1. Modelling coral’s interactions with other species on a coral reef.
Talya Mellor: 1336294
1 Introduction
Coral reefs are deteriorating at an alarming rate with 70% of coral reefs now classed as threatened
[1]. It is possible that mathematical models could aid reef Managers in their effort to protect these
habitats. Sometimes known as rainforests of the sea, coral reefs cover less than 1% of the ocean floor
[2]. Despite this small percentage they are thought to accommodate between 1 and 9 million different
species [3] making them some of the most biologically diverse habitats on earth. Alongside the threat
to biodiversity, reef decline has significant implications for local communities that rely on them for
food, income and coastal defences. A variety of factors are known to cause reef decline. These include:
• Overfishing of reef grazers allowing algae to overgrow the reefs.[2]
• Increase’s in extreme weather, such as hurricanes which can cause lasting structural damage to
the reef.
• The breakdown of corals relationship with the algae leaving it unable to gain the nutrients it
needs.The most widely researched cause being increases in sea temperature. [4].
• Increases in the sea’s uptake of C02 decreases the seas pH. This can cause the dissolution of
calcium carbonate which is the primary building block of coral reefs [2] [4].
Coral reefs are made up of hard and soft corals, sponges and algae [3]. Hard corals are known
as reef builders. They are responsible for developing the multicoloured limestone structures that
categorise a coral reef. Soft transparent coral polyps utilise calcium carbonate in seawater to create a
hard limestone exoskeleton to protect their bodies[3] . When the polyps die they decompose leaving
this limestone rock behind for new polyps to build on [2]. Thus only the outer layer of coral is alive
[5]. Corals attribute their bright colours to an algae named zooxanthella which live inside the polyps’
bodies. This algae provides the coral with nutrients and in return the coral offers the algae protection
[2].
Overfishing is of particular interest as it is directly caused by human behaviour. As in any ecosys-
tem coral competes with other species on the reef for space, nutrition and light. One of these com-
petitors is an algae known as macroalgae. It is usually kept at bay by species of fish that graze the
reef. Grazed macroalgae can provide a good environment for new coral polyps to develop. However
overfishing of grazing fish can allow the macroalgae to overgrow the reef causing coral death [6].
In this project we look to understand how system’s of ordinary differential equations have been
developed to model this relationship between grazers, corals, and macroalgae. For the most part this
project is based around a model described by Mumbey et al. in [9]. We look to understand how
the model is established and consider how analytical methods can be used to gain an understanding
of the behaviour of the system. The later sections of this project go on to consider the work done
in subsequent papers analysing and extending the model. We then go on to introduce the basic
concepts of delay differential equations and consider how scientists have used them to analyse whether
a change in fishing strategy can increase reef resilience to alternative stresses, such as a natural
disaster. Throughout this analysis we are particularly interested in whether shifts in reef conditions
are reversible [7].
1

2. Figure 1: Left: A healthy coral reef [2]. Right: A coral reef overrun with macroalgae [8].
2 Modelling corals relationship with macroalgae
The model created by Mumby et al. considers the interactions between coral and algae on coral reefs
in the Caribbean. Caribbean reefs are of particular interest because they are seen as less resilient
than other reefs due to their low species diversity. The model considers the interactions between three
different groups; coral, algal turf and macroalgae. Algal turf is a thin layer of algae that remains on
the reef after macroalgae has been grazed. It provides good conditions for coral growth but if left un-
grazed will grow back into macroalgae. We begin by defining important parameters and assumptions.
For simplicity the reef is modelled as a reasonably flat structure enclosed in a limited area of seabed.
Thus the area covered by coral, algal turf, or macroalgae, is perceived as if from a birds eye view
and taken as a fraction of the available space. The parameters defined below are kept consistent
throughout this project:
• C: Coral cover: The of area covered by coral as a fraction of the available seabed.
• T: Algal turf cover: The area of seabed covered by a algal turf given as a fraction of the available
seabed.
• M: Macroalgae: The area of macroalgae cover as a fraction of the available seabed.
• r: The rate that corals colonise algal turfs.
• d: The natural mortality rate of corals.
