A Theoretical Model of Pattern Formation in Coral Reefs

Susannah Mistr and David Bercovici*

Department of Geology and Geophysics, School of Ocean and Earth Science and Technology, University of Hawaii, Honolulu, Hawaii 96822, USA

ABSTRACT We present a mathematical model of the growth of analysis reveals that the nonlinear growth creates coral subject to unidirectional ocean currents using sharp fronts of high solid fraction that, as predicted concepts of porous-media flow and nonlinear dy- by the linear stability, advance against the predom- namics in chemical systems. Linear stability analysis inating flow direction. The findings of regularly of the system of equations predicts that the growth spaced growth areas oriented perpendicular to flow of solid (coral) structures will be aligned perpendic- are qualitatively supported on both a colonial and a ular to flow, propagating against flow direction. regional reef scale. Length scales of spacing between structures are selected based on chemical reaction and flow rates. In the fully nonlinear system, autocatalysis in the Key words: coral reefs; pattern formation; self- chemical reaction accelerates growth. Numerical organization; colonial organisms.

INTRODUCTION coral colonies and reef systems subject to one-dimensional nutrient-transporting water currents Coral reefs are composed of small, calcifying coral and construct a theoretical model of coral growth. polyps that together build complex architectures. We simplify the growth process to a hypothetical That these architectures can be different for colo- chemical reaction between a limiting nutrient sup- nies of the same species in different environmental plied by the water column and solid material man- conditions suggests that quantifiable, nongenetic ifest as a porous skeleton. The chemical reaction, chemical and physical controls are important in the represented mathematically using the law of mass organization of the system. Although the complex action, produces additional solid matrix. The self- interactions controlling the chemistry of coral cal- enhancing growth of the solid matrix is said to be cificaltion are difficult to model, it is possible to autocatalytic and reflects the need for an existing model some aspects of growth using fundamental organism in order to sustain growth. The density of principles of physics and chemistry. Important fac- solid matrix influences the flow of the nutrient- tors in the growth of coral include light, which bearing current by changing the constriction and/or influences the rate of calcification, and flow, which tortuosity of fluid pathways or by inducing subcriti- influences the delivery and uptake of the nutrients cal turbulence by means of enhanced surface essential for growth. roughness. Overall, we assume that an increase in Herein we consider the simplest examples of matrix density impedes flow, creating a nonlinear feedback mechanism between matrix growth and nutrient supply. To consider how natural patterns Received 1 October 2001; accepted 22 May 2002. *Corresponding author’s current address: Department of Geology and Geo- may arise, we consider the behavior of perturba- physics, Yale University, PO Box 208109, New Haven, Connecticut tions in an initially uniform model system. 06520-8109, USA; email: [email protected] Alan Turing was one of the first to describe a Current address for S. Mistr: College of Medicine, Medical University of South Carolina, 96 Jonathan Lucas Street, PO Box 250617, Charleston, theory for the onset of biological form and to con- South Carolina 29425, USA sider theoretically the dynamics of spatially varying

61 62 S. Mistr and D. Bercovici

Figure 2. Bunker Reef group, Great Barrier Reef, Aus- tralia (after Jupp and others 1985).

regional scale, reef islands orient in series (Figure 2) along the predominating direction of large-scale currents (Figure 3). Figure 1. Aggregations of common reef coral Agaricia Well-developed coral reefs are thought to be tenuifolia, Carrie Bow Cay, Belize. The colony consists of present in otherwise low-productivity waters due to a series of upright plates oriented perpendicular to flow. the tight recycling of nutrients between symbiotic The sand channel is parallel to the flow direction (after zooxanthellae within the tissues of the coral organ- Helmuth and others 1997). ism and the coral organisms themselves (Lewis 1973). Classically, these photosynthetic zooxanthellae provide oxygen and food for the coral or- nonlinear chemical systems (Turing 1952). The key ganism through photosynthesis, while the coral in point in Turing’s work relevant to many aspects of turn provides essential nutrients to the zooxanthel- pattern formation is that asymmetry could be am- lae through metabolic waste (Muscatine 1973). Cal- plified by means of competing chemical reactions, cification can be positively correlated with light ex- and, as mitigated by diffusion, the asymmetry could posure, indeed calcification rates can be about three be spatially distributed. By considering autocata- times faster in light than in dark (Kawaguti and lytic, or self-enhancing, chemical species Turing was Sakumoto 1948; Baker and Weber 1975a, b; see able to show how steep gradients could result from a also Gattuso and others 1999). It is therefore widely small perturbation to uniform concentrations. suspected that the photosynthesis in zooxanthellae Patterns in coral reefs range from the small-scale contributes to increased rates of calcification. For a architecture of a particular colony to large-scale reef basic understanding of how photosynthesis may distribution patterns. Because of the many complex enhance calcification, we consider the overall reac- factors that can contribute to the development of tions of both. these patterns, examples for theoretical study are For photosynthesis, carbon dioxide (CO2) and taken from environments where one particular en- water (H2O) are used to produce glucose and oxy- vironmental parameter appears to play a distinctly gen: significant role. In particular, the influence of ocean sunlight currents appears to affect coral reef orientation over ϩ O¡ ϩ CO2 H2O C6H12O6 O2 (1) many scales. On the colony scale, certain coral colonies orient as a series of plates perpendicular to the predominating direction of flow (Figure 1). On the For calcification, calcium and bicarbonate ions pre- Theoretical Model of Coral Reefs 63

