ABSTRACT
HALL, TESSA ELIZABETH. Size-Structured Population Model for Ocean Acidification Impacts Through Effects on Demographic Processes. (Under the direction of Dr. Kevin Gross).
Size-Structured Population Model for Ocean AcidificationImpacts Through Effects on Demographic Processes
by
Tessa Elizabeth Hall
A thesis submitted to the Graduate Faculty of North Carolina State University
in partial fulfillment of the requirements for the degree of
Master of Science
Biomathematics
Raleigh, North Carolina 2018
APPROVED BY:
Dr. Kevin Gross
Chair of Advisory Committee
Dr. Alun Lloyd
DEDICATION
BIOGRAPHY
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OF TABLES . . . vi
LIST OF FIGURES . . . vii
Chapter 1: Introduction . . . 1
Chapter 2: Methods . . . 5
Model Formulation . . . 5
Numerical Implementation: Escalator Boxcar Train . . . 11
Analysis . . . 16
Equilibrium behavior . . . 16
Sensitivity anlaysis . . . 17
Overview . . . .17
Local sensitivity analysis . . . 18
Global sensitivity analysis . . . 19
Stochastic analysis . . . 21
Chapter 3: Results . . . 24
Chapter 4: Discussion . . . 33
LIST OF TABLES
Table 1: State variables, units, and dimensions. . . 22
Table 2: Model inputs and trial values. . . 23
Table 3: Escalator Boxcar Train Method for Coral Population Model. . . 23
LIST OF FIGURES
Figure 1: Equilibrium values for the proportion of coral cover were evaluated for varying demographic rates. Demographic rates are centered around an estimated value for ambient conditions. The value of ”p” is associated with rates for both internal and external recruitment. Note that scales
axes differ between panels. . . 26 Figure 2: Size distributions are shown for demographic rates associated with
growth, mortality and recruitment (internal and external). Size distributions are seen for demographic rates ranging from 50% to
150% of the demographic rate under ambient conditions. . . 27 Figure 3: Box and whisker plots displaying 10th, 25th, 50th, 75th and 90th
percentiles for coral sizes (radius) at equilibrium. Plots are for varying demographic rates associated with growth, mortality and recruitment. Rates range from 50% to 150% of the demographic rate
under ambient conditions. . . 28 Figure 4: Proportion of coral cover is shown over time for a stochastic per capita
morality rate. Coefficient of variation is found from values of coral cover
at quasi-equilibrium (values occurring after 300 years). . . 29 Figure 5: Coefficient of variation for different rates of maximum linear extension
are shown for a per capita mortality rate with varying degrees of
associated standard deviation. . . 29 Figure 6: Coefficient of variation for different mean rates of per capita mortality
are shown for a per capita mortality rate with varying degrees of
associated standard deviation. . . 30 Figure 7: Coefficient of variation for different rates of recruitment are shown
for a per capitamortality rate with varying degrees of associated
standard deviation. . . 30 Figure 8: Coefficient of variation for different rates of maximum linear extension
are shownfor an external recruitment rate with varying degrees of
associated standard deviation. . . 31 Figure 9: Coefficient of variation for different rates of per capita mortality are
shown for anexternal recruitment rate with varying degrees of
associated standard deviation. . . 31 Figure 10: Coefficient of variation for different rates of recruitment are shown for
an external recruitment rate with varying degrees of associated
1
Introduction
A decline in coral reef health (C`ot´e et al., 2005; Lough et al., 2018; necessitates a greater understanding of the components that affect the dynamics of a coral reef system. Coral reef ecosystems deliver distinct and irreplaceable goods and services that are endangered by human activities. They provide fisheries and material goods, as well as structural, bi-otic, biogeochemical, informational (monitoring and pollution records) and social services (Moberg & Folke 1999). There has been a global trajectory of reef degradation, with mass coral mortality events occurring in recent decades (Pandolfi et al. 2006; Done et al. 1992; Hughes et al. 1994, Aronson et al. 2000; Bruno & Selig 2007; Edmunds & Elahi 2007). This decline has culminated in a 19% loss of the world’s original area of coral reef, and many reefs will be under serious threat of loss in future decades (Wilkinson 2008; Pandolfi et al. 2006). Anthropogenic activity is the principal source of this environmental deterioration and our action is required to avoid further loss (Hughes et al. 2003; Bruno & Selig 2007) .
increasing mechanical bioerosion through weakened substrate (Schonberg et al. 2017). Ad-ditionally, OA can negatively affect recruitment rates by depressing larval metabolism and reducing settlement by disrupting substrate communities that give settlement cues (Albright & Langdon 2011).
In this paper, I form a mathematical model to describe the dynamics of a coral popula-tion. Using this model, I am able to discern how potential effects of OA on the demographic processes of coral colonies can scale up to impact the dynamics of an entire population of coral. In recognizing how demography can affect coral dynamics, we can help guide future research toward understanding complex reef systems. Sensitivity analyses can reveal which aspects of demography are most important, which will help focus future experiments. An appreciation of how an environmental change affects the demographics of coral colonies, along with the knowledge of demographies that have the greatest impact on the population dynamics, can also help to inform remediation and reformation efforts.
Demography also plays an important role in the ability of a coral population to resist or recover from a disturbance, or in other words, the resilience of the coral population. If OA negatively affects coral demography, it could also hinder a reef’s resilience after a distur-bance. Corals live in an environment with disturbances of increasing frequency and intensity (Pandolfi et al. 2006). Disturbances range from coral bleaching caused by increased water temperatures, to storm events and consumption by crown-of-thorns starfish. These events lead to mass mortality of coral populations and could also result in decreased external re-cruitment. Stressors induced by climate change can lead to more frequent disturbances, such as bleaching, as our climate continues to warm (Roche et al. 2018). Therefore, the resilience of a coral reef is especially important in current climate conditions. Using the model formed in this paper, I hope to determine how the effects of OA on demography can impact reef resilience.
for the demographic rates of individual coral colonies to depend on both the density of the coral population, and the size of the colony. During the larval phase, recruits settle onto unoccupied substrate as newborns (or polyps), where they begin to asexually reproduce and grow outward. Upon reaching maturity, colonies sexually reproduce and form planktonic larvae. Larvae enter a pelagic pool, after which they will either return to its reef of origin to settle, or it will recruit in a new reef. Recruitment is proportional to the amount of free space in a reef (Hughes et al., 1985; Connell et al., 1997), which exerts a space-limitation. The amount of free space corresponds to the total density of coral, introducing density-dependent feedback into the system. Including size-dependence in the model is important because field research shows that demographic processes of coral depend upon the size of the coral colony (Hughes and Jackson, 1980; Kojis and Quinn, 1985).
The density-dependence and size-dependence of a coral system can be captured with a physiologically structured population model (PSPM). A PSPM enables the investigation of a population’s dynamics through the behavior of its constituent individuals and considers the environment as a component of the system (de Roos, 1997). This framework is appropriate for a coral population because colonies have unique demographic rates associated with indi-vidual size, and the amount of free space can be considered the environmental component of the system. Using the PSPM, the life history of each individual is tracked along with its effect on the environment, which subsequently determines its future and that of all other colonies. In this way, the dynamics of the entire population is established. The form of the PSPM is that of a partial differential equation (PDE), which allows for clear specifications of demographic processes (Muko et al., 2001) and is better equipped than matrix models to manage density dependence (Caswell et al. 1997). A continuous-time model is also appro-priate for the aseasonal environments of the tropics.