• a: The rate that macroalgae overgrows corals.
• q: The rate that macroalgae spreads over algal turfs.
• g: The grazing rate of fish.
Here the lower-case parameters are treated as constants which can be estimated from historical
data, whilst the parameters in capitals are variable’s. It is important to note that in this model
grazing (g) is taken as a constant. In reality this is unlikly to be the case as grazing is dependent on
the population of fish. We assume a triangular relationship between coral, macroalgae, and algal turf
as seen in Figure 3. Thus, algal turf grows over dead coral and coral grows over algal turf. Macroalgae
grows over both algal turf and coral whilst grazing reduces it back to algal turf. Algal turf grows over
dead coral and is produced via grazing. It is assumed that there is no time delay to take account of
growth. This means, for example, that whilst in a real world environment dead coral might remain
undisturbed for a period of time, in this model it is immediately colonised by algal turf. The final
assumption is that C+T +M = 1 so that all available space is covered at all times [9]. This assumption
reflects the understanding that an increase in macroalgae leads to a decrease in coral and vice versa.
Note, also that due to the assumed triangular relationship this system is dependent on grazing in this
model graving (g) is taken as a constant.
2

3. Figure 2: A diagram to show the relationship between coral, macroalgae and algal turf.
Mumby et. al builds up differential equations for the changes in cover of the three variables; M,
C, T as follows. The change in macroalgae cover over time dM
dt is given by
dM
dt
= aMC
Macroalgae growth over coral
−
gM
M + T
Macroalgae removed due to grazing
+ qMT.
Macroalgae growth over turf
The change in coral cover over time dC
dt is given by
dC
dt
= rTC
Coral growth over turf
− dC
Coral mortality
− aMC.
Macroalge growth over corals
The change in algal turf cover over time dT
dt is given by
dT
dt
=
gM
M + T
Turf added due to grazing
− qMT
Macroalgae growth over un-grazed turf
− rTC
turfs colonized by coral
+ dC.
Coral mortality
The assumption that no area of seabed is left uncovered means that T = 1 − C − M and thus dT
dt
can be found in terms of dC
dt and dM
dt [6]. Therefore we only require two of the three equations stated
above to describe changes in C, M and T. It makes sense to use the equations for C and M since we
are interested the relationship between grazers, corals, and macroalgae. Choosing these two equations
also reduces the number of terms, simplifying the analysis. Therefore we have the model:
dM
dt
= aMC −
gM
M + T
+ qMT, (1)
dC
dt
= rTC − dC − aMC. (2)
3 Analytical methods for Solving this model
Li et al. provides a detailed global analysis of the above ODE model in [6]. In this project we consider
the methodology used to interpret a system of ODE’s of this type. Dynamical Systems are a ‘means
of describing how one state develops into another state over a course of time’ [10]. The System of
ordinary differential equations above considers how a reef might shift from a coral dominated state
to a macroalgae dominated state and back again over a course of time. Therefore it can be described
as a Dynamical system. It is clear to see that this system is non-linear, demonstrated by the terms
3

4. where M, C, and T are multiplied together, and autonomous, because the system does not depend on
time [11].
We wish to eliminate T by substituting T = 1 − C − M from above. In order to make the system
more familiar let M=x and C=y [6]. On rearranging the system becomes:
dx
dt
= x q − qx + (a − q)y −
g
1 − y
, (3)
dy
dt
= y(r − d − (a + r)x − ry). (4)
We begin by noting that there is a singularity at y=1. In order to utilise much of the theory for
analysing non-linear dynamical systems our equations must be analytic at all points. Therefore define:
Ω = {(x, y) : 0 < x, 0 < y, x + y < 1} as the region of interest [6]. We are interested only in the
positive quadrant because it is not possible to have a negative population size [11].
Our task now is to analyse and understand what this system shows about corals relationship with
macroalgae. Arguably the best way to do this is to find solutions to our problem as flow lines of an
associated vector field V:
V =
F(x, y)
G(x, y)
=
x(q − qx + (a − q)y − g
1−y )
y(r − d − (a + r)x − ry)
.