cellularly controlled process that requires metabolic energy for the pumping of materials into or out of the cell (see Gattuso and others 1999). Finally, the introduction of a mineral inhibitor can hold calcification to 1% of normal incorporation of calcium into the skeleton without affecting rates of photosynthesis (see Gattuso and others 1999). The combination of these observations suggests that photosynthesis-driven calcification involves more complex chemical schemes than Eqs. (1) and (2); however, since the aim of this paper is to consider the relationship of growth and shape of coral colonies in response to differing environmental factors, it is not entirely necessary to understand the explicit mechanism of calcification. What we must consider is that skeletogenesis is an active process that occurs in the presence of the organism, which must be supplied with some form of food or energy. Because corals are known to feed on zooplankton and remove food fragments from the water column (Yonge 1930; Porter 1976), we devise very simplified “calcification” reactions where preexisting calcium carbonate “reacts” with some limiting nutrient Figure 3. Southwest Pacific Circulation (after Maxwell to produce more calcium carbonate, representing 1968). “X” denotes the location of the Bunker Reef coral growth. The nutrient is supplied at the sea sur- Group. face and diffuses or settles downward. We then consider the effect of currents circulating through the reef by imposing a unidirectional flow on the system. An interesting characteristic of coral reef mor- cipitate calcium carbonate and evolve carbon diox- phology is that there is often a strong pattern of ide (Ware and others 1991, from Stumm and Mor- vertical zonation along the reef (see Jackson 1991). gan 1981): The absorption of light as it penetrates through the ϩϩ ϩ Ϫ 3 ϩ ϩ water column and the change in flow regime are Ca 2HCO3 CaCO3 CO2 H2O (2) suspected to contribute to the change of dominating The calcification reaction produces carbon dioxide, form with depth (Baker and Weber 1975a, b; Graus which can be used in photosynthesis. This removal and Macintyre 1989). Although the difference in of carbon dioxide by photosynthesis provides a dominating forms with depth is often related to a mechanism by which the calcification reaction in change in the dominating coral species and is thus Eq. (2) can be driven toward the right; it has been more of an evolutionary problem, there are in- proposed that this mechanism explains enhanced stances where corals of the same species exhibit calcification (Goreau 1963). Although this is a via- morphological plasticity in accordance with flow ble mechanism, the details of calcium supply and regime. Using an example of Pocillopora damicor- bicarbonate usage are poorly constrained; there- nis, we see that as flow increases, morphology fore, few conclusions can be drawn as to the signif- changes from more space-filling knobby forms in icance of these reactions alone in enabling organis- high-flow regimes to more delicately branching mal calcification. forms in low-flow regimes (Figure 4). Since the For example, complicating the simple photosyn- same species can exhibit different growth forms thesis-driven calcification hypothesis, some species under different flow regimes, it is evident that the of coral possess an organic matrix on the outside of ambient environmental conditions profoundly in- cells lining the skeleton that contains calcium-bind- fluence morphology. The effect of water and nutri- ing substances (Isa and Okazaki 1987). The organic ent flux in controlling biological pattern formation matrix provides a way to lower the energy for pre- is obviously important in other environments as cipitation and thus drive calcification (Borowitzka well, such as vegetation in semiarid areas (HilleRis- 1987). Additionally, there is evidence that the in- Lambers and others 2001) corporation of calcium into the coral skeleton is a It is our goal to understand how the calcification 64 S. Mistr and D. Bercovici

