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A comparison of model structures for the
simulation of amphipod (Talitrus saltator)
population dynamics.
P. M. Anastácio
S. C. Gonçalves
M. A. Pardal
J. C. Marques
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A comparison of model structures for the simulation of
amphipod (Talitrus saltator) population dynamics.
P.M. Anastácioa, S.C.Gonçalvesb, M.A. Pardalb, J.C. Marquesb
a IMAR c/o Department of Ecology, University of Évora, Rua Romão Ramalho, 59
7000-671 Évora, Portugal (anast@uevora.pt)
b IMAR c/o Department of Zoology, University of Coimbra 3004-517 Coimbra, Portugal.
Abstract: A dynamic model is proposed in order to simulate sandhopper (Talitrus saltator) population
dynamics under various scenarios. It was built under the Stella simulation software and so far it has attained
the calibration and verification stages. The basic structure of the model depends on the simulation of the
number of animals in each age class (45 days). A von Bertalanffy growth curve has been adjusted to the
treated field data. From this equation another one expressing the instantaneous growth rate was extracted and
used to calculate the growth. The number of individuals in each age class increases due to the ageing process
i.e. new individuals coming from the previous age classes. The number of animals in the first age class
increases due to recruitment. Fertility is related to the weight of females, therefore the recruitment is
predicted by knowing the average weight and the number of females present. The outputs from each age class
are the ageing and mortality. Three types of recruitment were tested: 1. A single and long reproduction period
only dependent on the day of the year. 2. More than one reproduction period but also dependent on the day of
the year. (2 and 3 periods). 3. Reproduction dependent on the temperature and day length. Each of these types
involved a different set of equations therefore the comparison of the results of each type acts (sensu lato) as a
form of sensitivity analysis. Satisfactory results were accomplished by some of the model variants described
above. A simple alternative approach to these model variants was also tested with the best results. No size
dependent fertility and no parallel structures to simulate growth in size or weight were considered. In
addition the use of shorter age classes (28 days) was also experimented.
Keywords: sandhopper; recruitment; population density; photoperiod; temperature.
1. INTRODUCTION
In the literature some studies on T. saltator
mention reproductive biology and population
dynamics. These studies are absolutely
fundamental but with few exceptions [e.g.
Williams, 1985] the causality of the processes
affecting population dynamics is not thoroughly
studied. Ecological models lead to a higher
degree of self-conscience in what concerns the
gaps in our knowledge. Moreover they allow us
to test some hypothesis regarding the mechanics
behind the observed processes. By building a
population dynamics model of Talitrus saltator
we are not only building a tool for predictions
under various scenarios but we are also forced to
produce a state of the art on factors affecting its
population dynamics. It was therefore our
purpose that the model would accomplish the
following goals: a) a correct simulation of the
population dynamics of Talitrus saltator b)
incorporation in the model of the most important
ecological processes involved in the population
dynamics of the species and c) indications on the
relative importance of each parameter or process on
the dynamics of the population. This will aid the use
of the species as a possible indicator of beach
disturbance.
2. DATA BACKGROUND
The calibration of the model was performed using
data from Talitrus saltator population structure on
the coast of Lavos obtained by Marques et al.
[subm.]. Data used included, life span, density (per
m coast), recruitment periods, growth, sex ratio and
age of sexual maturation. Mortality rates were
calculated by fitting exponential decay equations to
the data of the densities of each cohort during the
period when it was identified. This procedure
assumes a constant value for the mortality of each
cohort throughout its entire life. No consistent
length or weight dependent daily mortality rates
were found, therefore an average value of 0.00993
(S.D. of 0.00757) and a median value of 0.00638
503
were found for the cohorts identified. Other
important information, such as fertility, was
obtained from previously published papers [e.g.
Williams, 1978] and then checked for model
performance. In the case of the fertility some of
the versions of the model used a weight vs.
fertility regression but the final one relied on an
average value of 13 per female.
3. CONCEPTUALISATION OF THE
MODEL
Sandhopper (T. saltator) population was
subdivided into age classes, each one
constituting a state variable. This division would
account for differences on the fertility of the
females and would allow us to calculate the
number of individuals in each age class, their
average weight and the biomass of the entire
population. Air temperature, lunar periodicity
and day length are the main forcing functions
for the model (Figure 1). The moisture content
of the sand seems to affect only the spatial
distribution of the animals and therefore it was
not considered a relevant forcing function.