1967; VanSickle 1977; Murphy 1983). Models that are continuous in time and consider space limitation effects can be seen in Artzy et al. (2007) and Muko et al. (2001). Neither of these models, however, consider internal recruitment. They do not consider as wide a range of de-mographic inputs as I wish to consider here. Muko et al. find right-skewed size distributions for steady state solutions, but Artzy et al. show that certain growth functional forms can give distributions that do not decrease monotonically with size. This motivates the question of how demography can affect the form of size distributions.
2
Methods
2.1
Model Formulation
I use the coral taxon Pocillopora verrucosa in Moorea, French Polynesia, to motivate the formation of the model’s demographic functional forms. The Moorea Coral Reef Long Term Ecological Research project sustains ecological and physical monitoring that has given rise to OA experimental work (Edmunds et al. 2013). Moorea has also experienced several large perturbations in the past four decades, making it a suitable site for studies of reef resilience (Han et al. 2016). P. verrucosa is a common branching coral in Moorea and has been studied in experiments there (Evensen & Edmunds 2016; Edmunds & Burgess 2018; Done et al. 1991).
Our model conveys the dynamics of a biological system consisting solely of coral and algal turf. Development of our model closely follows de Roos (1997). Coral colonies are assumed to be of a single species and grow without competition from any other species. Coral colonies expand onto algal turf, which is thought of as free space. Spatial distribution is neglected so that all colonies are assumed to experience the same environment. Furthermore, the growth and development of individuals is deterministic, and only a large population is considered so that demographic stochasticity does not come into play. In this way, the formulation of the PSP model that follows is a deterministic approximation of a stochastic process.
rate of radial growth of a colony declines linearly with increasing radius. Coral colony growth also decreases with an increased coral density (Karlson et al. 1996; Griffin et al. 2015). It is therefore assumed that radial growth scales linearly with the amount of free space. There is a maximum radial growth rate of l when a colony is newly settled, and a radial growth rate of 0 when the radius attains its maximumrmax. Space constraints and the likelihood of mortality make it unrealistic for a colony to grow indefinitely. The values of l used in our model, as well as all other model inputs for ambient conditions, can be found in Table 2. The rate at which the radius of an individual colony grows is then:
g(C, r) =l(1− r
rmax
)(1−C) (1)
where C is the proportion of the total area covered by coral.
The model considers both internal and external recruitment. Although larval settle-ment is thought to be seasonal, Artzy et al. 2006 point to cases where recruitsettle-ment occurs consistently for 8 months and the entire year. Also, in Muko et al.’s study, they find that a model with discrete settlement was well approximated by a model with constant recruitment. Internal recruitment is a result of sexual reproduction of coral colonies within the system, and is therefore determined by the size of coral colonies within the system. The settlement of larvae is proportional to the amount of free space so that space limitation drives the density dependence of coral recruitment (Vermeij & Sandin 2008). The rate at which an individual colony produces recruits, b(C, r), is proportional to the surface area of the hemispherical colony, 2πr2. The rate at which new recruits are produced per unit surface area of coral colony, multiplied by the probability that a recruit successfully attempts to settle, is given by the parameter β. I assume that the proportion of these recruits that successfully settle and form a new colony is proportional to the amount of free space. Internal recruitment is then given by:
Note that the value of 2, from surface area proportionality, is folded into the parameter β. External recruitment occurs when larvae enter a pelagic pool far from the reef they were produced in, and eventually settle in a new reef. In this way, coral reefs are demographically open systems. External recruitment is independent of local density but depends on available free space for settlement. The rate of settlement of new recruits from an external source into an area of free space (turf) will be noted by s. External recruitment is defined as
S(C) = s(1−C). (3)
I consider a case of mortality that is independent of colony size. I do not consider partial mortality or mortality to be density dependent. The rate at which individual coral colonies die, µ(C, r), is a constant per capita rate δ:
µ(C, r) =δ (4)
This concludes the specification of the individual behavior of coral. I now look at how the behavior of coral colonies scale up to the population level, continuing to follow the methods put forth by de Roos (1997). PSP models account for the difference between individuals in a population. This leads to the concept of individual state, or i-state. The individual state is the individual physiological property that determines behavior (growth, reproduction, mortality). The individual trait that influences behavior, and is therefore classified as the i-state, is the size (radius) of the coral colony. For an individual, the rest of the population is considered part of its environment, and the environment constitutes a form of feedback. Feedback is present because a statistic of the population, proportion of coral cover, influences individual behavior and therefore regulates population density.
at time t. Also, let rb be the radius of newly settled coral polyp and rmax be the maximum achievable radius for a coral colony. The proportion of substrate covered by coral at t is given by
C(t) =
Z rmax
rb
n(t, r)πr2dr (5)
The fraction of seabed covered by algal turf (free substrate) is represented as T, so that
T = 1−C.
To derive a PDE that describes the coral dynamics, consider a small size interval with a width of ∆r. The size interval is around a length ofr=r0, and within a small time interval ∆t, the size of the coral population in the interval ∆r changes. Fluxes across the bounds of this size interval are examined to describe coral growth and death.
Coral growth leads to a flux across the lower bound at r = r0 − ∆r
2 . The flux is the
rate at which individuals enter the the interval and is proportional to rate at which a single coral colony grows across the boundary and the density of coral colonies doing so:
g(C, r0− ∆r
2 )n(t, r
0 −∆r
2 ) (6)
Where g(C, r) is the rate at which the radius of an individual colony grows. The flux across the upper bound, at r = r0 + ∆2r, is the rate at which individuals leave the interval due to growth and is given by:
g(C, r0+ ∆r 2 )n(t, r
0 +∆r
2 ) (7)
The death of coral colonies within the size interval is represented by the product of the death rate (rate at which individual colonies die) and the density of coral colonies with a radius within the interval:
A balance equation can be formed for the difference between the number of coral coloniesn(t+ ∆t, r0)∆r at time t+ ∆t and the number of coral colonies n(t, r0)∆r at time
t. This is accomplished by multiplying all rates by the time interval ∆t:
n(t+ ∆t, r0)∆r−n(t, r0)∆r=
−g(C, r0+∆r 2 )n(t, r
0+∆r
2 )∆t+g(C, r
0− ∆r
2 )n(t, r
0− ∆r
2 )∆t
−µ(C, r0)n(t, r0)∆r∆t
Divide through by ∆r∆t:
n(t+ ∆t, r0)−n(t, r0)
∆t =
− g(C, r
0+∆r
2 )n(t, r0+∆2r)−g(C, r0−∆2r)n(t, r0− ∆2r)
∆r −µ(C, r
0
)n(t, r0)
Take the limits ∆t → 0 and ∆r → 0 such that g(C, r)∆t ∆r and drop the prime to obtain the partial differential equation (PDE) for the density function n(t, r):
∂n(t, r)
∂t =−
∂g(C, r)n(t, r)
∂r −µ(C, r)n(t, r) (9)
To supplement with a boundary condition that specifies density at the lower end (r=rb) of the interval at every time t, integrate left and right sides of the PDE from r = rb to
r=rmax:
Z rmax rb
∂n(t, r)
∂t dr =−
Z rmax rb
∂g(C, r)n(t, r)
∂r dr−
Z rmax rb
µn(t, r)dr d
dt
Z rmax rb
n(t, r)dr=g(C, rb)n(t, rb)−g(C, rmax)n(t, rmax)−
Z rmax rb
µn(t, r)dr
An individual coral colony will have a growth rate of 0 at rmax, so g(C, rmax)n(t, rmax)≡ 0. Therefore, g(C, rb)n(t, rb) should equal the total population reproduction rate. Total population birthrate must take into account both internal and external recruitment. The total internal birthrate is the integral of the product of the internal, per capita reproduction rate b(C, r), and the population density n(t, r), over the size interval [rb, rmax). For external recruiment, I assume that new recruits arrive at a constant rates. Of these, a proportion (1 - C) settle on to available free space and enter the population, while the remainder die. So the boundary condition has the form:
g(C, rb)n(t, rb) =S(C) +
Z rmax rb
b(C, r)n(t, r)dr (10)
The initial states of coral must be defined. The initial state of the coral population specifies the initial size distribution and initial population size.