formulation to model our skeletal framework. Mak- ing use of a porous matrix allows us to describe flow of fluid by Darcy’s law in porous materials. Aha- ronov and others (1995) used a porous-medium formulation to address feedbacks between dissolution of host matrix and fluid infiltration for understanding melt-focusing beneath midocean ridges. Figure 4. Three specimens of the coral Pocillopora dami- Their model considered the interaction of dissolu- cornis from sites increasingly sheltered from water move- tion, changing porosity, and enhanced fluid infiltra- ment. Specimen a is most exposed to flow; specimen c is tion and revealed that larger channels develop at most protected from flow. After Kaandorp and others 1996 (from Veron and Pichan 1976). the expense of smaller channels, thus providing a mechanism for focusing melt beneath the ridge. The basic premise of their work is that changing porosity can affect flow velocity. This idea will be rate and the flow regime affect the growth of coral retained in our model; however, rather than con- reefs. We hypothesize that colonies and reefs de- sidering a positive feedback between infiltration velop their complex structures as they obstruct flow and dissolution, we examine the interaction of the and intercept available nutrients, either by en- autocatalytic growth of the matrix and the advec- hanced surface roughness or by increased complex- tive supply of a limiting nutrient. ity of the path for the flow field. Complex feedbacks The model system is considered to take place in a among turbulent flow, nutrient supply, and calcifi- layer of thickness H that rests atop the sea floor but cation likely operate to generate the elaborate above which no coral grows (Figure 5). The coral reef three-dimensional colony architectures. These mass is considered to be a porous, immobile, and solid complex interactions are unwieldy in theoretical matrix that reacts chemically with dissolved nutrient continuum models and thus have largely been ap- to form more solid matrix. The nutrient itself is sup- proached using idealized lattice Boltzman or cellu- plied by a uniform source from the overlying water lar-automata models of aggregation to explain the column and by flux of the nutrient-bearing water, increase of space-filling forms with increased flow forced to flow horizontally through the solid matrix regime (Preece and Johnson 1993; Kaandorp and by an imposed pressure gradient. The solid matrix has others 1996). molar concentration ␺ (that is, moles in a unit volume Here we attempt to provide a simple, physically containing solid, fluid, and nutrient); the nutrient has based model using basic carbonate chemistry in molar concentration ␣. Both values are given in units conjunction with a porous-flow model to describe of mol/m3 to facilitate the provisions of a stoichiomet- the overall decrease in flow rate that accompanies ric relation for their chemical reaction (see Appendix the increase in coral cover that occurs as corals take 1). The pressure whose gradient drives flow is de- nutrients from the water column. We represent noted by P and has units of Pa. All three dependent increasing coral cover in the face of nutrient supply variables (␺, ␣, and P) change in space and time and by the autocatalytic construction of a porous matrix obey equations of mass conservation and fluid flow infiltrated with fluid subject to a one-dimensional (Darcy’s law); however, these equations are vertically pressure gradient. The nutrient, supplied from the averaged in z so that they can be posed in terms of surface, diffuses through and is carried by the fluid. horizontal position (x, y) only, as well as time t (see We arrive at the model by considering the conser- Appendix 1). vation of mass for the fluid, dissolved nutrient, and The growth, loss, and transport of the nutrient is the solid. We then analyze the model to explore the determined by its advective flux with water dynamic interaction of nutrient supply, solid and through the solid matrix, its diffusion through the nutrient diffusion rates, flow rates, and chemical water, and the mass lost through chemical reaction reaction rates and to find cases where small depar- with the solid matrix; when all dependent and in- tures from the uniform system are amplified and dependent variables are nondimensionalized by the patterning phenomena arise. natural mass, length, time, and molar scales of the system (see Appendix 1), this relation is expressed as: THEORETICAL MODEL ␣ץ P͔ ϭ ٌ2␣ ϩ ␣ Ϫ ␣ Ϫ ␤␣␺2 ١␣⅐ ͓͑1 Ϫ ␺͒2 ١ Although it is a simplification, considering that tis- Ϫ t h h h sץ sue, and thus growth, occurs only on the outer surface of the coral, we employ a porous-medium (3) Theoretical Model of Coral Reefs 65