We begin by considering the equilibrium points of the system. These are points in the field where the
solution remains unchanged and we have ˙x = 0 and ˙y = 0 [11]. A simple calculation establishes that
there are three boundary equilibria at (0,0), (0,r−d
r ), and (q−g
q ,0) and an additional interior equilibrium
point where 0 = (q − qx + (a − q)y − g
1−y ) and 0 = (r − d − (a + r)x − ry) [6]. These two equations
are difficult to solve for x and y. Therefore we will classify the boundary equilibria only and look to
gain greater incite of the behaviour at this internal equilibria via analysis of the nullclines
It is possible to establish the behaviour of our system in a region surrounding the three boundary
equilibrium points using the method of linearisation.
Theorem 1. For a simple analytic function a taylor series expansion is given by
˙y = ˙x = f(y + x0) = f(x) + Df(x)y + o(|y|2
), where
D =
Fx(x, y) Fy(x, y)
Gx(x, y) Gy(x, y)
is the Jacobian. When a taylor series expansion of a function exists at all points the linearization of
a system F at x0 is given by L = DF(x0) [11].
Calculating the Jacobian for the system gives:
DV (x, y) =
q − qx + (a − q)y − g
1−y − qx x(a − q + −g
(1−y)2 )
y(−(a − r)) r − d − (a + r)x − ry − ry
.
and so
DV (0, 0) =
q − g 0
0 r − d
has eigenvalues λ1 = q − g and λ2 = r − d.
DV 0,
r − d
r
=
a − d
r (a − q) − g r
d 0
−(r−d)(a+r)
r d − r
4

5. has eigenvalues λ1 = a − d
r (a − q) − g r
d and λ2 = d − r and finally,
DV
q − g
q
, 0 =
g − q (1 − g
q )(a − q − g)
0 −(d + a) + g
q (a + r)
has eigenvalues λ1 = g − q and λ2 = −(d + a) + g
q (a + r).
To determine the behaviour of the flow near each of these equilibrium points we need to determine
the sign of the eigenvalue’s. We could take a case by case approach but this would be time consuming
due to the number of unknown parameters. Fortunately the model is based on a real-world environment
and observational studies can provide insight into what these parameters could be. Using a variety of
different studies, highlighted in [7], it is estimated that a=0.1, q=0.8, r=1, d=0.44 and g is taken to
be variable in the region 0.1 ≤ g < 0.8. This enables us to assume that generally a < d < q < r < 2q
[6].
Therefore we have that at V(0,0), both values of λ are greater than 0 and for all values of g.
Therefore the eigenvalues are real, different and positive which suggests there is an unstable node for
the linearised system at (0,0).
At V (0, r−d
r ), λ2 < 0 at all points because d < r. The sign of λ1 is dependent on the value of g.
We know that q > a so a − ar
d (a − q) > 0. Now we have two cases:
• Case A: If g is small (i.e g=0.1) then ar
d (a − q) > g r
d . This implies that λ1 > 0 and λ2 < 0. The
eigenvalues are real and of different sign. This implies a saddle point for the linearised system.
• Case B: If g is large (i.e g=0.4) then ar
d(a − q) < g r
d so λ1 < 0. Therefore the eigenvalues are
both real, different and negative; λ1 < 0 and λ2 < 0. This implies a stable node in the linearised
system.
Finally we classify V (q−g
q , 0). λ1 < 0 because q > g. λ2 on the other hand depends on the value
chosen for g. Therefore, as before, we have two cases:
• Case A: If g is small (i.e g=0.1) then g
q (a + r) < d + a. This implies that λ2 < 0. Therefore the
eigenvalues are both negative. As above this implies a stable node in the linearised system.
• Case B: If g is large (i.e g=0.4) then g
q (a + r) > d + a. Therefore λ2 > 0. Thus the eigenvalues
are of different sign which implies there is a saddle point in the linearised system.
From above it is clear that changing the value of g between 0.1 and 0.8 creates two different phase
portraits. Now we turn our focus towards the equilibrium point that satisfies 0 = (q−qx+(a−q)y− g
1−y )
and 0 = (r − d − (a + r)x − ry). To classify this point using the above method is challenging as it
requires solving for x and y. Fortunately considerable insight into the behaviour of the system close
to this point is gained through studying the nullclines of the system [6].