4. MODEL STRUCTURES
Each sandhopper entering an age class takes 45 or
28 days to reach the next one depending on the
model version. At the end of the final age class
sandhoppers are considered to have reached the
maximum possible life duration. Recruitment
happens during one or more than one period of
the year depending on the version of the model
being used.
4.1 First approach
Average sandhopper weight was calculated for
each of eight age-class time intervals as described
by Anastácio et al. [1999]. The increase in weight
for each 45-day period was considered dependent
on the temperature. At very low winter
temperatures [David, 1936] but also at extremely
high summer temperatures [Scapini et al., 1992],
sandhoppers retreat to burrows therefore it was
assumed that growth was arrested. Fertility was
considered dependent on the weight and several
mechanisms to start recruitment were tested.
4.2 Second approach
Considerably simpler than the previous one.
Eleven, instead of eight, age classes, were used
lasting 28 days each. This value was chosen
taking into account the need to test smaller
intervals in the age classes and also the lunar
influence [Williams, 1979] in the life cycle of the
animal. The weight of the animals was not calculated
and therefore fertility was considered as a constant
value per female, independently of its age. This
approach uses a minimum and a maximum
temperature and a minimum day length to proceed
with the recruitment. Recruitment takes place on a
semi-lunar periodicity and it was considered that
maturity is attained only near the end of age class 4.
This was accomplished by considering age classes 1
to 3 as totally constituted by juveniles (or immatures)
and age class 4 as having only a small percentage of
adults capable of reproduction.
Figure 1. Conceptual model for Talitrus saltator
population dynamics.
Table 1. Components of the model and their values
when applicable. ª - initial values are presented.
F.F.=Forcing function. Par.=Parameter. S.V.=State
Variable. C.R.=Calculated rate. C.V.=Calculated
value
Type
Name
Value
Units
F.F.
temperature
time series
ºC
F.F.
daylength
time series
hours
F.F.
timenew
1 to 364
dimensionless
Par.
mortality_rate
0.178
per 28 days
Par.
pctg_age 4
0.9
proportion
Par.
fertility
13
recruits/female
Par.
max_recruit temp
27
ºC
Par.
min_recruit_dayl
13
hours
Par.
min_recruit temp
14
ºC
Par.
recruit_pulse_period
14
days
Par.
sex_ratio
0.455
females/total
S.V.
age 1
0ª
ind./m coast
S.V.
age 2
0ª
ind./m coast
S.V.
age 3
0ª
ind./m coast
S.V.
age 4
0ª
ind./m coast
S.V.
age 5
253ª
ind./m coast
S.V.
age 6
0ª
ind./m coast
S.V.
age 7
193ª
ind./m coast
S.V.
age 8
235ª
ind./m coast
S.V.
age 9
153ª
ind./m coast
S.V.
age 10
0ª
ind./m coast
S.V.
age 11
0ª
ind./m coast
C.R.
recruit
variable
ind./m coast/ day
C.V.
juveniles
variable
ind./m coast
C.V.
adult_females
variable
ind./m coast
C.V.
total_number
variable
ind./m coast
5. EQUATIONS
Sun
Air
temperature
Moon
(Day length)
504
Stella (version 5.1.1) with Euler integration
method was used as software to run the model.
5.1 First approach
a) Growth:
The daily increase in dry weight was determined
and used to fit Equation 1:
DW0.006845DW0.002496
d
d
3
2
t
w
¥-¥=
(1)
Having a series of values, W1 to W8,
corresponding to the weights after each 45 days
period (age class) it is possible to calculate the dry
weight by keeping track of the value of the
previous W, 45 days ago (Equation 2):
)1)- W(i0.006845-1)- W(i(0.002496
daysT+1)- W(i= Wi
(-45)
(2/3)
(-45)
eff(-45)
¥¥
¥¥
(2)
Wi - dry weight at the end of age class “i”
W(i-1)(-45) - dry weight at end of age class “i-1”
Teff - temperature regulator for growth, dependent
on the average temperature during growth
days - number of the previous 45 days in which
growth was possible
The temperature regulator for growth, “Teff” (f(T)
in the equation) was calculated with the aid of the
following equation from Lehman et al. [1975 in
Bowie et al., 1985], which has the shape of a
skewed normal distribution (Equation 3):
(3)
Topt - temperature for maximum growth
Tx - =Tmin when T<=Topt
=Tmax when T>Topt
Tmin - Minimum temperature for growth.