␣ ٌ where h is the horizontal gradient operator, s is Eqs. (3), (4), and (5) are the dimensionless gov- the uniform supply of nutrient through the top erning equations that we use to analyze the stability surface of the layer, and ␤ is a chemical reaction of our autocatalytic, nutrient-limited porous ma- parameter (that is, the dimensionless number that trix. The complete derivation of these equations contains the reaction rate; see Appendix 1). The and details about nondimensional parameters and terms on the left side of Eq. (3) are (from left to nondimensionalizing scales are discussed in Appen- right) the growth rate of nutrient density and the dix 1. advective nutrient flux carried by Darcy flow that is ٌ itself forced by a pressure gradient hP through a ANALYSIS OF THEORETICAL MODEL matrix with permeability proportional to (1 Ϫ␺)2 (such that when the solid density ␺ϭ1, the matrix Stability Analysis is impermeable). The terms on the right side repre- Philosophy. One of the most powerful tools for ␣2ٌ sent horizontal diffusion of nutrient ( h ), vertical elucidating self-organization and pattern formation diffusion relative to the overlying source concentra- in natural systems is linear stability analysis. Most ␣ Ϫ␣ ␣Ͻ␣ tion ( s , such that if s nutrient diffuses systems have a simple equilibrium state that is es- ␣Ͼ␣ down into the layer, and if s it diffuses up out sentially uniform and patternless. However, this of the layer), and mass loss due to chemical reaction equilibrium state might be unstable, so that distur- (term proportional to ␤). A coefficient of diffusivity bances or perturbations would grow and force the for ␣ does not explicitly appear in Eq. (3) because it system to a different state with some inherent pat- is absorbed into the nondimensionalizing time scale tern. A ball atop a hill is a simple zero-dimensional (see Appendix 1). example: Undisturbed, the ball can be balanced and The equation for change of solid matrix concen- remains at rest; but with a sufficient disturbance tration is controlled only by mass addition through (for example, wind, earthquake, a light kick), the chemical reaction with nutrients and by loss ball will roll off the hill and not stop until it reaches through minor diffusion; such solid diffusion is used some other equilibrium state (for example, a valley to represent both slight dissolution of coral reef deep enough to capture it). mass and—perhaps more importantly—erosion The classic paradigm of instability and pattern and damage near the perimeter of coral reef con- formation in physics is that of thermal convection, centrations, which are here represented as steep whereby a fluid layer is uniformly heated along its gradients in ␺. This leads to the following equation base and cooled along its top. This system has a (again dimensionless): static and patternless thermally conductive state whereby heat is transported from the hot base to a ␺ץ -ϭ ␭ٌ2␺ Ϫ ␭␺ ϩ ␤␣␺2 cooler surface by transmission of molecular vibra h (4) t tions, or phonons. But since fluid near the hot baseץ is less dense than fluid near the colder top, it is where ␭ is the solid matrix diffusivity (scaled by the gravitationally unstable. Thus, perturbations or dis- nutrient diffusivity) and will be referred to as “the turbances such as slight fluid motion can jostle the diffusion parameter” (see Appendix 1 for more de- system and initiate overturn such that light fluid tails). The terms in Eq. (4) show growth of solid moves to the top and heavy fluid moves to the concentration (left side), diffusion both horizontally bottom; with the continuous heat supply through and vertically (terms proportional to ␭), and mass the bottom and heat loss through the top, the mo- addition from chemical reactions with the nutrient tion persists as convective circulation. The pertur- (term proportional to ␤). bation of a given length scale that can cause the Lastly, the mass conservation of water flowing layer to turn over most efficiently, or for the least through the matrix leads to an equation for pres- amount of heat input, is generally the one that will sure itself: grow the fastest and dominate the pattern of con- ␺ץ ١ ␺ ⅐ ١ 2 h hP 1 vection (that is, it will determine the length scale of (2P ϭ Ϫ (5 ٌ .(t the convection cellsץ h ͑1 Ϫ ␺͒ ͑1 Ϫ ␺͒2 Thus, in stability analysis, one seeks the charac- where spatial adjustments in pressure (left side of teristics (length scale and temporal behavior, such the equation) are controlled by flow through con- as growth and propagation) of this dominant per- strictions in the matrix—for example, a “nozzle” turbation. The formalism of stability analysis is well effect (first term on right side) and a growth of the explained in various texts (see, for example, Chan- matrix that would squeeze fluid out of pores (last drasekhar 1961; Drazin and Reid 1989; Nicolis term on the right side). 1995). Here we employ the principles of linear sta- 66 S. Mistr and D. Bercovici bility analysis to build a framework for interpreting pattern formation and self-organization in our model coral reef system (see Appendix 2 for full details). Equilibrium States. Given the cubic nonlinearity of the system of governing equations (in particular, the reaction term proportional to ␤ in Eqs. [3] and [4]), there are three uniform equilibrium states. One such state is the trivial state, wherein there is no solid mass; the two other states involve small Figure 5. Schematic of the model. Fluid infiltrates the solid-mass fraction and large solid-mass fraction, porous matrix, and flow is maintained by a pressure respectively. The trivial state is stable and does not gradient in one direction. Limiting nutrient is supplied lead to any pattern formation, implying that at least from the surface, diffuses through the fluid, and reacts some minimal amount of initial solid is necessary to with the matrix for the autocatalytic production of addi- initiate instability and pattern growth; hence, we do tional matrix. The model assumes that the system is con- not consider this state further. Likewise, the high tinuous in x and y with finite depth z ϭ H. solid fraction state is stable under almost all conditions and is also not very susceptible to pattern formation, implying that little more solid structure of our system instead of the x, y, and t dependence. can be built if the solid fraction is near its maximum We thus assume that each perturbation to ␺ , ␣ , allowed value; we will therefore not consider this 0 0 and P0 is a sinusoid in x, with wavenumber k (that state either. (For discussion of these states, see Ap- is, wavelength 2␲/k) and growth rate s. The growth pendix 2.) rate s can be complex; the real part Re(s) implies The low solid fraction state is unstable to pertur- growth (if positive) or decay (if negative), while the bations and leads to the growth of structures. This ␺ ␣ imaginary part Im(s) implies wavelike propagation. state is denoted by ( 0, 0, P0) where: (As shown in Appendix 2, there are in fact two ␣ ␣2 possible growth rates for each k, although only one 1 s s 4 ␺ ϭ ͩ Ϫ ͱ Ϫ ͪ , ␣ ϭ ␣ Ϫ ␭␺ (6) allows growth of perturbations; this growth rate is 0 2 ␭ ␭2 ␤ 0 s 0 referred to in Appendix 2 as sϩ, but for simplicity we have dropped the “ϩ” subscript here; however, and the equilibrium pressure has a constant gradi- ϭϪ␯ in all the figures, we still refer to sϩ for consistency ent given dP0/dx , which represents an im- with the figures in the appendixes.) posed background current velocity or flow rate. If we consider an infinite number of perturba- (Note that in Appendix 2, this equilibrium state is ␺Ϫ tions with different wavelengths, the one that has referred to by 0 to keep it distinct from the other Ϫ the largest growth rate max(Re(s)) will dominate the equilibrium states; here we drop the “ ” super- pattern that evolves. The wavelength of this fastest script.) ␲ growing perturbation is 2 /kmax, and its propaga- Instability of Perturbations. We next address the tion rate, or wavespeed, is: question of perturbations to the equlibrium state ϭ Ϫ ͑ ͒ given by Eq. (6) and consider how their properties cmax Im s max/kmax (7) (growth rate, spatial structure) are affected by the model parameters. To maintain a manageable pa- where Im(s)max is the imaginary part of s whose real rameter space, we examine only the effect of vary- part is max(Re(s)). ing the flow rate ␯ and reaction rate ␤, and we hold Figure 6 shows the maximum growth rate solid diffusion fixed at ␭ϭ10Ϫ3 (that is, much max(Re(s)) along with the corresponding wave- smaller than nutrient diffusion) and surface nutri- number kmax and wavespeed cmax as functions of ␣ ϭ ϫ Ϫ4 ␯ ␤ϭ ent supply fixed at s 9 10 (much smaller flow velocity and for three reaction rates 5, than the molar concentration of pure solid by 10, and 100. The maximum growth rate increases which all molar concentrations have been nondi- monotonically with both ␯ and ␤ (Figure 6a), al- mensionalized; see Appendix 1). though for the large reaction rate ␤ϭ100, the As shown in Appendix 2, the only perturbations growth rate is nearly constant over ␯. Because ␤ϭ that grow are completely uniform in the y (cross- 100 corresponds to a very fast chemical reaction, current) direction and have nonuniform structure there is little influence from current velocity since in the x (along-current) direction. We therefore the reaction is nearly instantaneous for any flow only need to consider the x and t (time) dependence rate. On the other hand, for slower reactions, such Theoretical Model of Coral Reefs 67

