Appendix I: Technical Overview of Model
An individual-based model of Undaria pinnatifida growth and reproduction was developed in
the C++ programming language based on a generic, computational framework for representing large
populations of autonomous agents in a discrete, two-dimensional environment (Murphy et al. 2013).
It is optimised for parallel processing (by domain decomposition) using the Message Passing Interface
in order to be able to efficiently process large populations of independent agents (Gropp et al. 1996).
The main life history stages (gametophytes and sporophytes; Fig. S1) of U. pinnatifida are represented
as distinct agents in the model with unique properties based on experimental data from the literature.
Here we present an overview of the latest version of the UndariaGEN model (v0.6.4), which
incorporates several significant new features/revisions to improve its accuracy and robustness as
compared to the previous version (Murphy et al. 2016a). A more detailed description of the base
model (v1.0) and the process of selection/parameterisation of the input values can be found in
(Murphy et al. 2016a). Empirical data (from a survey in Brest Harbour) was used to validate the model
(Murphy et al. 2016b).
There are two principal types of agents in the model, corresponding to the gametophyte and
sporophyte stages in the life cycle of U. pinnatifida respectively. These have distinct growth
parameters and differ in their response to environmental cues. The growth rate and maturation of the
gametophyte and sporophyte agents are functions of local environmental parameters such as
irradiance, day length and temperature. In order to quantify this relationship, a review of the literature
was carried out to gather quantitative data and build mathematical descriptions of these interactions
at the individual level. The overall population dynamics is therefore an emergent property of the
interactions between these components and the environmental parameters.
1.1. Program structure
The model is constructed using the object-oriented programing paradigm of C++. The initial phase
of the program involves the creation and initialisation of an array of U. pinnatifida agents which are
stored in an array data structure. The input parameters for the simulation are entered via a text input
file (Table S1). These specify physical parameters such as the size and scale of the environment as well
as parameters for the U. pinnatifida agents such as the initial number of seedlings and their responses
to environmental cues.
The main program loop consists of a series of steps representing the main biotic and abiotic
functions of the system (Fig. S2). Each loop represents a discrete hour of real-time during which the
agents grow and interact with their environment. The first step of the loop involves updating the
current environmental conditions (temperature, irradiance and day length). These values are updated
on a daily basis (i.e. every 24 loops of the simulation). Step 2 involves the implementation of Fick’s
first law of diffusion for dispersal of spores. Spore germination to form new gametophyte agents can
occur in the presence of an appropriate substrate for attachment. Steps 4 and 5 involves the growth
and reproduction of sporophyte/gametophyte agents according to their particular life cycle rules.
The simulation will exit the program loop when certain exit conditions occur. These include when
all agents have died or when a set (user-defined) period of time has elapsed. There are also processes
for displaying a real-time graphical display of the locations of the agents in the environment, and for
recording various diagnostic parameters to text files for later analysis and graphing of results. The
following sections include an overview of the main components of the model: the environment, and
the main agent types (gametophytes/sporophytes).
1.2. The environment
The environment is represented as a discrete, two-dimensional grid with each grid element
(or lattice point) corresponding to an area of 0.25 m2. For this study the modelled environment
represented a port/marina with floating pontoons as potential attachment points for the macroalgae.
A 2-dimensional representation of the environment was chosen since in this situation the species is
established close to the surface, although depth is taken into account when calculating light
availability for photosynthesis. The dispersal of particles (e.g. spores) between lattice positions in the
environment is calculated using a discretised implementation of Fick’s First Law of diffusion (Ginovart
et al. 2002). This is sufficient for modelling dispersal at a local scale since U. pinnatifida spores lose
their fixation ability shortly (a few hours) after release, and thus long distance transport mediated by
hydrodynamic processes can be ignored when the focus is on local population dynamics (Suto 1950;
Thiébaut et al. 1998).