Tmax - Maximum temperature for growth.
In the model the following values were used
according to unpublished results from Collombini
and Chellazi: Maximum temperature 28.8ºC,
minimum temperature 9ºC and optimum
temperature 24.7ºC. This is in accordance with
data showing that the species was active from
10 to 28.8ºC [Scapini et al., 1992].
b) Mortality:
Analysis of field data did not demonstrate any
age, size or weight dependent mortality rates. For
this reason a common value was used for all the
age classes.
c) Fertility and recruitment:
Fertility is known to depend on the size of T.
saltator. The equation from Williams [1978] was
used in which we have (Equation 4):
"number of embryos per brood" =
= -2.58 + 1.17 * "body length" (4)
Three types of recruitment were tested. 1. A single
and long reproduction period only dependent on the
day of the year. 2. More than one reproduction
period but also dependent on the day of the year:
2a) two periods or 2b) three periods. 3.
Reproduction dependent on the temperature and day
length
Each of these types involved a different set of
equations therefore the comparison of the results of
each type will act (sensu lato) as a form of
sensitivity analysis. Type 1 used Equation 5 for the
regulation of recruitment:
†
Fert reg
=e
-2.3¥Timenew -Max_day
Phase-Max_day
Ê
Ë
Á ˆ
¯
˜
2
Ê
Ë
Á
Á
ˆ
¯
˜
˜
(5)
The variable “Phase” will assume different forms
when the following conditions are met:
IF Timenew >Max_day THEN Phase = End_day
ELSE Phase = Start_day
Variables in Equation 5 are: Fert reg = regulator for
recruitment. Timenew = Julian day. Max_day = day
when the highest recruitment happens. Start_day =
day when recruitment starts. End_day = day when
recruitment ends.
This equation provides an adequate shape of the
recruitment curve but it needs to be multiplied by
two other values in order to obtain the number of
new recruits each day. These are the number of new
recruits that would be released if all the mature
females were giving birth and the maximum
percentage of mature females releasing new
recruits.
Type 2 (a and b) used Equation 6 modified from the
normal distribution curve presented in Sokal and
Rohlf [1987] for each wave of recruitment:
˜
˜
¯
ˆ
Á
Á
Ë
ʘ
¯
ˆ
Á
Ë
Ê
¥
¥¥=
2
sd
day-Timenew
0.5-
e
sd
0.39894
a Wave
(6)
In which:
Wave = Recruitment wave. a = correction factor for
the adjustment of the curve. sd = standard
deviation. Timenew = Julian day. day = day of the
maximum recruitment value.
“Wave” multiplied by the number of new recruits
that would be released if all the mature females
were giving birth will provide the value for the new
recruits at each day.
†
f
(T)
=e
-2.3 T-T
opt
T
x
-T
opt
Ê
Ë
Á
Á
ˆ
¯
˜
˜
2
È
Î
Í
Í
˘
˚
˙
˙
505
In case 2a for the first recruitment wave a = 1.9,
day = 140 and sd = 20 and for the second
recruitment wave, a = 0.15, day = 223 and sd =
21.0236. In case 2b for the first recruitment
wave a = 0.8, day = 110, sd = 6, for the second
recruitment wave, a = 1.3, day = 160, sd = 4 and
for the third recruitment wave, a = 0.3, day =
235 and sd = 6.
Recruitment type 3 used the following logic (7)
to produce recruits:
IF (Temperature > min_recruit_temp AND
Temperature < max_recruit_temp AND
Daylength>min_recruit_dayl) THEN PULSE(1,
1, recruit_pulse_period) ELSE 0
It means that if the temperature is within the
limits for recruitment and if the day length is
long enough to start reproduction then we will
have recruitment on a periodical basis e.g. based
on a lunar cycle. Otherwise recruitment will be
zero. The model used a value of 14ºC for the
minimum temperature for recruitment, 27ºC for
the maximum temperature for recruitment, and
13 for the minimum day length.