sion (as occurs in many classic problems of diffusion-limited aggregation [DLA]). Beyond the critical ␯, diffusion destroys the advantage of having sharp pressure gradients and thus the system’s dominant length scale becomes determined by the recovery length—that is, the distance traveled by a nutrient-depleted parcel of fluid before it recovers ␣ its nutrient content via the surface supply s; this recovery distance is obviously influenced by the fluid velocity, as demonstrated in Figure 6b. In this latter regime (that is, for ␯ greater than the critical value), increased reaction rates (larger ␤) also cause

the spacing of structures to grow (kmax decreases) because the fast reaction depletes the nutrients more thoroughly and rapidly, necessitating a longer recovery. Figure 6. Maximum growth rate max(Re(sϩ)) of least Our model also shows that fastest-growing per- stable perturbation (a), corresponding wavenumber k max turbation also propagates into the incoming flow (where perturbation wavelength is 2␲/k) (b), and corre- Ͻ (cmax 0). By supplying more abundant nutrients sponding propagation velocity c ϭϪIm(sϩ)/k (c) as max ␯ functions of imposed flow velocity (or pressure gradient), upstream, increasing drives the system toward ␯, for three reaction rates, ␤ϭ5, ␤ϭ10, and ␤ϭ100. upstream growth—that is, the structure propagates toward the upstream accumulation fronts (for example, the direction a surface travels while being, in effect, spray-painted). As ␯ affects the nutrient sup- ␤ϭ as 5, we see increased growth rates as the flow ply, it affects the slowest reaction cases the most— speed increases, suggesting that the growth of solid that is, those with small ␤—since for fast reactions, structures is more sensitive to nutrient supply from nutrients stay more ubiquitously depleted. A con- ␯ the current for slower chemical reaction rates. As stant propagation value is achieved at a critical flow 3 ϱ ␤ϭ , the growth rates for 5 approach those for rate ␯, since, above a certain flow rate, the nutrient ␤ϭ 10 and 100 as the growth rate of the dominant supply must still be affected by the rate at which it perturbation becomes primarily controlled by the is supplied by nutrients from the surface. nutrient flux from the moving current. The important features of this linear stability This one-dimensional model also reveals that the analysis predict that perturbations to our nonlinear system develops solid growth maxima at character- system will grow, and that the system will be char- ␲ istic, finite wavelengths (2 /kmax) over a wide range acterized by the development of gradients in the ␯ ␤ ␯ϭ of and (Figure 6b). For zero fluid flow, 0, solid and nutrient fields that propagate upstream. spatial variability leads only to diffusive loss and thus decay of perturbations to the solid and nutri- Nonlinear Analysis ent fields; therefore, the wavelength or spacing for ϭ the fastest-growing structure is infinite (kmax 0) For the nonlinear analysis, we build on the results for all ␤—that is, growth is spatially uniform with from the linear stability analysis. We assume that no fluid motion. However, as the background flow derivatives in y are zero for Eqs. (3), (4), and (5), speed is increased (␯Ͼ0), nutrient parcels become thus effectively considering the evolution of the depleted asymmetrically as upstream reaction af- one-dimensional nonlinear system. fects downstream supply; thus, a finite wavelength We integrate the nonlinear equations using finite or spacing for fastest-growing patterns occurs. How- differences. We take the fully explicit forward-time ever, the wavelength of the fastest-growing struc- difference for time derivatives and a space-centered

tures at first decreases (kmax increases) with increas- difference for second-order derivatives; for first-or- ing flow ␯; then, at a critical ␯, it increases again der spatial derivatives, which reflect advection, we (Figure 6b), which suggests competition between use upwind-differencing (Jain 1984; Press and oth- effects. In particular, a decrease in wavelength with ers 1992). The Courant condition on the time-dif- increased flow permits sharper pressure gradients ferencing is met by time-stepping at 10% of the with which to impede flow and more efficiently Courant time-step (Press and others 1992). For the capture nutrients; however, at too sharp a wave- evaluation of the pressure field, we assume an in- length, the growth of structures is limited by diffu- stantaneous response to changed solid and nutrient 68 S. Mistr and D. Bercovici