There is a parameter called the light attenuation coefficient for photosynthetically available
radiation (KdPAR) which determines the amount of light attenuation, due to dissolved matter, with
increasing depth in the water column (Saulquin et al. 2013). This is used to calculate the residual
energy available for photosynthesis by U. pinnatifida agents according to the following equation:
𝐸(𝑧) = 𝐸(0)𝑒 −𝑧𝐾𝑑𝑃𝐴𝑅
(1)
where z is the depth (m) of the seaweed below the surface, E(0) is the level of irradiance at the water
surface, and E(z) is the energy available for photosynthesis at depth z. For the case studies in this
paper, the depth (z) was set to 1.0 m and the attenuation coefficient (KdPAR) to 0.4 for conditions
representative of the coastline of Brittany, France (Saulquin et al. 2013).
1.3. Sporophyte agents
1.3.1. Growth
Sporophyte agents are modelled from their initial microscopic, cellular scale up to a maximum
length of 1-3 metres. This represents a particular modelling challenge due to the range of scales
involved (covering several orders of magnitude). Therefore, the growth rate is calculated as a function
of the length of the sporophyte, in order to take into account inherent scaling effects. Studies from
the literature with growth rate estimates of sporophytes in different size classes (from microscopic
sporophytes in culture to mature sporophytes 79 cm in length) were used to calculate the basic
relationship between length and the relative growth rate (Choi et al. 2007; Pang, Lüning 2004; Shaojun, Chao-yuan 1996). A power law functional relationship was then fitted to this data to calculate the
baseline relative growth rate (RGR_Sbase) per day for sporophyte agents (where conditions are
assumed to be constant with irradiance = 40 mol m-2 s-1, temperature = 15oC, day length = 12 hours):
𝑅𝐺𝑅_𝑆𝑏𝑎𝑠𝑒 = 3.615𝑙 −0.407
(2)
Photosynthetic efficiency has also been shown to decrease with increasing sporophyte size,
due to increased thallus complexity/density and increased dry weight:fresh weight ratio (Campbell et
al. 1999; Gómez, Wiencke 1996). Therefore, when determining the photosynthetic rate of sporophyte
agents, plant size must also be taken into account (Table S1). This relationship was estimated using
data from the literature on the photosynthetic efficiency of sporophytes of different size classes in
comparison with microscopic U. pinnatifida gametophytes (Campbell et al. 1999; Choi et al. 2005).
These relationships were used to calculate the parameters (Pmax, , Ic) as a function of sporophyte
length (Fig. S3) in order to generate a unique photosynthesis-irradiance curve for each agent based
on the hyperbolic equation of Jassby, Platt (1976):
𝑅𝐺_𝑆𝐼 = 𝑃𝑚𝑎𝑥 ∙ (1 − exp [−𝛼 ∙
𝐼 − 𝐼𝑐
])
𝑃𝑚𝑎𝑥
(3)
where RG_SI is the relative effect of irradiance on the growth rate of the sporophyte agent, I is
irradiance (mol m-2 s-1), Pmax is the maximum rate of photosynthesis,  is the slope of the curve, and
Ic is the compensation point.
In order to determine growth as a function of water temperature, parameters for a thermal
performance curve were estimated by fitting to experimental data on U. pinnatifida sporophytes (Fig.
S4a) (Morita et al. 2003b). This is used to estimate the relative effect of temperature on the growth
rate of the sporophyte:
𝑅𝐺_𝑆𝑇 = 𝑆 [
1
(1 + 𝐾1 𝑒 −𝐾2 (𝑇𝑏
] × [1 − 𝑒 𝐾3 (𝑇𝑏 −𝐶𝑇max) ]
− 𝐶𝑇𝑚𝑖𝑛 )]
(4)
where RG_ST is the relative effect of temperature on the sporophyte growth rate, K1, K2 and K3 are
constants, CTmin and CTmax are the lower and upper critical temperature limits respectively, and S is a
scaling factor (Stevenson et al. 1985).