5.2 Second approach
Equations defined in the previous approach for
Growth (and its temperature regulation), female
fertility and recruitment do not apply in the
present case. Recruitment i.e. the number of
recruits being added to the population each day
at age class 1 is defined by Equations 8, 9, 10
and 11:
recruitment = recruitment_pulse _
adult_females _ fertility (8)
recruitment_pulse =
IF (temperature > min_recruit_temp AND
temperature <max_recruit_temp AND
daylength>min_recruit_dayl) THEN (9)
PULSE(1, 1, recruit_pulse_period) ELSE 0
adult_females = sex_ratio _
(total_number - juveniles) (10)
juveniles = age_1 + age_2 + age_3 +
pctg_age4 _ age_4 (11)
The average number of eggs produced by each
female (“fertility”=13) was determined by
Williams [1978]. The equation for
“recruitment_pulse” uses an IF THEN ELSE
logical statement and it means that a minimum
and a maximum temperature and a minimum day
length are needed to produce a certain output
value. If these conditions are met then a value of
one will be produced at a periodicity defined by
“recruit_pulse_period” if they are not met then zero
is the output value.
In the equations most of the names of the
components are self-explanatory. Table 1 provides
the values used in each parameter, the initial values
of each state variable and also an explanation for
the meaning of each model component.
6. SENSITIVITY ANALYSIS
Table 2. Sensitivities of the density of Talitrus
saltator (total number) to several model parameters.
Sensitivities of “total number”
+50%
+10%
-10%
-50%
mortality rate
-1.041
-1.460
-1.764
-2.678
pctg age 4
-0.228
-1.028
-1.028
-1.028
fertility
0.810
0.765
0.742
0.696
max recruit temp
0.000
0.000
0.847
1.278
min recruit dayl
-1.278
-3.038
-45.036
-44.113
min recruit temp
-1.031
-2.748
-11.261
-2.252
recruit pulse period
-0.397
13.962
-12.159
-2.868
sex ratio
0.810
0.765
0.742
0.696
Table 3. Sensitivities of the density of Talitrus
saltator (total number) to several initial values.
Sensitivities of “total number”
50%+
10%+
10%-
50%-
age 5
0.708
0.708
0.708
0.708
age 7
0.172
0.172
0.172
0.172
age 8
0.082
0.082
0.082
0.082
age 9
0.036
0.036
0.036
0.036
The method described by Jorgensen [1988] and by
Haefner [1996] was used to perform a sensitivity
analysis. The effects caused on the density by ±10%
and ±50% changes in the parameters and initial
values values were tested (Table 2 and 3). When
initial values were zero it was impossible to use the
formula.
The total densities of Talitrus saltator are deeply
affected by variations in the minimum recruitment
day length (min_recruit_dayl), the minimum
temperature for the recruitment (min_recruit_temp),
and the period between each recruitment wave
(recruit_pulse_period). These parameters have a
strong influence in the timing of reproduction and
population density as opposed to the maximum
temperature for recruitment (max_recruit_temp),
which seems rather unimportant for model
performance. Also very important is the “mortality
rate”, showing also an inverse reaction pattern to
changes in the parameter.
506
Figure 2. Field data and simulations results
(dark line) obtained under the first approach
using four different model versions to simulate
recruitment: a) one long reproduction period
dependent on the day of the year b) two
reproduction periods dependent on the day of
the year. c) three reproduction periods
dependent on the day of the year. d)
reproduction dependent on the temperature and
day length. “a” and “b” use smoothed field data.
“c” and “d” present simulation results after an
initiation run of ca. one year.
The sensitivity of the population densities to
each initial value of the state variables was
equal for all the levels of change. The density of
T. saltator was most sensitive to the initial
number of individuals in age class 5 and least
sensitive to the initial number in age class 9.