Figure 7. Solutions for ␺, the solid matrix, versus x for the time integration of the nonlinear system for pertur- Figure 8. Values of ␺, the solid matrix, versus time t for ␤ϭ ␺ bations to the steady state for 100. Contours repre- linear stability solution for perturbations to 0 (dashed sent different stages in the time evolution; the maximum line) and for nonlinear solutions (solid line) after 2800 amplitudes reflect the most recent time integration and time steps. Result is for perturbations to the steady state are approximately evenly spaced in time. The indicated for ␤ϭ100, ␯ϭ100. ␺ ␺ ␺ peak and trough values of , max, and min are for the ␺normalized ϭ ␺Ϫ␺ first and last contour, and ( min)/ ␺ Ϫ␺ ␺ ( max min). From top to bottom, solutions for slow uniform steady state, 0 (Eq. [6]), as in the linear flow (␯ϭ4), moderate flow (␯ϭ10), high flow (␯ϭ50), stability analysis. We assume the steady state of the very high flow (␯ϭ100). Perturbations to the steady system is perturbed at 10% of the steady-state state were at the wavelength of the least stable mode value, and we integrate the equations over time. predicted by the linear stability analysis at 10% of the We assume that the system selects the scale pre- steady-state value ␺ ϭ 0.0113; however, the magnitude 0 dicted by the linear stability, and we perturb the of the perturbation for the ␯ϭ4 case is 1% of ␺ . 0 system at that selected wavelength. We show the results of the numerical integration of the nonlinear system for the fastest reaction rate fields. We solve for P at each grid point using a ␤ and for four different values of the pressure gra- tridiagonal matrix solver (Press and others 1992). dient ␯, which is defined here, for our finite do- ␯ϭ Ϫ We then use the adjusted pressure field in the next main, as (Pl Pr)/X, where X is the length of time-step integration of the nutrient and solid domain in x and Pl, Pr are the presssures at the left fields. Convergence tests and comparisons with lin- and right boundaries, respectively (in the cases con- Ͼ ear stability analytic solutions suggest that the divi- sidered here, Pl Pr). In these numerical calcula- sion of the domain into 201 grid points provides tions, we consider how the nonlinearities in the sufficiently precise solutions. autocatalysis affect the shape and propagation of an Because the system is open, we expect the nutri- infinitesimal perturbation. For ␤ϭ100, the fastest ent and solid fields to obey Eqs. (3), (4), and (5) at reaction parameter examined, we show the devel- the boundaries. The first derivatives of pressure, P, opment of solid growth fronts for a range of four at the endpoints are approximated by making use of flow parameters (Figure 7). These results show the second-order Taylor expansions of the two strong asymmetry between the peaks and troughs, points interior to the boundary, expanded around suggesting that the autocatalysis in the chemical the boundary itself. For the nutrient ␣ and solid ␺, reaction acts to accelerate the growth of the solid derivative expressions are approximated by the de- maxima. With respect to the linear system, growth rivatives of the nearest interior point. Because we is greatly accelerated (Figure 8). cannot specify ␣ and ␺ at the endpoints, we cannot Examining the behavior of the associated nutri- make use of the Taylor expansion about the bound- ent fields for ␤ϭ100, ␯ϭ50 permits a good aries, as in the case of pressure. illustration of the system’s behavior (Figure 9). Since there are an infinite number of nonlinear Here we find that the sharply growing solid fronts solutions, here we seek only to extend the results of deplete the nutrient field and that the downstream the linear stability analysis into the nonlinear re- recovery of the nutrient field controls the spacing of gime. Thus, we consider small perturbations to the the solid fronts. Also, we see that the fronts prop- Theoretical Model of Coral Reefs 69