n(0, r) =ψ(r)
Where ψ(r) is the density function characterizing the initial coral population. The initial proportion of available free space (algal turf), is then given by:
T(0) = 1−
Z rmax rb
ψ(r)πr2dr
as:
∂n(t, r)
∂t = −
∂l(1− r
rmax)(1−C)n(t, r)
∂r −δn(t, r) l(1− rb
rmax
)(1−C)n(t, rb) = s(1−C) +
Z rmax rb
βπr2(1−C)n(t, r)dr (11)
2.2
Numerical Implementation: Escalator Boxcar Train
The size-structured model (equation 10) has an uncommon form: it is a PDE with a boundary condition that contains an integral of the solution itself, n(t, r). Therefore, find-ing the numerical solution requires an unconventional method, the Escalator Boxcar Train (EBT) method (de Roos, 1997). Our development of the EBT method, along with notation, closely follows de Roos (1997). This method is derived by generalizing an age-structure Leslie-matrix model (Leslie, 1948) with size as the i-state variable, and a continuous-in-time reproduction process that results in a continuous size distribution. It approximates the PSPM by allowing for continuous reproduction and the formation of distinct cohorts that are described by the number and average size of coral colonies within the cohort.
Similar individuals are grouped into cohorts that are described by the number of in-dividuals they contain and their average size. Note that inin-dividuals in the same cohort are not identical, but are enough alike to be realistically characterized by their average. It is assumed that individuals remain in the same cohort and can leave only through death. The fate of the cohorts is tracked through time, with new cohorts arising through reproduction. To form the cohorts, the range of coral radii [rb, rmax) is subdivided into small intervals such that each interval consist of similar individuals. I choose a set of points ri so that the entire range [rb, rmax) is covered by the intervals
Where r1 =rb and rN+1 =rmax. The total density of individuals with a size in the interval Ωi is given by
λi(t) =
Z ri+1 ri
n(t, r)dr (13)
and the average size of the individuals is given by
µi(t) =
Rri+1
ri rn(t, r)dr
λi(t)
. (14)
It is assumed that the cohort with index iis fully characterized by its density of individuals,
λi(t), and their average size (radius), µi(t).
We can write the integral in the boundary condition of equation 10 as a sum of integrals over the intervals Ωi:
Z rmax rb
b(C, r)n(t, r)dr= N
X
i=1
Z ri+1 ri
b(C, r)n(t, r)dr. (15)
Using a Taylor expansion aroundr =µi(t), we can obtain an approximation of the integral up to second-order precision:
Z rmax rb
b(C, r)n(t, r)dr = N
X
i=1
b(C, µi(t))λi(t). (16)
We have obtained a weighted sum that can be used to compute the population reproduction rate.
The dynamics of the density of individuals in a cohort, λi(t), and the average size of individuals in the cohort, µi(t), must be understood to describe the overall population dynamics. It is assumed that individual coral colonies remain in the same cohort throughout their life. Therefore, the boundary values ri are adjusted over time in congruence with the rate of coral colony growth:
dri
This assures that coral colonies never move into a different cohort and allows intervals Ωi to expand or shrink based on a changing growth rate.
The dynamics ofλi is given by the ODE:
dλi dt =
d dt
Z ri+1 ri
n(t, r)dr. (18)
Utilizing methods outlined in de Roos 1997, the following form of the ODE can be derived:
dλi dt =−
Z ri+1 ri
δ(C, r)n(t, r)dr. (19)
Similarly, the ODE for the dynamics of µi(t),
dµi dt =
d dt
Rri+1
ri rn(t, r)dr
λi(t)
, (20)
can be shown to have the form:
dµi dt =
Rri+1
ri g(C, r)n(t, r)dr
λi(t)
−
Rri+1
ri (r−µi(t))δ(C, r)n(t, r)dr
λi(t)
. (21)
To obtain closed and solvable systems for the dynamics of λi and µi, the functions δ(C, r) andg(C, r) are approximated by their Taylor expansions aroundr =µi(t). Neglecting higher order terms, we obtain the following approximate dynamics:
dλi
dt =−δ(C, µi)λi
dµi
dt =g(C, µi) i= 1, ..., N
(22)
The ODE’s (21) describe the dynamics for internal cohorts, or cohorts that are already present in the system. To account for reproduction we must look at the dynamics of a boundary cohort for newborn individuals (polyps). The total density of individuals in the boundary cohort is given by:
λ0(t) =
Z r1 rb
n(t, r)dr. (23)
Equation (14) cannot be used to describe the size of individuals in the boundary cohort because the average radius is initially undefined. The density of colonies within an interval is initially zero, so the denominator of equation (14) could be zero. Instead, we use:
π0(t) =
Z r1 rb
(r−rb)n(t, r)dr. (24)
which informs how the distribution of coral colonies within the interval [lb, l1(t)) changes over time. Following the methods of de Roos et al. 1997, we obtain the following dynamics for the boundary cohort:
dλ0
dt =−δ(C, rb)λ0− ∂
∂rδ(C, rb)π0+
N
X
i=1
b(C, µi)λi dπ0
dt =g(C, rb)λ0+ ∂
∂rg(C, rb)π0−δ(C, rb)π0
(25)
The dynamics of the internal (21) and boundary (24) cohorts approximate the PDE and boundary condition of (10) with a system of ODE’s. This approximation, however, will eventually fail because the interval of the boundary cohort [rb, r1(t)) would become large over time. The EBT method, therefore, uses a renumbering operation that progresses the boundary cohort into an internal cohort and creates a new, empty boundary cohort. Internal cohorts are also renumbered to make room for the new cohort; their order is retained and cohorts that contain a negligible density of individuals are removed.
and µ0(t∗) = 0) and internal cohorts will be characterized byλi(t∗) and µi(t∗) (i= 1, ..., N). Between the time t∗ and t∗ + ∆t, ODE’s of (21) and (24) are solved numerically, and the renumbering operation is performed att∗+ ∆t. The transformations are shown below, where
t∗+ ∆t+ denotes values of variables after the transformation.