Definition 3.1. 0-Isoclines are given by F(x,y)=0 and G(x,y)=0 where F(x,y)=0 gives the x isocline
and G(x,y) gives the y isocline, 0-isocline are also known as nullclines [11].
Using this definition the x nullcline’s are given by x = 0 and
q − qx + (a − q)y −
g
1 − y
= 0
=⇒ y2
(q − a) + y(a − 2q + qx) + q − qx − g = 0
Solving for y gives
=⇒ y =
(2q − a − qx) + q2x2 + a2 − 2aqx + 4gq − 4ga
2(q − a)
5

6. The y nullcline’s are given by y = 0 and
r − d − (a + r)x − ry = 0
=⇒ y = − 1 +
a
r
x + 1 −
d
r
These lines can be plotted using a package called P-plane. To do this the values of a, q, r and d
are taken as the estimated values stated above.
Figure 3: The nullclines plotted for different values of g. Left g=0.4, middle g=0.3 and right g=0.1.
Equilibrium points occur when nullclines cross. In the left and right graph the lines only cross at
the boundary’s. Therefore the equilibrium point which satisfies 0 = q − qx + (a − q)y − g
1−y and
0 = (r − d − (a + r)x − ry) does not occur in the region of interest when g=0.4 or larger or when
g=0.1. In the middle graph, however this internal equilibrium point exists where the nullclines cross.
Clearly the grazing level of fish completely changes the system.
The red and yellow arrows on the graphs signify the direction of flow. Using the information gained
through the classification of the boundary equilibria above and this knowledge of the direction off flow
phase lines are added to create a full phase portrait at different values of g in figure 4.
Figure 4: Phase planes of the system plotted for different values of g. Left g=0.4, middle g=0.3 and
right g=0.1
The graph for g=0.4 (case B above) has only one stable steady state at x=0 suggesting that all
macroalgae is kept at bay by the grazers. Likewise in the graph for g=0.1 on the right the only steady
6

7. state is at y=0, indicating the death of all coral. For a grazing rate of 0.3 both macroalgae, and coral
dominated states are stable and the outcome is dependent on the initial ratio of coral:macroalgae.
Clearly the preferable environment is presented in the graph on the left however this is clearly reliant
on grazing being high enough to keep macroalgae at bay.
4 Dynamic Grazing
An obvious problem with the model above is that grazing is set as constant. Grazing intensity is
dependent on the population size of the grazers. This inevitably fluctuates over time and can be
dramatically influenced by fishing intensity and changes in coral and macroalgae populations over
time. We consider how the method used by Blackwood et al. in [7] allows grazing intensity as a
function of parrot fish population g(P) to be introduced into the model. We need to make assumptions
about the type of fish grazing the reef. The Parrotfish is one of the more common fish found grazing
Caribbean coral reefs. Different species of parrotfish exhibit different grazing behaviour. We require
that the parrotfish graze macroalgae and algal turf evenly. Such behaviour is exhibited by certain
species in the genus Sparisoma which Blackwood et al. assumes to dominate the community. To
introduce dynamic grazing into the model we must consider the change in the number of parrotfish
over time dP
dt . Blackwood et al. models the growth of this population using the logistic equation.
Definition 4.1. The logistic equation is a continuous model for population growth given by dN
dt =
sN(K−N)
K1
where s represents the maximum rate of population growth and K is the maximum sustainable
population. (carrying capacity) [10].