Spacing of the fronts is invariably controlled by the competition among (a) the propensity for solid structures to introduce sharp pressure gradients through obstructions that impede flow and thus more effectively capture nutrients; (b) the diffusion of sharp gradients in both the solid and (especially in the cases considered here) nutrient fields; and (c) both the decay and recovery length scales over which the nutrient content in a parcel of fluid is, while being swept downstream, either lost to reactions with the solid or replenished by surface supply. We find that the slowest-reacting systems are the most sensitive to changes in the imposed fluid flow (or pressure gradient) in terms of the growth rates Figure 9. Normalized solutions for (from top to bottom) and selected wavelength for the growing structures ␺, the solid matrix, and ␣, the nutrient field, versus x for (Figure 6a and b), suggesting that slow-growing the time integration of the nonlinear system for pertur- corals are most succeptible to environmental plas- ␤ϭ ␯ϭ bations to the steady state for 100, 50. See Figure ticity. Also, we expect this plasticity to be the most 7 for information about contours and normalization. evident over slow flow regimes. We further find that for the fastest reactions, as well as the fastest flow rates, long wavelengths dominate (Figure 6b), agate upstream. For the ␤ϭ10 and ␤ϭ5 cases the suggesting that faster-growing systems might ex- results are very similar, although growth is much hibit broader structures. slower and the asymmetry is not as extensively As we have mentioned, there are morphologies exaggerated by the end of the time integrations of coral reefs that orient perpendicular to flow, with (Mistr 1999). regular spacing of the growth plates parallel to flow (Figure 1). Our theory suggests that this type of reef DISCUSSION OF THE MODEL is organizing to best intercept nutrient supply, whereas nonlinear growth permits the develop- The main result from the nonlinear analysis is the ment of sharp growth fronts in the face of a one- development of pronounced nutrient and solid fronts that propagate upstream. Sharp gradients in dimensional flow. Additionally, the regular spacing the nutrient field develop as the solid depletes the of reef cays off the coast of Queensland parallel to available nutrient by reacting autocatalytically to the main direction of the East Australian Current create additional solid that can further deplete the leads us to anticipate a similar mechanism for reef nutrient. Growth of the solid due to the autocatal- organization (Figure 2). Evidence of upstream ysis is greatly accelerated with respect to the linear propagation in each of the colonial and large-scale system, and this rapid growth must far exceed dif- cases is difficult to ascertain, although further in- fusive loss to allow for the development of the solid vestigation of these areas would reveal clues. Ex- fronts. These fronts are most pronounced for in- amining the paleohistory of Bunker Reef would creased reactions rates and increased flow rates. We help to establish the validity of the hypothesis of can conclude from this analysis that without the nutrient-controlled growth patterns. Additionally, nonlinearities of the autocatalysis in the reaction it would be enlightening to determine whether A. chemistry, the development of sharp fronts is not tenuifolia advances into the flow as it grows addi- possible. The self-enhancing behavior of the solid tional plates. Because the nondimensional scales acts to deplete local nutrient concentrations include hypothetical reaction components, it is not sharply, so that growth occurs most rapidly in the informative to consider the physical scaling of the upstream direction, where nutrient supply is great- spacing. est. To some extent, the solid structures act as nu- The model presented here is quite simplified, and trient filters; thus, it is not surprising that they thus many possibilities exist for further theoretical accumulate and absorb nutrients faster on the sup- modeling of reef systems. As information comes to ply side of the structures. The areas of high solid light regarding the relationship of photosynthesis to concentration act to impede flow, increasing the calcification, a simple model of the chemical dynamics potential for nutrient depletion and further growth. of calcification would be very appropriate. 70 S. Mistr and D. Bercovici

Appendix 1. Derivation of the Theoretical chemical diffusion and advection of mobile species Model and therefore make the following fundamental as- sumptions. First, we consider that the dissolved spe- Let us suppose a system wherein we have a porous cies (nutrients) do not contribute to the volume of ␺ ⅐ Ϫ3 matrix of solid material, in mol m ,infiltrated the fluid infiltrating the porous matrix. Second, the ␣ with fluid containing dissolved nutrient in solid is considered immobile (with respect to advec- ⅐ Ϫ3 mol m . This dissolved nutrient undergoes a reaction) although subject to diffusive loss. tion with the solid material that generates addi- Loss of solid mass is accounted for solely by ef- tional solid material (Figure 5). In other words, the fective diffusion of ␺ representing, for example, system is subject to the autocatalytic production of disintegration and erosion near the perimeters of ␺ . Stated symbolically; coral reef concentrations that are represented by

k sharp gradients in ␺; thus, its time rate of change is: l␣ ϩ m␺ ¡ n␺ (A1) ␺ץ 2 2 (ϭ D␺ٌ ␺ ϩ k␣␺ (A5 tץ with a rate constant, or activity coefficient, k, where units depend on the stoichiometric coefficients, l, m, We relate the velocity of the fluid to the evolving and n, where n ϭ m ϩ l. The stoichiometric coeffi- pressure gradient by Darcy’s law (Bear 1988): Ϫ3 cients specify the number of mol ⅐ m that react and K ͑1 Ϫ ␥␺͒2 ٌ ϭ Ϫ 0 are produced. The law of mass action states that the q ␩ P (A6) rate of a chemical reaction is proportional to the product of the concentrations of the reactants raised where q is the Darcy velocity vector, ␩ is fluid to their stoichiometric coefficients (Bailyn 1994; viscosity, ␥ is the volume that one mole of pure Nicolis 1995). In an ideal system, this rate would be solid occupies such that 1 Ϫ␥␺is the matrix poros- 3 Ϫ␥␺ 2 given by the product of the number of moles per m ity, and K0(1 ) is the porosity-dependent per- of the reactants alone, which gives the statistical meability in which K0 is a reference permeability. probability of the two species coming into contact; Conservation of fluid mass (assuming the fluid is however, in reality, a rate constant, or activity co- incompressible) prescribes that: efficient, such as k must be used as a proportionality ץ (factor to account for the gap between the statistical ͑1 Ϫ ␥␺͒ ϩ ٌ ⅐ q ϭ 0 (A7 tץ -probability and the actuality of the reaction occur ring (Nicolis 1995). To constrain the stoichiometric (see Mistr 1999; also Bercovici and others 2001 and coefficients for the hypothetical reaction, we picked references therein), which leads to: l ϭ 1 for simplicity and investigated the stability of ␺ץ a system for various m and found that the system is 2␥ ␥␩ ϭ ٌ␺ ⅐ ٌ Ϫ 2ٌ (A8) ץ unstable, yielding solid growth only in cases where P Ϫ ␥␺ P ͑ Ϫ ␥␺͒2 1 K0 1 t m Ն 2 (Mistr 1999). We chose the lowest-order unstable case for our model, given by: With an expression for the velocity of the fluid, we may then consider the evolution of the nutrient k ␣ ␣ ϩ 2␺ ¡ 3␺ (A2) field. The time rate of change for the nutrient, ,is controlled by advection, diffusion, and reactive loss, leading to: Ϫ Ϫ ␣ץ ,in which case k’s units are m6 mol 2 s 1. Thus 2 2 (according to the law of mass action, for rate of ϩ ٌ ⅐ ͑q␣͒ ϭ D␣ٌ ␣ Ϫ k␣␺ (A9 tץ change of fluid and solid material (with no flow or other sources of mass): Our governing equations are (A5), (A6), (A8), and (A9), which interrelate advection of our nutrient to ␣ץ ϭ Ϫ ␣␺2 growth of the solid. We consider the system to be (k (A3 ͪ ץ ͩ t reaction continuous horizontally, in x and y, with finite vertical thickness, H. We supply nutrient to the system ␺ץ ϭ ␣␺2 by holding the nutrient density at the top surface k (A4) ␣ϭ␣ ϭ ͪ ץ ͩ t constant, s at z H. Holding the left entrance reaction ϭ to the channel (x 0) at a constant pressure, Pl, These two expressions provide the chemical source and maintaining the right exit at a constant lower