Meanwhile, experimental data from Pang, Lüning (2004) were used to define a hyperbolic
relationship between the growth rate and day light hours (Fig. S4b). In the model, the growth rate of
the sporophyte is also assumed to be proportional to its rate of photosynthesis. Therefore, a
photosynthesis-irradiance curve was defined based on empirical data from Campbell et al. (1999):
𝑅𝐺_𝑆𝐷𝐿 = 𝐺𝑚𝑎𝑥 ∙ (1 − exp [−𝛼 ∙
𝐼 − 𝐼𝑐
])
𝐺𝑚𝑎𝑥
(5)
where RG_SDL is the relative effect of day length on the sporophyte growth rate, I is the irradiance
(mol m-2 s-1), Gmax is the maximum rate of growth,  is the slope of the curve and Ic is the
compensation point (Jassby, Platt 1976).
In the latest version of the model, an additional parameter has been incorporated to represent
scale effects on photosynthetic performance between individual plants and communities. It has been
demonstrated that the light saturation point (Ik) for macroalgal communities can be several times
higher than for individual thallus pieces tested in the laboratory (Binzer, Middelboe 2005). Therefore,
in order to take this into account in the model, we have included an additional parameter to vary the
light saturation point according to the community structure. For the test cases described in this paper,
this value was calculated by fitting to field data collected by Voisin (2007) from Brest harbour, France
(see Table S1, “Community light saturation factor”). The parameter , in equation (5) above, is divided
by the light saturation factor in order to represent the effect of reduced photosynthetic efficiency on
the sporophyte growth rate.
In conclusion, the relative daily growth rate of the sporophyte agent (RGR_S) is calculated as
a function of the combined effects of the three environmental parameters (irradiance, temperature
and day length), described in equations (3)-(5), on the base growth rate, equation (2):
𝑅𝐺𝑅_𝑆 = 𝑅𝐺𝑅_𝑆𝑏𝑎𝑠𝑒 × 𝑅𝐺_𝑆𝑇 × 𝑅𝐺_𝑆𝐼 × 𝑅𝐺_𝑆𝐷𝐿
(6)
1.3.2. Maturation and spore release
In order to determine the mean size at which sporophytes reach maturity, field data from
Brest harbour, France was used (mean sporophyte length at maturity = 32.7 cm) (Voisin 2007). This is
used in the model to determine the mean size at which maturity is reached (i.e. the formation of a
spore-producing structure called the sporophyll). The probability of spore shedding by mature
sporophytes is described as a function of sea water temperature (Saito 1975). This relationship was
calculated for the model by fitting a logistic function to field data from Suto (1952). Once released
spores are subject to diffusion in the environment according to Fick’s First Law of diffusion (Fick 1855).
When they come into contact with a suitable substrate they can attach and germinate to form new
gametophyte agents according to a specified (substrate-specific) probability of fixation/germination.
Otherwise, the spores in the water column are subject to a pre-defined half-life for degradation over
time (Table S1).
1.3.3. Death of sporophyte
The death of a sporophyte agent is determined either when it reaches the natural end of its
lifespan, assumed to be after it has matured and released all its spores, or through premature
death/removal. Field studies have shown that up to 70% of sporophyte recruits die/disappear within
one month of their appearance (Voisin 2007). Therefore, premature death (by various means such as
competition or physical dislodgement) represents a significant proportion of the deaths in a
population. To account for this, field data collected by Voisin (2007) from a population of U. pinnatifida
in Brest, France, was used to create an age to mortality curve (Weibull distribution) which determines
the probability of premature death as a function of the age of the sporophyte.
In the earlier version of the model, there was a single age to mortality curve used to describe
the probability of early death among recruits. However, in the latest version, it has been updated to
take into account seasonal variation in the probability of premature mortality. When the deaths of
young sporophytes recorded in Brest harbour are plotted according to the month they occur in, it is
possible to observe a clear seasonal trend in the age to mortality (Fig. S7). For example, in July/August
there is a peak in the mortality of young recruits (<1 months old). This is probably due to increased
competition/shading from mature macroalgal assemblages inhibiting the establishment of new
recruits.