λi(t∗+ ∆t+) = λi−1(t∗+ ∆t)
µi(t∗+ ∆t+) = µi−1(t∗+ ∆t)
λ1(t∗+ ∆t+) = λ0(t∗+ ∆t)
µ1(t∗+ ∆t+) = rb+
π0(t∗+ ∆t)
λ0(t∗+ ∆t)
λ0(t∗+ ∆t+) = 0
π0(t∗+ ∆t+) = 0
(26)
The first three equations perform simple renumbering of cohorts, and the equation forµ1(t∗+ ∆t+) gives the average size of individuals for a cohort of index 1. The last two equations allow for an empty boundary cohort following the transformation. The renumbering operation adds to the total number of cohorts with the addition of new boundary cohorts, but cohorts are also removed when they become negligible (density of individuals in the cohort is considerably small). In this way, the number of ODE’s to be numerically solved is changing throughout time.
2.3
Analysis
2.3.1 Equilibrium behavior
I implemented the EBT method in the programming language R to obtain population dynamics of the size-structured coral model. I identified equilibrium values for the proportion of coral cover by running simulations of the deterministic model until a stable (unchanging) value of coral cover was reached.
The formulation of the EBT method enabled me to examine the size distribution of coral density for the deterministic model at equilibrium. Environmental changes or dis-turbances could impact one size class of coral more than another. For example, Leray et al. (2012) found that Pocillopora spp. populations in Moorea had a positive relationship between coral head size and survival after an outbreak of a corallivorous sea star. They concluded that the size structure of Pocillopora populations at the time of the outbreak influenced the extent of coral mortality. For this perturbation, if the impacts of OA on demography resulted in a size structure that was skewed more toward smaller coral colonies than under ambient conditions, the outbreak would cause a greater magnitude of mortality. Conversely, other types of disturbance, such as cyclones, could have a higher degree of mor-tality on larger corals (Harriot & Fisk, 1987). Hence, it is important to determine how effects on demography influence size structure to predict how great of an impact disturbances could have on a coral population of a certain demography.
The density of coral for certain radii was approximated by assuming that boundaries of the cohorts are roughly halfway in between the average size in each cohort and that λi (density of individuals in each cohort i) is approximately equal to the density times the width of the cohort:
λi =
Z
r∈cohort i
So that the approximate density for coral colonies of radius µi is given by
n∗(µi) =
λi
width(i). (28)
To visualize this size distribution, I produced box plots based on the 10th, 25th, 50th, 75th, and 90th percentiles of coral size. Box plots can reveal differences in the degree of skewness between size distributions that might not be apparent from simply observing the distributions. To approximate these percentiles from the EBT output, I first found the cumulative sum ofλ at each cohort index and divided the cumulative sums by the total sum of λ0is.
cumulative proportionj =
Pj
i=1λi
PN
i=1λi
j = 1,2,3, ...N (29)
The quantiles are then found at the minimum cohort index where the cumulative sum divided by the total sum is greater than the percentile of interest. Let q ∈ (0,1) denote a quantile. The index of the cohort that contains that quantile is given by
jq=min{j :cumulative proportionj > quantile}. (30)
The average size of the cohort for a quantile is then
µq =µjq. (31)
2.3.2 Sensitivity analysis 2.3.2.1 Overview
stable state, then it is essential that the functional form and parameter values used are accurate. OA has been observed to affect aspects of coral demography, thereby changing rates of demography (growth, recruitment, mortality), and generating new predicted steady states of coral cover. If equilibrium coral cover is sensitive to a demographic rate that OA affects, then we could expect OA to have a significant effect on coral cover. Furthermore, a sensitivity analysis can indicate model error if the relationship between input and output variables are unexpected.
I implemented forms of local and global sensitivity analyses. Local sensitivity views how a small perturbation near a specific point influences the equilibrium proportion of coral cover. The perturbations are centered around parameter values corresponding to ambient conditions. This allows us to view how a previously unexposed coral population will be affected by changes in demographic rates via environmental changes (such as OA). A local sensitivity analysis does not, however, fully explore the parameter space. It only investigates the behavior of the model in the region of ambient conditions. Parameter values for the demographic rates can be uncertain and can depend on coral species or location. Also, local sensitivity only considers one rate at a time while all other rates are fixed, but it is likely there are interactions between demographic rates. For these reasons I also performed a sampling based global sensitivity analysis to calculate parameter sensitivities in a wider parameter space. The global sensitivity analysis can quantify the sensitivities of the equilibrium coral cover with respect to large variations in demographic rates. The range of parameter values are specified by probability densities. Local sensitivities are informative for a reference set of parameters but do not explore the full parameter space as do global sensitivities.
2.3.2.2 Local sensitivity analysis
was found at a small perturbation above and below the ambient value of the demographic rate, while all other demographic rates were kept at ambient values. The average of these steady state solutions is then taken as the approximated sensitivity of equilibrium coral cover with respect to the demographic rate at ambient condition:
∂C ∂r ≈
Cr+−Cr−
2 (32)
where r represents the value of a demographic rate, Cr+ is the equilibrium proportion of coral cover with a demographic rate equal to r+, and >0 is a small number.
To compare sensitivities between demographic variables, I consider elasticity. The demographic rates have different scales of values, so the rate of change of equilibrium coral cover with respect to one demographic rate cannot be compared to the rate of change with respect to another. Elasticity can be thought of as a proportional sensitivity and is calculated as:
E = ∂lnC
∂lnr (33)
which gives a proportional and dimensionless sensitivity that can be compared to those of other demographic variables.
2.3.2.3 Global sensitivity analysis
I performed simulations to view changes in equilibrium coral cover over a range of de-mographic rates. The dede-mographic rates were varied individually while holding the other demographic rates constant at rates associated with ambient conditions. Internal and exter-nal recruitment rates were varied simultaneously and by the same proportion (p). OA might affect recruitment through reduced settlement success. The algae on which larvae settle is hindered by OA, limiting settlement cues and thereby settlement rates. Rates of settlement would be similarly affected for both internal and external recruitment, which is why I use
ranged from 50% to 150% of the demographic rate under ambient conditions. A sampling based global sensitivity analysis was completed through a series of steps. First, I defined distributions to characterize uncertainties in the demographic rates. I used the distributions to generate samples for each parameter. Next, the model was evaluated for all parameter sets to obtain outputs (proportion of coral cover at equilibrium). Finally, I quantified the relationship between uncertain inputs and output uncertainty.
