants,
an indicator of free radical damage to cell membranes under
stress conditions whereby the content depends upon temperature and soil moisture (Xu and Zhou, 2006). Water Stress
and soil salinity are two important factors for reduced yield
in semiarid regions. Because soil water pressure and osmotic
potential are additive in reducing the free energy of soil water,
it was assumed that their effect on transpiration is also additive (Meiri, 1984).
If plants are subjected to adverse conditions such as high
temperature, drought, or salinity stresses, the scavenging system may lose its function, resulting in oxidative damage of
cell membranes. Malondialdehyde (MDA) is a product of peroxidation of unsaturated fatty acids in phospholipids and it is
a maker for cell membrane damage (Xu and Zhou, 2006).
2.
Materials and methods
2.1.
Measurements of gross nitrification
Model approach
(2)
Litter samples were collected in October 2004 from a mature
beech-tree stand at the “Dübener Heide”, located 30 km north
of Leipzig, Germany. The typical soil type in this region is an
Eutric Cambisol (FAO system). The beech stand has a temperate continental climate with cool winters and hot summers,
an annual precipitation value of 539 mm, and a mean annual
temperature of 9.3 ◦ C.
The litter was homogenized carefully by hand and filled
according to the natural bulk density in seven cylinders (100 ml
each). Measurements of gross nitrification (duplicated) were
carried out using the barometric process separation (BaPS)
technology (UMS, Germany) (Ingwersen et al., 1999). The litter
was equilibrated at each temperature (1 ◦ C, 10 ◦ C, 20 ◦ C, 30 ◦ C
and 35 ◦ C) for different moisture steps (130%, 180%, 250%, 300%
440% g H2 O g−1 dry litter) for 36 h before measurements. Litter
pH-values were measured in 0.01 M CaCl2 (litter:CaCl2 1:5) with
a pH-Meter (Hanna Instruments Inc., USA). Litter moisture was
determined gravimetrically after each moisture step and by a
dried sub-sample at the end of the experiment. The samples
In most concepts, response functions are combined by multiplication (e.g., N-turnover in soils: Rodrigo et al., 1997).
The presented approach is the inverse function of the sum
of reciprocal response functions. The mathematical formula
is:
g(x1 , . . . , xn ) =
n
i=1
n
(3)
1/f (xi )
with g(x1 , . . ., xn ) is the response function; n the number of
considered factors; f(xi ) is the response function for one factor
(e.g., temperature or moisture).
This approach can be expanded to three or more factors.
For the evaluation of the applicability of the approach in ecological modeling, (1) a data set of gross nitrification at different
soil temperatures and moisture conditions, (2) plant transpiration measurements under different water and salinity stress
and (3) plant production of malondialdehyde by water and
temperature stress were used.
2.3.
Temperature and moisture response on gross
nitrification
The influence of temperature on N-transformation processes
has been investigated in numerous studies, and many papers
and reviews disclosed the best relationship (e.g., Stark and
Firestone, 1996; Kätterer et al., 1998; Dalias et al., 2001).
Since the measured rate at 35 ◦ C is lower than at 20 ◦ C
and 30 ◦ C, the optimum function from O’Neill (Diekkruger
et al., 1995) was used. This function is more suitable to
characterising microbiological processes at high temperatures than the monotonically increasing Arrhenius function.
The O’Neill function increases quasi-exponentially at lower
temperatures and decreases at temperatures beyond the
optimum:
f (T) =
Tmax − T
Tmax − Topt
a
ea((T−Topt )/(Tmax −Topt ))
(4)
with T is the soil temperature (◦ C); Tmax the maximum temperature of the O’Neill function (◦ C); Topt the optimal temperature
of the O’Neill function (◦ C); a is the shape parameter of the
O’Neill function.
A variety of functions are presented for soil moisture as
well (Paul et al., 2003). High soil water content often leads to
decreasing turnover rates, due to the limitation of one or more
reactants. The same interrelation was found in this study
for the gross nitrification rate, which was decreasing at high
moisture conditions. Therefore, an optimum function, first
presented in Diekkruger et al. (1995), was used to describe the
influence of soil moisture on the gross nitrification rate:
f (M) =
M
Mopt
b
e1−(M/Mopt )
b
(5)
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e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 547–552
with M is the soil water content (g water g−1 soil); Mopt the optimal water content (g water g−1 soil); b is the shape parameter
of the function
In order to demonstrate the applicability of the approach
that combines the temperature and soil moisture response
functions and to estimate the parameters, the data set of
gross nitrification of a beech litter horizon measured by the
barometric process separation technique (BaPS) was used.
Estimations of parameters using the three models (multiplicative; minimum; new) for gross nitrification were carried out
with the non-linear parameter estimation procedure (nlin) in
SAS® software (SAS Institute Inc, Carn, NC).
2.4.