Therefore, a series of 12 individual Weibull functions (for each month of the year) were fitted
to the mortality data in order to capture the seasonal correlation in the probability of early death.
Cosine curves were used to describe the change in the shape (k) and scale () parameters of the 12
Weibull curves as a function of the day of the year (Fig. S8):
𝑦 = 𝐴 cos[𝜔(𝑥 − 𝛼)] + 𝐶
(7)
where x is the day of the year, A is the amplitude,  is the horizontal phase shift, C is the vertical offset
and  is the angular frequency (2/365). This functional relationship could then be used in the model
to generate a probability of early mortality based on the time of the year and the age of the
sporophyte, rather than assume a constant probability.
1.4. Gametophyte agents
1.4.1. Growth
Gametophytes represent the haploid phase of the life cycle of U. pinnatifida with their own
unique growth requirements (Fig. 1 in the main text). They are represented in the model as distinct
agents with independent biological properties. The responses of the gametophytes to temperature,
light and day length were modelled using the same methodology as for the sporophytes, but with
unique parameters estimated from the literature (Fig. S5 & Table S1 ) (Choi et al. 2005; Morita et al.
2003a). However, since gametophytes generally do not grow beyond a few millimetres in size, scaling
effects on photosynthetic efficiency and growth were not applied.
The base growth rate (relative daily increase in size) was calculated using the thermal
performance equation (see eq. 4) of Stevenson et al. (1985) fitted to empirical data from the literature
on the growth of gametophytes (Fig. S5b) (Morita et al. 2003a). Similarly, the relative effect of day
length and data was calculated by fitting to empirical measurements from Choi et al. (2005). The
hyperbolic equation of Jassby, Platt (1976) (see eq. 3) was fitted to this data (Fig. S5a). The parameters
(Pmax, , Ic) for the hyperbolic equation are themselves functions of day length (see Table S1).
1.4.2.Gametogenesis
The process of gametogenesis (maturation and production of gametes) in gametophytes is
also sensitive to environmental parameters such as temperature and day length. The probability of a
new sporophyte forming is assumed to be a function of the relative proportion of fertile gametophytes
present. Data from the literature on the effects of temperature and different day length regimes on
gametophyte fertility were used to calculate the relative probability of gametogenesis and subsequent
sporophyte formation (Choi et al. 2005; Morita et al. 2003a).
The relationship between temperature and gametophyte fertility is defined by the following
logistic relationship which was fitted to empirical data from Morita et al. (2003a) (Fig. S5a):
𝑇𝑔 = 1 − [
1
]
(1 + 𝑒 −𝑘(𝑡−𝑡0 ) )
(8)
where Tg is the relative temperature effect on gametogenesis, k is the steepness of the curve, t is the
current water temperature, and to is the temperature at the midpoint of the sigmoid. Similarly, the
effect of day length on fertility was determined by fitting a Weibull curve to data from Choi et al.
(2005) (Fig. S6b).
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Table S1: Input parameters for UndariaGEN simulations of Undaria pinnatifida in simulated
port/marina environment. l = sporophyte length (m), d = daylight hours, loop = simulation loop.