I was unable to find sufficient literature to form data-driven probability distributions of the demographic variables. I therefore chose uniform distributions for all demographic rates to avoid putting greater significance on any one value, and to account for the range of demographic values of different coral species and locations. Latin Hypercube Sampling (LHS) was chosen for the sampling technique. LHS allows input parameters to be varied simultaneously, but samples those parameters in a structured and efficient way, so that each value of each parameter is used only once (Blower Dowlatabad, 1994). LHS has also been shown to be more efficient than simple random sampling (Stein, 1987; Handcock, 1989). Par-tial rank correlation coefficient (PRCC) was used to measure the strength of the correlation between the input and output variables, while allowing for a non-linear relationship between them. With LHS, input parameters are often interdependent, and PRCC can determine the relationship between an input and output variable while keeping all other input parameters constant (Conover, 1980). In this way, PRCC examines the statistical relationship between an input parameter (demographic rate) and the outcome variable while controlling for their mutual dependence on the other input parameters.
After uniform distributions were chosen for all demographic parameters, the number of simulations to run, S, was chosen to be 100. This satisfies the inequality (S > (4/3)K), where K is the number of uncertain variables. The inequality has been empirically estab-lished for the lower limit of the number of simulations to be run (Blower Dowlatabadi, 1994; McKay, Conover Beckman, 1979). Next, I divided the range of each parameter value intoS
interval of each variable was sampled once, and a SxK matrix was created (in our case a 100x3 matrix). The sampling indices in the matrix were then replaced by parameter values by applying their PDFs. Finally, 100 runs of the model were generated to give values of coral cover at equilibrium for all parameter sets. The PRCCs were determined to identify which input parameters are important to the variability of the outcome variable (equilibrium proportion of coral cover).
2.3.3 Stochastic analysis
Stochasticity was first added into the model via the mortality rate,µ(C, r). The change in mortality rate reflects environmental fluctuations and is only one of many methods that could be used to add stochasticity to the model. Mortality was chosen as a source of stochas-ticity because Moorea has experienced severe, episodic mortality from disturbances of coral bleaching (Hoegh-Guldberg 1999; Penin et al., 2007), predatory starfish outbreaks, and at least one cyclone (Adjeroud et al., 2009). Stochasticity was then included in the model through the external recruitment rate. The rate at which external recruits arrive,s, depends on the dispersal ability of other regional coral populations. Dispersal can be highly variable due to mass spawning events combined with seasonal wind and current variability (Thomp-son et al, 2018).
different demographic forms will have unique equilibrium means, necessitating the unitless measurement of coefficient of variation for comparison. The coefficient of variation is the ratio of the standard deviation to the mean, and will be considered as an indication for reef fragility.
I ran simulations for a changing per capita mortality rate (δ), followed by simulations for a variable external recruitment rate (s). Rates were altered at time intervals of ∆t, coin-ciding with the renumbering operation. I set ∆tto be 0.5, so thatδandsare re-defined every six months. The values of δ and s were drawn from a gamma distribution for varying mean values and standard deviations. The coefficient of variation was observed for the proportion of coral cover at equilibrium. I accomplished this by running the model simulation for a long period of time (2,000 years), and extracting values of the proportion of coral cover at time intervals (every 10 years) after dynamics reach a state of quasi-equilibrium. The term quasi-equilibrium is used because the stochasticity in the system does not allow for a true equilibrium to be reached; however, the proportion of coral cover will reach a state where values remain close to some average proportion of coral cover. Simulations were run for demographic rates of varying values, and different levels of standard deviation. Coefficient of variation was calculated as:
CV = SD
µC
. (34)
Where SD is the standard deviation of the extracted values of the proportion of coral cover at equilibrium, and µC is the mean of those values.
Table 1. State variables, units, and dimensions.
state variable dimensions units
n(t, r) individuals length−1 area−1 individuals m−1 m−2
Table 2. Model inputs and trial values.
sym. Parameter definition Value dimensions units
rb radius of new coral colony 0.002 length m
rmax max. radius of a coral colony 1.0 length m
l max. coral growth rate 0.029 length time−1 m yr−1
δ per capita coral mortality rate 0.06 time−1 yr−1
β coral fecundity per surface area 0.3 indiv. area−1 time−1§ indiv. m−2 yr−1
s external recrutiment 0.01 indiv. area−1 time−1§ indiv. m−2 yr−1
§ Note that the areas inβandsare different. βis the rate at which new larvae are produced
per unit surface area of living coral tissue. s is the rate at which external larvae recruit per unit area of substrate.
Table 3. Escalator Boxcar Train Method for Coral Population Model
Continuous-time dynamics dλ0
dt =−δ(C, rb)λ0− ∂
∂rδ(C, rb)π0+
PN
i=1b(C, µi)λi
dπ0
dt =g(C, rb)λ0+ ∂
∂rg(C, rb)π0−δ(C, rb)π0
dλi
dt =−δ(C, µi)λi
dµi
dt =g(C, µi)
Transformations at end
of cohort cycle λ1(t∗+ ∆t+) =λ0(t∗+ ∆t)
µ1(t∗+ ∆t+) = rb+π0(t ∗+∆t) λ0(t∗+∆t)
λ0(t∗+ ∆t+) = 0
π0(t∗+ ∆t+) = 0
λi(t∗+ ∆t+) =λi−1(t∗+ ∆t)
3
Results
Simulations for the deterministic model allowed us to find equilibrium values for the pro-portion of coral cover. Equilibrium values for a range of demographic rates can be seen in Figure 1. The equilibrium value of the proportion of coral cover is seen to increase with an increasing rate of maximum linear extension of the coral. The rate of increase is greater for lower maximum linear extension rates and then decelerates for greater values of maxi-mum linear extension rate. The equilibrium values decrease with an increasing per capita mortality rate and have a more consistent rate of change than those associated with growth and recruitment. Finally, the equilibrium values increase with increasing larval settlement success. The rate of increase is greater for lower levels of recruitment rates and decreases as recruitment rates increase.
Elasticities of the demographic rates were computed for local sensitivities at ambient conditions. Subsequently, global sensitivities were observed for the demographic rates with PRCC’s (Table 4). The signs for elasticity and PRCC values are as expected (positive for recruitment and growth, negative for mortality). Both local and global sensitivity mea-surements reveal mortality to have the highest absolute value and recruitment to have the lowest absolute value. The difference between demographic inputs is greater for elasticity than PRCC. PRCC values are high for all demographic variables, suggesting all demographic rates are highly correlated with the steady state proportion of coral cover.
recruit-ment rate, on the other hand, causes a notable decrease in the average coral colony size. The box-and-whisker plot also confirms an increasing average colony size with increasing growth rate.
Looking at the stochastic model, coefficient of variations are observed for varying demo-graphic rates and standard deviations. Figure 4 gives an example of the solutions from which the coefficient of variation is found. Figures 5-7 show coefficient of variation for stochasticity added through mortality, and Figures 8-10 represent stochasticity added through external recruitment. Variable mortality and external recruitment show similar trends for changing demographic rates. For both modes of stochasticity, the coefficient of variation decreases with increasing rates of growth and recruitment. Increasing mortality rate, on the other hand, results in an overall increase in the coefficient of variation. Figures 5 and 8 reveal CV to be more sensitive to changes in growth rate when growth rate is lower than the ambient value. Figures 6 and 9 show CV to be more sensitive to changes in per capita mortality rate when per capita mortality rate is higher than the ambient value. Additionally, the sensitivity of CV to changes in settlement success is seen to be greater for settlement success below ambient conditions (Figures 7 and 10). For all demographic variables, and both forms of stochasticity, the magnitude of environmental variation (standard deviation) does not have a substantial effect on the sensitivity of the coefficient of variation.