Salinity and water stress response to
transpiration
Homaee et al. (2002) experimental results indicated that for
both water and salinity stress the well-known linear response
functions can be used, but combined water and salinity stress
on root water uptake is neither additive nor multiplicative.
Therefore, I use this data set by combining the reduction factors for water stress measured without salinity stress and for
salinity stress measured without water stress to predict the
plant response to combined water and salinity stress.
2.5.
Malondialdehyde content in plants with water
and temperature stress
Table 1 – Soil parameters of the beech litter
Horizon
Bulk density (kg l−1 )
Corg (%)
Nt (%)
C/N
pH-value
Beech litter
0.10
45.7
1.78
25.6
4.6
optima in the investigated range (Fig. 1). For parameter estimation Tmax of the O’Neill function was fixed at 50 ◦ C, since no
measurements were completed at high temperatures. Using
the r2 value as an indicator, the best agreement could be found
with the new approach (r2 = 0.860) followed by both the minimum approach (r2 = 0.851) and the common multiplicative
approach (r2 = 0.851). Results of the parameter estimation are
given in Table 2.
The estimated parameters Kmax , Topt and Mopt vary
marginally, for example the optimum temperature (Topt )
varies between 25.22 ◦ C and 25.47 ◦ C (new and multiplicative approach, respectively), whereas the shape parameter of
the temperature (a) and moisture (b) response functions are
strongly influenced by the different approaches. Both a and b
increased from the multiplicative, to the minimum, and to the
new approach.
The differences between the three approaches increase
with wider ranges of the temperature and soil moisture, cor-
The results from Xu and Zhou (2006) show a strong response
to decreasing soil water moisture, but more at 29 ◦ C and 32 ◦ C
than at 23 ◦ C. The soil moisture response at the two higher
temperatures can be described very well with a potential function:
f (M) = a(M)
b
(6)
with M is the soil moisture (% field capacity (FC)); a and b are
the shape parameters of the function.
Whereas the temperature response at low soil moisture can
be described adequately with a linear function:
f (T) = cT + d
(7)
with T is the soil temperature (◦ C).
The multiplicative and presented approaches are tested.
Please note that in the multiplicative approach parameters
a and c are redundant and either can be omitted. Parameter
estimations were carried out with the non-linear parameter
estimation procedure (nlin) in SAS® software (SAS Institute
Inc, Carn, NC).
3.
Results
3.1.
Temperature and moisture response to gross
nitrification
Selected litter characteristics are described in Table 1. The
amount of mineral soil in the litter was negligible. Both the
temperature and the moisture response functions showed
Fig. 1 – Comparison of measured and simulated
temperature response on gross nitrification rates at
different litter moisture. Symbols represent measurements,
fine line the multiplicative approach, bold line the proposed
approach.
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e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 547–552
Table 2 – Parameter optimization results (estimated
value and standard error) for the three approaches given
by Eqs. (1)–(3) for gross nitrification in a beech litter
horizon. Also indicated are the residual sum of squares
(RSS) and the coefficients of determination (r2 )
Multiplicative
approach
Minimum
approach
New approach
kmax
Topt
a
Mopt
b
3684.3, 198.6
25.47, 0.45
6.43, 0.75
2.93, 0.07
2.55, 0.16
3775.1, 200.3
25.27, 0.39
8.38, 0.84
2.95, 0.06
3.01, 0.15
3674.9, 189.8
25.23, 0.42
10.73, 1.01
3.01, 0.06
3.49, 0.19
RSS
r2
7,106,666
0.851
7,086,514
0.851
6,668,005
0.860
respondingly the differences are greater at the borders than
in the centre. Discrepancies between the approaches in the
r2 values appear to be marginal, but the high residual sums
of squares values are caused by measurement uncertainties.
If mean values of the two duplicates were used for parameter estimation, r2 values increased clearly over 0.9, but also
the differences between the approaches in the r2 values and
residuals sum of square increased.
3.2.
Salinity and water stress response to
transpiration
Fig. 2 – Comparison of measured and simulated soil water
response on malondialdehyde content in plants at different
temperature. Symbols represent measurements from Xu
and Zhou (2006), fine line the multiplicative approach, bold
line the proposed approach. Please note that two axis of
ordinates are use to separate the graphs at 29 ◦ C and 32 ◦ C.
the applied water was only 70%, as compared to the other
treatments of the severe water stress level (Homaee et al.,
2002).
3.3.
Malondialdehyde content in plants with water
and temperature stress
This example demonstrates that the proposed approach is not
fixed on combining functions as a matter of course; it is practicable for factors. Table 3 shows that the multiplicative model
underestimated the real transpiration, except the treatment
at the highest water and salinity stress (SWS 5.0). Except this
special treatment, where the novel approach overestimated
the measurement clearly, the approach is in good agreement
with the measured results. The RSS (residual sum of square)
decreased from 0.246 for the multiplicative to 0.119 for the
novel approach. It must be noted that more than the half of
the RSS for the novel approach is caused by the one treatment (SWS 5.0). It must be concluded that the water stress in
this treatment was most likely higher than intended, because
Both soil moisture and soil temperature influence the drought
stress for plants and their oxidative damage. Nevertheless, the
plants react more sensitively to the combination of water and
temperature stress than to one factor alone.