Ecophysiological parameters derived from literature review (Choi et al. 2005; Morita et al. 2003a, b;
Pang, Lüning 2004; Shao-jun, Chao-yuan 1996)
Parameter Type
Parameter (units)
Input Value
General
Length of Simulation Loop (hours)
1
Environment Size (No. of Cells)
514 x 482
2
Cell Area (m )
0.25
Substrate depth in water (m)
1.0
Attenuation coefficient (KdPAR)
0.4
Community light saturation factor
3.33
Sporophyte
Initial length, l0 (m)
20.0
agents
Base growth rate
3.615 l-0.407
Day length response (hyperbolic curve):
Pmax
1.56
a
0.13
Ic
0.0
Thermal performance curve :
K1
21.09
K2
0.213
K3
0.006
CTmin
1.62
CTmax
28.28
Scale
3031
Photosynthesis-irradiance curve:
Pmax
0.4ln(l) - 0.596
a
0.5l-0.33
Ic
2.5ln(l) – 19.9
Mean length at maturity (cm)
32.66
Cosine curves
Weibull shape parameter (k):
A
0.205
(see Fig. 3):

248.5
C
2.607
Age to Mortality
Weibull scale parameter ():
A
0.2

363.2
C
1.165
Gametophyte
agents
Thermal performance curve :
K1
35.67
K2
0.158
K3
0.015
CTmin
4.45
CTmax
28.24
Scale
10.63
Photosynthesis-irradiance curve:
Gametogenesis
Pmax
0.29e0.11d
a
0.029d – 0.2
Ic
0.0
Prob. of fertilisation (loop-1)
0.0002
Temperature response curve (log):
x0
17.6
k
0.82
Day length response (Weibull):
Spores

4.5

10.96
Half-life (hours)
24
Release rate (agent-1 loop-1)
2.0 x 107
Spore stock (agent-1)
1010
Diffusion coefficient
0.15
Prob. of germination (loop-1)
10-9
Fig. S1: Main stages in the heteromorphic biphasic life cycle of Undaria pinnatifida: Diploid (2N)
sporophyte stage (can grow up to 3 metres in length) which matures to form a sporophyll structure
that releases spores. The microscopic haploid (N) gametophyte stages (male and female) form after
fixation and germination of spores on a suitable substrate. They reproduce sexually to form a new
diploid sporophyte generation. Photos of Gametophytes/Spores: Daphné Grulois-Station Biologique
Roscoff
Fig. S2: Program flow structure for agent-based model of Undaria pinnatifida population dynamics
coded in C++ programming language. One program loop = 1 hour real-time.
Fig. S3: Photosynthesis-irradiance curve parameters (Pmax, , Ic) for hyperbolic equation of Jassby &
Platt (1976) calculated as a function of sporophyte length (m).
(a)
(b)
Fig. S4: Relative effect of (a) water temperature (oC) and (b) day length (hours of day light) on the
growth rates of sporophyte agents.
(a)
(b)
Fig. S5: Effects of (a) irradiance (mol m-2 s-1) and day length (h), and (b) water temperature (oC) on
the relative growth rates of gametophyte agents.
(a)
(b)
Fig. S6: Effects of (a) water temperature (oC), and (b) day length (h) on the relative fertility of
gametophyte agents.
(a)
Mortality (Age <1 month)
1
R2=0.85
0.9
0.8
0.7
0.6
R2=0.73
0.3
0.2
0.1
0
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Mortality (Age 1-2 months)
(b)
Fig. S7: Probabilities of premature mortality among juvenile sporophytes of U. pinnatifida calculated
from field data in Brest harbour, France (Murphy et al. in press). (a) Probability of mortality among
recruits <1 months old. (b) Probability of mortality for sporophytes 1-2 months old. Data points
represent 3-month moving average values. Cosine functions were fitted to the data.
(a)
y = 0.2cos[(x-248.5)] + 2.6
R2 = 0.89
Weibull Scale ()
2.8
2.7
2.6
2.5
2.4
2.3
0
100
200
lambda
Weibull shape (k)
(b)
1.4
300
Cosine
y = 0.2cos[(x-363.2)] + 1.2
R2 = 0.97
1.3
1.2
1.1
1
0.9
0
100
200
300
Day of Year
k
Cosine
Fig. S8: The shape (k) and scale () parameters for monthly Weibull distributions (fitted to the age to
mortality data for juvenile sporophytes from Fig. 2 in the main text) as a function of the time (day) of
year. Cosine functions fitted to the data ( = 2/365).