Table 4. Sensitivity Analyses with Elasticity and PRCC. Growth Mortality Recruitment Elasticity 0.7155 -1.2332 0.5175
0.015 0.025 0.035 0.1 0.2 0.3 0.4 0.5
Maximum Linear Extension Rate
Propor
tion of Cor
al Co
ver at Equilibr
ium
0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Per Capita Mortality Rate
Propor
tion of Cor
al Co
ver at Equilibr
ium
0.6 0.8 1.0 1.2 1.4
0.20 0.25 0.30 0.35 0.40 0.45 p Propor
tion of Cor
al Co
ver at Equilibr
ium
0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 Growth Rate Size Density −50% −25% la +25% +50%
0.0 0.2 0.4 0.6 0.8 1.0
0 2 4 6 8 Mortality Rate Size Density −50% −25% δa +25% +50%
0.0 0.2 0.4 0.6 0.8 1.0
0 2 4 6 8 Recruitment Rate Size Density −50% −25% pa +25% +50%
0.1
0.2
0.3
0.4
0.5
Growth Rate
Siz
e
−50% −25% la +25% +50%
0.1
0.2
0.3
0.4
0.5
Mortality Rate
Siz
e
−50% −25% δa +25% +50%
0.1
0.2
0.3
0.4
0.5
0.6
Recruitment Rate
Siz
e
−50% −25% pa +25% +50%
0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time (years)
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Proportion of Coral Cover
Proportion of Coral Cover Over Time - Stochastic Mortality
Figure 4. Proportion of coral cover is shown over time for a stochastic per capita morality rate. Coefficient of variation is found from values of coral cover at quasi-equilibrium (values occurring after 300 years).
0.015 0.02 0.025 0.03 0.035 0.04
Max. Linear Extension Rate 0
0.05 0.1 0.15 0.2
Coefficient of Variation
CV for Varying Growth Rate and SD - Stochastic Morality
SD = 0.01
SD = 0.02
SD = 0.03
0.03 0.04 0.05 0.06 0.07 0.08 0.09
Per Capita Mortality Rate
0 0.02 0.04 0.06 0.08 0.1 0.12
Coefficient of Variation
CV for Varying Mortality Rates and SD - Stochastic Morality
SD = 0.01 SD = 0.02 SD = 0.03
Figure 6. Coefficient of variation for different mean rates of per capita mortality are shown for a per capita mortality rate with varying degrees of associated standard deviation.
0.5 1 1.5
p 0.02
0.04 0.06 0.08 0.1 0.12 0.14
Coefficient of Variation
CV for Varying Recruitment Rates and SD - Stochastic Morality
SD = 0.01 SD = 0.02 SD = 0.03
0.015 0.02 0.025 0.03 0.035 0.04
Max. Linear Extension Rate
0 0.01 0.02 0.03 0.04 0.05 0.06
Coefficient of Variation
CV for Varying Growth Rate and SD - Stochastic External Recruitment
SD = 0.01
SD = 0.02
SD = 0.03
Figure 8. Coefficient of variation for different rates of maximum linear extension are shown for an external recruitment rate with varying degrees of associated standard deviation.
0.03 0.04 0.05 0.06 0.07 0.08 0.09
Per Capita Mortality Rate
0 0.02 0.04 0.06 0.08 0.1
Coefficient of Variation
CV for Varying Mortality Rate and SD - Stochastic External Recruitment
SD = 0.01 SD = 0.02 SD = 0.03
0.5 1 1.5 p
0 0.02 0.04 0.06 0.08 0.1
Coefficient of Variation
CV for Varying Recruitment Rate and SD - Stochastic External Recruitment
SD = 0.01 SD = 0.02 SD = 0.03
4
Discussion
In this study, I hoped to understand how potential changes in coral colony demog-raphy due to ocean acidification scale up to impact the dynamics of an entire population. I was able to accomplish this by creating a physiologically structured population model, implementing the Escalator Boxcar Train numerical method, and analyzing solutions from numerical simulations. Analyses were conducted to view the relationship between demo-graphic processes and steady state size distributions, appreciate the relative importance of each demographic variable to coral cover, and recognize how potential demographic effects of OA can impact reef resilience.
Local and global sensitivity analyses revealed the importance of per capita mortal-ity. A change in per capita mortality rate will have the greatest impact on coral cover for both small perturbations from ambient conditions as well as large perturbations. If OA has the greatest impact on the mortality rate of individuals, the consequences of OA on coral cover will be particularly sizable. Future experiments should put emphasis on identifying accurate values for mortality. However, high PRCC values for growth and recruitment sug-gest a strong relationship between coral cover and all demographic rates when considering a large parameter space. Location dependent sensitivity can be seen in Figure 1; there is a non-linearity of the proportion of coral cover for changing growth rates and recruitment rates. Coral cover becomes more sensitive to changes in recruitment and growth rate as the rates decrease. If OA decreases growth and recruitment rates, we could see disproportional changes in coral cover as the demographic rates decrease. A chronic disturbance (OA) can make a coral population more sensitive to an acute disturbance.
of OA on demography are accurate, OA could have contrasting effects on the steady state size distribution of coral. The overall impact on size distribution would then depend on which demographic process has the greatest influence on size distribution. However, a study by Mollica et al. (2018) has shown that OA does not hinder the linear extension rate of coral, but instead affects the density component of growth. If this is the case, a change in recruitment rate from OA effects would have the greatest impact on the shape of the steady state size distribution, resulting in a higher density of larger corals. Figure 3 shows the largest jump in percentile values between the lowest and second lowest inputs of growth and recruitment rates. This means the relative change in coral size distribution, due to a change in growth or recruitment rate from OA, would depend on the current values of the demo-graphic rates. Changes in the size structure of a coral population could have implications for the magnitude of impact a disturbance will have on the population. It is therefore impor-tant to understand the relationship between changes in demography and size distribution to better predict how a disturbance can affect a coral population.
size distributions, and coefficient of variation measurements, one parameter is varied while all other parameters are maintained at values of ambient condition. Demographic rates can be highly uncertain and vary between species and location. This study is limited in that it focuses on one set of parameter values and does not consider concurrent changes in param-eter values for size distribution and coefficient of variation effects.