Rearranging the logistic equation and letting N=P where P is the number of parrotfish in the
population gives dP
dt = sP 1 − P
K1
. The maximum sustainable population (K1) is inevitably limited
by habitat conditions. To incorporate this Blackwood et al. introduces a term 0 < K(C) < 1
that ‘limits carrying capacity as a function of coral cover’[7]. Therefore if K1 = β is the maximum
sustainable population then βK(C) is the maximum sustainable population when limited by coral
cover. Finally we represent the parrotfish removed from the system due to fishing by fP where f is the
constant rate of fishing. Subtracting this term from dp
dt and adding this new differential equation to
the system in Section 2 gives:
dM
dt
= aMC −
g(P)M
M + T
+ qMT, (5)
dC
dt
= rTC − dC − aMC, (6)
dP
dt
= sP 1 −
p
βK(C)
− fP. (7)
In order to define g(P) Blackwood et al. let grazing intensity be proportional to P
β . Therefore
assuming that α is a positive constant g(P) = αP
β . Previously the upper limit for g was defined as
0.8. Blackwood et al. extends this to 1 for convenience and lets α = gmax = 1. This results in
g(P) = P
β which means that if the number of parrotfish reaches the maximum sustainable population
then grazing will be equal to 1. This can only be the case when there is no limitation on the parrotfish
population. Fortunately the only way to obtain P = β is to let K(C)=1 and f=0. These values
correspond to there being no fishing and no limitation from coral. Therefore our value for g(P) makes
sense.
Finally we can non-dimensionalise in order to study the ‘dynamics relative to the maximum car-
rying capacity β’ [7]. This is done by letting P = P∗β which becomes dP
dt = P∗
dt β on differentiating.
7

8. Substituting this into the equation gives:
dP∗
dt
= sP∗
1 −
p∗
K(C)
− fp∗
[7].
The introduction of dynamic grazing into the system is a step towards making the model more
consistent with the real world environment. It allows us to consider the system under different reef
conditions. Having already established the importance of grazing Blackwood et al. goes on to consider
whether the introduction of a controlled fishing strategy would increase coral’s resilience to a natural
disaster [7]. Natural disasters, such as hurricanes, can cause significant structural damage to corals
limestone exoskeletons. Parrotfish rely directly on coral for protection from predation. Therefore at a
very low coral cover more fish will be removed from the system due to predation. We incorporate this
into the model via the K(C) term. At low coral cover the carrying capacity will be low. An increase
in coral will result in an increase in carrying capacity as less fish will be removed due to predation.
Under this assumption Blackwood et al. lets K(C) = C.
Previously it was assumed that shifts in the system happen immediately with no time delay. In
reality growth delays occur throughout the system however, we are particularly interested in any delays
associated with grazing. We note that there will inevitably be an delay between increased carrying
capacity and an increase in parrotfish population. This is because an increase in coral reduces the
number of parrotfish removed due to predation which allows more parrotfish to go on to reproduce
which then increases the population size. This is a substantial delay that could potentially alter the
entire system. In the next section we look at different methods for introducing a delay into the model.
5 Delay Differential Equations (DDE)
In this section we provide a brief introduction into delay differential equations and the different ways in
which delays can be incorporated into a system. We demonstrate; how a system of delay differential
equations can be reduced to a system of ordinary differential equations (ODE’s) and consider the
impact that introducing a delay into the system in section 4 has. A system which incorporate’s a time
delay is known as a delay differential equation (DDE).
Definition 5.1. A differential equation is known as a delay differential equation when its time deriva-
tives depend on its solution and potentially its derivatives at previous times [16]. A linear differential
equation with continuous delay is given as:
dx
dt
= Ax(t) − B
∞
0
F(θ)x(t − θ)dθ
where A and B are real coefficients, F the delay kernal is an integrable function and θ is the delay.
(Delay differential equations, 2016) [12]
For certain functions of F, such as Dirac’s delta function, this delay can be simplified to the linear
discrete delay differential equation.
Definition 5.2. The Dirac delta function is a ‘non-physical, singularity function’ defined as:
δ(x) =
0 x = 0
∞ (undefined) x = 0
where
∞
−∞
δ(x)dx = 1 [17] (8)
Definition 5.3. A linear discrete delay differential equation is given by;
dx
dt
= −Ax(t) − Bx(t − σ) where A, B ∈ , and σ = the delay. [12] [16]
8

9. Fixed time delays of this type are often used in mathematical models. However they produce
infinite dynamical systems and thus their analysis quickly becomes complicated. Introducing a con-
tinuous distributed delay is more realistic because it allows for variation in delay time [15]. In this
project we consider distributed delays of gamma type as defined below.