As Fig. 2 demonstrates, both approaches described the
experimental results very well, but the new approach is in
better agreement with the soil moisture response at low temperature (23 ◦ C). Hence the goodness of fit, corresponding to
the r2 values, is better for the proposed approach (0.986) than
for the multiplicative approach (0.968), and even the adjusted
r2 AR, which also considers the different number of estimated
parameters, is better (0.980 against 0.960).
Table 3 – Relative transpiration rates under stress condition
Water
Without salinity stress
Salinity (dS m−1 )
1.5
2.0
3.0
4.0
5.0
Without water stress
1
0.92
0.85
0.78
0.67
0.59
Moderate water stress (MWS)
0.66
0.74
0.77
0.61
0.72
0.74
0.56
0.65
0.72
0.51
0.6
0.66
0.44
0.5
0.62
0.39
Severe water stress (SWS)
0.5
0.67
0.65
0.46
0.64
0.63
0.43
0.62
0.61
0.39
0.4
0.57
0.34
0.29
0.54
0.30
Relative transpiration of the separate water stress (second column) and separate salinity stress (second line) and comparison between measured and modelled relative transpiration of combined water and salinity stress (italic measurement; bold the proposed approach; normal the
multiplicative approach). Data from Homaee et al., 2002.
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 547–552
4.
Discussion
4.1.
Gross nitrification
The measured gross nitrification rates were much higher as
compared with the results from Stark and Hart (1997)), which
used the 15 N pool dilution technique to determinate the gross
rates. Breuer et al. (2002) compared the BaPS method and 15 Npool dilution technique for measuring gross nitrification rates
and found no statistical differences. Gross rates in a tropical
rain forest site determined by Breuer et al. (2002) were comparable to the observed gross rates in this study. Litter is the
most productive horizon of soil, especially considering the
weight. Nevertheless, field gross rates are likely to be different from the observed rates, due to sample preparation and
exclusion of root N-uptake. The main interest of this study
was to investigate the relative differences caused by changing temperature and moisture conditions. However, it should
be considered that subsurface horizons are significantly more
sensitive to increases in temperature than surface horizons
(Fierer et al., 2003). Decreasing Q10 values with increasing temperatures have been observed by many researchers (e.g., Stark
and Firestone, 1996; Dalias et al., 2001). The application of the
Arrhenius function to describe the temperature response is
limited by the temperature interval, which is well below the
optimum temperature. Typically, the optimum temperature is
near the maximum occurring temperature in the field (e.g.,
Stark and Firestone, 1996), and therefore derivations of field
data based on the Arrhenius function are rare.
The decreasing rates with increasing moisture at high
moisture conditions are due to oxygen limitation within the
soil rather than inhibition by water. Without manipulating the
oxygen content, it is impossible to distinguish between both
influences. Thus, the description of both factors was carried
out by one optimum function.
4.2.
The proposed approach
In most models, response functions for the effects of moisture
and temperature are combined by multiplication (e.g., Rodrigo
et al., 1997). But a number of studies show that especially
temperature and moisture are interconnected (e.g., Goncalves
and Carlyle, 1994; Zak et al., 1999; Knoepp and Swank, 2002).
For example, at lower temperatures, Maag and Vinther (1996)
found smaller responses to change in soil moisture than
at higher temperatures. This work presents three examples,
where the environmental factors interact among each other
and where the measured results can be better explained by
the proposed approach rather than the more commonly used
multiplicative approaches. The examples demonstrate that
the proposed approach is applicable with both more empirical functions and also more mechanistically based functions.
As Fig. 2 shows, the proposed approach, in contrast to the multiplicative approach, is able to change the characteristic of the
moisture response at different temperatures. Whereas, at high
temperatures, the moisture response is increasing exponentially, at low temperatures a saturation curve will be observed.
In contrast to an explicit modelling of the interaction of
factors (e.g., soil moisture and soil temperature) the proposed
551
approach does not need an additional parameter and separate
parameter estimation, as demonstrated in example 2 (salinity
and water stress). Perhaps it can be expected that this more
general approach is more universal than the direct interaction modelling. Additionally, the approach is not limited to
the interaction of two factors; it can also be used with three or
more factors and their interactions.
In ecology the majority of response functions do not act
independently; therefore the approach presented is a suitable
alternative for combining response functions without limiting
the number of functions.
Acknowledgements
I thank B. Apelt for his assistance in the laboratory and K.
Owen to improve the English text and style. The author thanks
the German Federal Ministry of Education and Research
(BMBF) for the financial support of the BaPS system.
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