REFERENCES
Adjeroud, M., Michonneau, F., Edmunds, P., Chancerelle, Y., de Loma, T., Penin, L., . . . Galzin, R. (2009). Recurrent disturbances, recovery trajectories, and resilience of coral assemblages on a south central pacific reef. Coral Reefs, 28(3), 775-780. doi:10.1007/s00338-009-0515-7
Albright, R., Langdon, C. (2011). Ocean acidification impacts multiple early life history processes of the caribbean coral porites astreoides. Global Change Biology, 17(7), 2478-2487. doi:10.1111/j.1365-2486.2011.02404.x
Albright, R., Takeshita, Y., Koweek, D., Ninokawa, A., Wolfe, K., Rivlin, T., . . . Caldeira, K. (2018). Carbon dioxide addition to coral reef waters suppresses net community calcification. Nature, 555(7697), 516-516. doi:10.1038/nature25968
Anderson, K. D., Heron, S. F., Pratchett, M. S. (2015). Species-specific declines in the linear extension of branching corals at a subtropical reef, lord howe island. Coral Reefs, 34(2), 479-490. doi:10.1007/s00338-014-1251-1
Aronson RB, Precht WF, Macintyre IG, Murdoch TJT (2000) Coral bleach-out in Belize. Nature 405: 36.
Artzy-Randrup, Y., Olinky, R., Stone, L. (2007). Size-structured demographic models of coral populations. Journal of Theoretical Biology, 245(3), 482-497. doi:10.1016/j.jtbi.2006.10.019
Baker, A. C., Glynn, P. W., Riegl, B. (2008). Climate change and coral reef bleaching: An ecological assessment of long-term impacts, recovery trends and future outlook. Estuarine, Coastal and Shelf Science, 80(4), 435-471. doi:10.1016/j.ecss.2008.09.003 Blower, S.M. and Dowlatabadi, H. (1994) Sensitivity and Uncertainty Analysis of
Bruno, J. F., Selig, E. R. (2007). Regional Decline of Coral Cover in the Indo-Pacific: Timing, Extent, and Subregional Comparisons. PLoS ONE, 2(8). doi:10.1371/journal.pone.0000711
Caldeira K, Wickett ME. 2003. Anthropogenic carbon and ocean pH. Nature 425:365 Caldeira K, Wickett ME. 2005. Ocean model predictions of chemistry changes from carbon
dioxide emissions to the atmosphere and ocean. J. Geophys. Res Oceans 110:C09S4 Caswell, H. 1997. Methods of matrix population analysis. Pages 19âĂŞ58 in S. Tuljapurkar
and H. Caswell, editors. Structured-population models in marine, terrestrial, and fresh-water systems. Chapman and Hall, New York, New York, USA.
Chan, N. C. S., Connolly, S. R. (2013). Sensitivity of coral calcification to ocean acidification: A meta-analysis. Global Change Biology, 19(1), 282-290. doi:10.1111/gcb.12011 Cheal, A., Macneil, M., Emslie, M., Sweatman, H. (2017). The threat to coral reefs from
more intense cyclones under climate change. Global Change Biology, 23(4), 1511-1524. doi:10.1111/gcb.13593
Conover, W.J. (1980). Practical Nonparametric statistics, 2nd ed., John Wiley Sons Inc., New York, NY.
C`ot´e, I. M., Gill, J. A., Gardner, T. A., Watkinson, A. R. (2005). Measuring coral reef decline through meta-analyses. Philosophical Transactions of the Royal Society B: Biological Sciences, 360(1454), 385-395. doi:10.1098/rstb.2004.1591
De Roos, A. M. 1997. A gentle introduction to physiologically structured population models. - In: Tuljapurkar, S. and Caswell, H. (eds), Structured-population models in marine, terrestrial, and freshwater systems. Chapman Hall, pp. 119-204.
Done, T. J., Dayton, P. K., Dayton, A. E., Steger, R. (1991). Regional and local variability in recovery of shallow coral communities: Moorea, French Polynesia and central Great Barrier Reef. Coral Reefs, 9(4), 183-192. doi:10.1007/bf00290420
Doney SC, Schimel DS. 2007. Carbon and climate system coupling on timescales from the Precambrian to the Anthropocene. Annu. Rev. Environ. Resour. 32:31-66
Edmunds, P. J., Burgess, S. C. (2018;2017;). Colony size and turbulent flow speed modulate the calcification response of the coral pocillopora verrucosa to temperature. Marine Biology, 165(1), 1-12. doi:10.1007/s00227-017-3257-z
Edmunds, P. J., Burgess, S. C. (2016). Size-dependent physiological responses of the branch-ing coral pocillopora verrucosa to elevated temperature and PCO2. The Journal of Experimental Biology, 219(Pt 24), 3896
Edmunds, P., Carpenter, R., Comeau, S. (2013). UNDERSTANDING THE THREATS OF OCEAN ACIDIFICATION TO CORAL REEFS. Oceanography, 26(3), 149-152. Retrieved from http://www.jstor.org/stable/24862076
Edmunds PJ, Elahi R (2007) The demographics of a 15-year decline in cover of the Caribbean reef coral Montastraea annularis. Ecological Monographs 77: 3-18.
Evensen, N. R., Edmunds, P. J. (2016). Interactive effects of ocean acidification and neigh-boring corals on the growth of pocillopora verrucosa. Marine Biology, 163(7), 1-11. doi:10.1007/s00227-016-2921-z
Feely RA, Orr J, Fabry VJ, Kleypas JA, Sabine CL, Landgon C. 2008. Present and future changes in seawater chemistry due to ocean acidification. In AGU Monograph on the Science and Technology of Carbon Sequestration. ed. B.J. McPherson, E.T. Sundquist. Am. Geophys. Union. In press
Griffin, J. N., Schrack, E. C., Lewis, K., Baums, I. B., Soomdat, N., Silliman, B. R. (2015). Density-dependent effects on initial growth of a branching coral under restoration. Restoration Ecology, 23(3), 197-200. doi:10.1111/rec.12173
Han, X., Adam, T. C., Schmitt, R. J., Brooks, A. J., Holbrook, S. J. (2016). Response of herbivore functional groups to sequential perturbations in moorea, french polynesia. Coral Reefs, 35(3), 999-1009. doi:10.1007/s00338-016-1423-2
Handcock, M.S. (1989). Latin Hypercube Sampling to improve the efficiency of Monte Carlo Simulations: Theory and Implementation in ASTAP, IBM Research Division, TJ Wat-son Research Center
Harriott, V.J. and Fisk, D.A., 1986. The effects of cyclone Win ifred on corals at Green Island Reef. In: Dutton, I. M. (ed.), The Offshore Effects of Cyclone Winifred. GBRMPA Work shop Series 7, pp. 52-58.
Hoegh-Guldberg O (1999) Climate change, coral bleaching and the future of the world‘s coral reefs. Mar Freshw Res 50:839-866 Hughes, T. P. (1994). Catastrophes, phase shifts, and large-scale degradation of a caribbean coral reef. Science, 265(5178), 1547-1551. doi:10.1126/science.265.5178.1547
Hughes, T. P., Baird, A. H., Bellwood, D. R., Card, M., Connolly, S. R., Folke, C., . . . Roughgarden, J. (2003). Climate change, human impacts, and the resilience of coral reefs. Science, 301(5635), 929-933. doi:10.1126/science.1085046
Hughes, T. P. (1994). Catastrophes, phase shifts, and large-scale degradation of a caribbean coral reef. Science, 265(5178), 1547-1551. doi:10.1126/science.265.5178.1547
Hughes, T. P., Connell, J. H. (1987). Population dynamics based on size or age? A reef-coral analysis. The American Naturalist, 129(6), 818-829. doi:10.1086/284677
Karlson, R. H., Hughes, T. P., Karlson, S. R. (1996). Density-dependent dynamics of soft coral aggregations: The significance of clonal growth and form. Ecology, 77(5), 1592-1599. doi:10.2307/2265554
Kojis, B. L., and N. J. Quinn. 1985. Puberty in Goniastrea favulus: age or size limited? Pages 289-293 in International Association of Biological Oceanography 1985, Vol. 4. Langdon, C., Takahashi, T., Sweeney, C., Chipman, D., Goddard, J., Marubini, F., . . .