Definition 5.4. Distributed delays of gamma type:
∞
0
x(t − s)gp
a(s)ds =
1
−∞
x(η)gp
a(t − η)dη
where the kernal gp
a is the density function of the gamma distribution: gp
a(u) = apup−1
(p−1)! e−au. Here p,
the shape parameter, divided by a, the rate parameter, give the mean delay time [18].
For this type of Kernal a method known as the chain trick can be used to reduce the DDE into a
system of ODE’s, simplifying the analysis process [18]. This method is used in [15] where Blackwood
and Hastings introduce a distributed delay of gamma type into the system in section 4. This method
can be demonstrated by introducing a delay into a generic model for species competition [15]. The
Lotka-Volterra model of competition is given by; dx
dt = x(1 − x − ay), dy
dt = by(1 − y − cx) where
a, b, c > 0 [14]. Although this model is much simpler than the coral model it demonstrates a similar
equilibrium structure with three boundary equilibria and one internal saddle point [15].
Introducing a distributed time delay into this Lotka-Volterra model gives:
dx
dt
= x(1 − x − ay),
dy
dt
= by(1 − y − c
t
−∞
x(η)gp
a(t − η)dη). (9)
If we let
zq =
t
−∞
x(η)gp
a(t − η)dη then, z1 =
t
−∞
x(η)αe−α(t−η)
dη. (10)
We now look to compute dz1
dt . To do this the following property, defined in [15], is required:
dz
dt
f(t, η)dη = f(t, t) + f(t, η)dη.
Applying this property gives
x(t)α + −
t
−∞
x(η)αe−α(t−η)
dη = α(x(t) − z1),
and the system reduces to:
dx
dt
= x(1 − x − ay), (11)
dy
dt
= by(1 − y − czp), (12)
dz1
dt
= α(x − z1). (13)
Note 5.1. If we were to solve this we would require boundary conditions. Let these be y(0) = y0,
x(0) = ψ(0) where x(t) = ψ(t) and t ∈ (−∞, 0] because the system depends on the history of the system.
After applying the chain trick they become y(0) = y0, x(0) = ψ(0) and z(0) =
0
−∞
ψ(η)g1
a(−η)dη [15].
Recall now the system of equations created in section 4. Blackwood and Hastings uses the above
method to introduce a time delay into the system. Whilst we do not study the specific mathematics it
is interesting to briefly consider their findings. Using the values for the parameter estimated through
9

10. Figure 5: a) a simulation of the model in section 4 without a time delay. b) a simulation of the model
in section 4 with a time delay introduced [15]. In both graphs the dashed line represents macroalgae
cover, the hard line represents coral cover and the dotted line represents the number of parrotfish.
historical data stated in section 3 and a computer simulation they are able to produce the graphs in
figure 5.
Note that this system was created under the assumption that a natural disaster had greatly reduced
the coral. Which led in turn to a reduction in parrotfish numbers. Clearly the time delay has a big
impact on the findings. In graph a), where no time delay is introduced both coral and parrotfish
populations die out while macroalgae overgrows the system. In contrast graph b) shows a different
result. The delay means the parrotfish population is able to keep the macroalgae at bay which
allows coral recovery. Through this system Blackwood and Hastings have successfully highlighted the
importance of incorporating a time delay. At the same time they have shown that if a structured
fishing plan is put into place then coral resilience to natural disasters could improve.
6 Conclusion
The above sections have considered how a system of ordinary differential equations can be created,
developed, and analysed to gather information about how corals compete with macroalgae. By plotting
the system using P-plane we were able to highlight the importance of maintaining a high grazing rate
to corals survival. Through considering two separate papers [7] and [15] we have considered the
different methods which have been used to extend the system. Whilst our original model highlighted
the importance of maintaining a high grazing rate, the extended model introduced a way to show
that a change in human behaviour, through the introduction of a controlled fishing system, could
improve reef resilience to alternate stresses. Therefore it is clear that the development and analysis
of mathematical models is useful when designing reef management programs and putting pressure on
institutions and local communities to change their fishing habits.