7. SIMULATION RESULTS
7.1 First approach
From the different types of recruitment chosen
different results were obtained (Figure 2). In
situation a and b the observed densities are
smoothed data, calculated as an average of three
points in order to eliminate abnormal density
oscillations in consecutive sampling dates. Situation
a corresponds to a simulation of a single and long
recruitment period, and we can see that two bumps
in the density line appear spontaneously. Situation b
corresponds to the simulation of two recruitment
waves and therefore the smoothed densities field
data and the simulated density present a closer
resemblance in behaviour.
0
500
1000
1500
2000
2500
0 100 200 300 400
Julian day
observed
simulated
Figure 3. Results obtained from the final model
version (second approach). Plot of the observed and
simulated data vs. time for each sampling date only.
Situations c and d in Figure 2 represent a
comparison of simulated density data and non-
smoothed field data. This last type of data presents
big oscillations and therefore it is more difficult to
model such behaviour. Situation c represents the
simulation obtained by the model in which three
different periods of reproduction were used. One
can see that in this case both the behaviour and the
values simulated closely match the field data.
Finally in the last situation (d) a dependence on
both the temperature and the day length was used,
and recruitment takes place on a lunar periodicity.
From the figure it seems that the behaviour of the
simulated data is close to the real data.
Unfortunately the model presented a very weak
long-term stability and therefore only the data from
two consecutive years are presented.
7.2 Second approach
This approach is in our opinion the best one. It
shows a good fit between the observed and
simulated densities of Talitrus saltator (Figure 3)
by using temperature and day length dependencies
for recruitment. There is a slight tendency to
0
500
1000
1500
2000
2500
0 200 400 600 800 1000
Days
Number of individuals
a
0
500
1000
1500
2000
2500
0 200 400 600 800 1000
Days
Number of individuals
b
0
500
1000
1500
2000
2500
3000
3500
350 550 750 950 1150
Days
Number of individuals
c
0
500
1000
1500
2000
2500
3000
3500
350 550 750 950 1150
Days
Number of individuals
d
507
overestimate the number of individuals at small
densities and this will probably occur during the
cold season.
8. DISCUSSION
Two main approaches were used: the first one
with four different versions and the second one
with a single and successful version. Although
the first approach resulted generally in a
reasonable match between simulated and
observed data it had some serious
inconvenients. The first 3 model versions
(section 5.1c) were insufficient for the model
purposes; in fact recruitment was simply started
by the day of the year. This has no biological
meaning although the same photoperiod is
found at the same Julian day. The fourth model
version considered mechanisms for the start of
reproduction in which day length and
temperature were now considered important.
Nevertheless the model had a poor long-term
stability. The use of continuous reproduction
during a period of the year or during a few
different periods provided non-satisfactory
results. In all the four cases in order to obtain a
good match between simulated and observed
values mortality rates had to be unrealistically
high. The consequence of this is the indication
that recruitment would be discontinuous and
therefore a different model structure had to be
tested. This is what led us to the implementation
of the second approach. This is in fact a simpler
but also an explanatory model. We believe that
the processes taking place in this model
correspond to mechanisms observed in nature.
Therefore this approach is the model being
discussed in the following paragraphs.
There is high model sensitivity to changes in the
structure and parameters of the recruitment
mechanism. This is notorious when several
model versions were sequentially tested using
different approaches for the recruitment process.
Also, as expected, the value of the mortality rate
is very important for the stability of the model.
A rate of 0.178 per 28 days was determined by
calibration and it is equivalent to 0.00726 per
day, which is close to both the average
(0.00993) and the median (0.00638) values for
the field data cohorts.
So far the model attained the calibration phase
and the question to be posed is if it will make
correct predictions under a different situation.
This could mean another year in the same or a
nearby location or a totally different and remote
T. saltator population. If one of the above is
achieved then the model can be considered
validated i.e. tested with an independent data set
[Jorgensen, 1988]. Nevertheless the process of
model building clarified some issues and some gaps
in the knowledge were identified.
9. ACKNOWLEDGEMENTS
This work was funded by the European Union
research project “From river catchement areas to
the sea: a comparative and integrated approach to
the ecology of Mediterranean coastal zones for
sustainable management”.
10. REFERENCES:
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Sokal, R.R. and Rohlf, F.J., Introduction to
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508