Atkinson, M. J. (2000). Effect of calcium carbonate saturation state on the calcifica-tion rate of an experimental coral reef. Global Biogeochemical Cycles, 14(2), 639-654. doi:10.1029/1999GB001195
Leray, M., Beraud, M., Anker, A., Chancerelle, Y., Mills, S. (2012). Acan-thaster planci outbreak: Decline in coral health, coral size structure modifica-tion and consequences for obligate decapod assemblages. Plos One, 7(4), e35456. doi:10.1371/journal.pone.0035456
Leslie, P. H. (1948). Some Further Notes on the Use of Matrices in Population Mathematics. Biometrika, 35(3/4), 213. doi:10.2307/2332342
Lough, J. M., Anderson, K. D., Hughes, T. P. (2018). Increasing thermal stress for tropical coral reefs: 1871-2017. Scientific Reports, 8(1), 6079.
Madin, J. S., Baird, A. H., Dornelas, M., Connolly, S. R., Cornell, H. (2014). Mechanical vulnerability explains size-dependent mortality of reef corals. Ecology Letters, 17(8), 1008-1015. doi:10.1111/ele.12306
Manzello, D. P. (2010). Coral growth with thermal stress and ocean acidification: Lessons from the eastern tropical pacific. Coral Reefs, 29(3), 749-758. doi:10.1007/s00338-010-0623-4
Moberg, F., Folke, C. (1999). Ecological goods and services of coral reef ecosystems. Eco-logical Economics, 29(2), 215-233. doi:10.1016/S0921-8009(99)00009-9
Mollica, N., Guo, W., Cohen, A., Huang, K., Foster, G., Donald, H., Solow, A. (2018). Ocean acidification affects coral growth by reducing skeletal density. Proceedings of the National Academy of Sciences of the United States of America, 115(8), 1754-1759. doi:10.1073/pnas.1712806115
MUKO, S., SAKAI, K., IWASA, Y. (2001). Dynamics of marine sessile organisms with space-limited growth and recruitment: Application to corals. Journal of Theoretical Biology, 210(1), 67-80. doi:10.1006/jtbi.2001.2298
Mumby, P. J., Edwards, H. J., Hastings, A. (2007). Thresholds and the resilience of caribbean coral reefs. Nature, 450(7166), 98-101. doi:10.1038/nature06252
Murphy, L.F., 1983. A nonlinear growth mechanism in size structured population dynamics. J. Theor. Biol. 104, 493-506.
Nadler, L. E., McNeill, D. C., Alwany, M. A., Bailey, D. M. (2014). Effect of habitat characteristics on the distribution and abundance of damselfish within a red sea reef. Environmental Biology of Fishes, 97(11), 1265-1277. doi:10.1007/s10641-013-0212-9 Orr JC, Fabry VJ, Aumont O, Bopp L, Doney SC, et al. 2005. Anthropogenic ocean
acid-ification over the twenty-first century and its impact on calcifying organisms. Nature 437:681-86
Pandolfi, J. M., Tudhope, A. W., Burr, G., Chappell, J., Edinger, E., Frey, M., . . . Newton, A. (2006). Mass mortality following disturbance in holocene coral reefs from papua new guinea. Geology, 34(11), 949. doi:10.1130/G22814A.1
Pascual, M., Caswell, H., 1989. The dynamics of a size-classified benthic population with reproductive subsidy. Theor. Popul. Biol. 39 (2), 129âĂŞ147.
Penin L, Adjeroud M, Schrimm M, Lenihan HS (2007) High spatial variability in coral bleach-ing around Moorea (French Polynesia): patterns across locations and water depths. Comptes Rendus Biol 330:171-181
Pereira, P., Munday, P. (2016). Coral colony size and structure as determinants of habitat use and fitness of coral-dwelling fishes. Marine Ecology Progress Series, 553, 163-172. doi:10.3354/meps11745
Roche, R. C., Williams, G. J., Turner, J. R. (2018). Towards developing a mechanistic understanding of coral reef resilience to thermal stress across multiple scales. Current Climate Change Reports, 4, 51-64. doi:10.1007/s40641-018-0087-0
Sabine CL, Feely RA. 2007. The oceanic sink for carbon dioxide. In Greenhouse Gas Sinks, ed. D Reay, N Hewitt, J Grace, K Smith, pp. 31-49. Oxfordshire: CABI Publishing Sabine CL, Feely RA, Gruber N, Key RM, Lee K, et al. 2004. The oceanic sink for
anthro-pogenic CO2. Science 305:367-71
Schonberg, C., Fang, J., Carreiro-Silva, M., Tribollet, A., Wisshak, M., Handling editor: Joanna Norkko. (2017). Bioerosion: The other ocean acidification problem. Ices Journal of Marine Science, 74(4), 895-925. doi:10.1093/icesjms/fsw254
Virginia G. W. Schutte, Selig, E. R., Bruno, J. F. (2010). Regional spatio-temporal trends in caribbean coral reef benthic communities. Marine Ecology Progress Series, 402, 115-122. doi:10.3354/meps08438
Sinko, W.J., Streifer, W., 1967. A new model for age-size structure of a population. Ecology 48 (6), 910-918.
Report of the Intergovernmental Panel on Climate Change. New York: Cambridge Univ. Press
Stein, M. (1987). Large sample properties of simulations using Latin Hypercube Sampling. Technometrics 29, 143-151.
Svensson, C.J., Jenkins, S.R., Hawkins, S.J., A berg, P., 2005. Population resistance to climate change: modelling the effects of low recruitment in open populations. Oecologia 142, 117-126.
Tanner, J. E. (1997). The effects of density on the zoanthid palythoa caesia. Journal of Animal Ecology, 66(6), 793-810. doi:10.2307/5996
Thompson, D. M., Kleypas, J., Castruccio, F., Curchitser, E. N., Pinsky, M. L., JÃűns-son, B., WatJÃűns-son, J. R. (2018). Variability in oceanographic barriers to coral larval dispersal: Do currents shape biodiversity? Progress in Oceanography, 165, 110-122. doi:10.1016/j.pocean.2018.05.007
VanSickle, J., 1977. Analysis of a distributed-parameter population modelbased on physio-logical age. J. Theor. Biol. 64, 571-586.
Mark J. A. Vermeij, Sandin, S. A. (2008). Density-dependent settlement and mortality structure the earliest life phases of a coral population. Ecology, 89(7), 1994-2004. doi:10.1890/07-1296.1
Wilkinson C, editor. (2008) Status of coral reefs of the world: 2008. Townsville, Australia: Australian Institute of Marine Science.