The mathematics in this project, however is far from exhaustive, and only scrapes the surface of
the mathematical research being carried out in this area. For the model described by Mumby et al.
alone far more advanced mathematics has been carried out to gain a more detailed understanding
of the way that the system behaves. Likewise there are papers which extend the work done in [7]
and [15] by looking into the creation of bifurcation diagrams. Moreover introducing a delay into the
system has been done in more than one way. For the system in section 3 the paper [6] goes on to
add a discrete time delay on grazing into the system and carries out in-depth analysis into the altered
system. Moreover mathematics is not only used to address the question of overfishing and competition.
Research has also been carried out into producing models that consider corals symbiotic relationship
10

11. with the algae it relies on to gain nutrients.
Therefore it is sensible to conclude that while this project has provided a brief introduction into
some of the techniques that mathematicians are using to model coral reefs, much more work would
need to be done in order to fully study the current research on this topic.
References
[1] Obura, D. and Grimsditch, G. (2008) Coral Reefs, Climate Change and Resilience An Agenda
for Action from the IUCN World Conservation Congress in Barcelona, Spain [online]. Available
from: http://cmsdata.iucn.org/downloads/resilience barcelona.pdf [Accessed 24 November 2015]
[2] Lallanilla, M. (2013) What are coral reefs? [online]. Available from:
http://www.livescience.com/40276-coral-reefs.html [Accessed 20 November 2015]
[3] Pilling, G.M., Davy, S.K. and Sheppard, C.R.C. (2009) The biology of coral reefs. United King-
dom: Oxford University Press
[4] Phinney, J., Hoegh-Guldberg, O. and Kleypas, J. (2007) Coral reefs and climate change: Science
and management (coastal and Estuarine sciences). 1st ed. Phinney, J. (ed.). Washington, DC:
American Geophysical Union
[5] Ridgell, R. (1988) Pacific nations and territories: The islands of Micronesia, Melanesia, and
Polynesia. 2nd ed. Honolulu, HI: Bess Press
[6] Li, X., Wang, H., Zhang, Z., et al. (2014) Mathematical analysis of coral reef models. Journal
of Mathematical Analysis and Applications, 416 (1): 352373
[7] Blackwood, J.C., Hastings, A. and Mumby, P.J. (2010) The effect of fishing on hysteresis in
Caribbean coral reefs. Theoretical Ecology, 5 (1): 105114
[8] Diaz-Pulido, G. and J. McCook, L. (2008) Environmental Status: Macroalgae (Seaweeds). Great
Barrier Reef Marine Park Authority.
[9] Mumby, P.J., Hastings, A. and Edwards, H.J. (2007) Thresholds and the resilience of Caribbean
coral reefs. Nature, 450 (7166): 98101
[10] Weisstein, Eric W. ”Dynamical System.” From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/DynamicalSystem.html4
[11] Taylor, M.E. (2011) Introduction to differential equations (pure and applied undergraduate
texts). Providence, RI: American Mathematical Society
[12] Atay, F.M. (2010) Complex time-delay systems: Theory and applications. Berlin: Springer-
Verlag Berlin and Heidelberg GmbH Co. K
[13] Weisstein, E.W. (2003) Logistic equation [online]. Available from:
http://mathworld.wolfram.com/LogisticEquation.html [Accessed 3 February 2016]
[14] Sternberg, S. (2009) Lecture 15 Lotka-Volterra [online]. Available from:
http://www.math.harvard.edu/library/sternberg/slides/11809LV.pdf [Accessed 3 February
2016]
[15] Blackwood, J.C. and Hastings, A. (2010) The effect of time delays on Caribbean coralalgal
interactions. Journal of Theoretical Biology, 273 (1): 3743
[16] Delay differential equations (2016). Available from: http://reference.wolfram.com/language/tutorial/NDSolv
[Accessed 3 February 2016]
[17] The Dirac delta function. Available at: http://www.nada.kth.se ∼ annak/diracdelta.pdf (Ac-
cessed: 22 March 2016).
[18] Smith, H. (2011) An introduction to delay differential equations with applications to the life
sciences. United States: Springer. (pp 119-130)
11