and sink terms for our model. We must consider pressure, Pr, we force fluid in the x direction, al- Theoretical Model of Coral Reefs 71 though the resulting flow with various obstructions tions and merely change the dimensional scales of due to heterogeneity in the matrix can be in any the system by a few tens of percent. direction. To simplify the system to a two-dimen- These equations are nondimensionalized accord- sional model, we integrate the equations over the ing to natural length, time, mass, and molar scales, H H2 ␩H5 H3 channel depth, z, assuming that vertical profiles of ϭ ϭ ϭ ϭ L 1/ 2 , T , M 5/ 2 , N ␥ 3/ 2 , the solid and the nutrient are parabolic to first order 3 3D␣ 3 D␣K0 3 in order to match the top and bottom boundary respectively, resulting in (dropping the overbars for conditions. Thus, the nutrient concentration ␣ is simplicity): constrained at the top by the surface nutrient sup- ␣ץ ply ␣ , and changes parabolically to zero gradient at s Ϫ ٌ ⅐ ͓͑1 Ϫ ␺͒2␣ٌ P͔ ϭ ٌ2␣ ϩ ␣ Ϫ ␣ Ϫ ␤␣␺2 t h h h sץ the bottom (that is, an effectively insulating bottom boundary through which neither flow nor particle (A16) diffusion can occur). The solid concentration ␺ is ␺ץ held to zero at the top and increases parabolically to (zero gradient at the bottom (again, the bottom is ϭ ␭ٌ2␺ Ϫ ␭␺ ϩ ␤␣␺2 (A17 t hץ effectively insulating). From these boundary condi- ␺ץ tions we assume that: 2ٌ ␺ ⅐ ٌ P 1 ϭ h h Ϫ 2 ٌ (A18) ץ h P ͑ Ϫ ␺͒ ͑ Ϫ ␺͒2 ␣͑ ͒ ϭ ␣ ϩ ͑␣៮ ͑ ͒ Ϫ ␣ ͒ ͑ ͒ 1 1 t x, y, z, t s x, y, t s f z (A10) where: ␺͑x, y, z, t͒ ϭ ␺៮ ͑ x, y, t͒ f͑ z͒ (A11) D␺ ␭ ϭ (A19) where: D␣

3 z2 kH2 ͑ ͒ ϭ ͩ Ϫ ͪ ␤ ϭ f z 1 2 (A12) 2 (A20) 2 H 3␥ D␣

1 ͵H ͑ ͒ ϭ ␣៮ ␺៮ The first parameter, ␭, is the ratio of diffusion coef- and since H 0 f z dz 1, then and are the values of ␣ and ␺, respectively, averaged over z. ficients of the solid to nutrient and will be referred Substituting the above functions into Eqs. (A5), to as “the diffusion parameter.” The second nondi- (A6), (A8), and (A9), integrating in z, and eliminat- mensional parameter, which contains k, the rate ing q, we find: constant of the reaction, will be called “the reaction parameter,” ␤. -៮ ͑ Ϫ ␥␺៮ ͒2 Eqs. (A16), (A17), and (A18) are the dimension␣ץ K0 1 Ϫ ٌ ⅐ ͩ␣៮ ٌ Pͪ less governing equations that we use to analyze the t h ␩ hץ stability of our autocatalytic, nutrient-limited po- 3͑␣ Ϫ ␣៮ ͒ rous matrix; these equations are reproduced in the 2 s ៮ 2 (ϭ D␣ٌ ␣៮ ϩ D␣ Ϫ k␣៮ ␺ (A13 h H2 main text as Eqs. (3)–(5).

␺៮ 3␺៮ Appendix 2. Linear Stability Analysisץ ϭ ٌ2␺៮ Ϫ ϩ ␣៮ ␺៮ 2 (D␺ h D␺ 2 k (A14 ץ t H Here we consider the growth of perturbations to a spatially constant system, characterized by uniform ␺៮ץ ␺៮ ⅐ ٌ ␥␩ ٌ␥ 2 h hP distribution of solid and nutrient and an imposed, (2P ϭ Ϫ (A15ٌ ץ h ͑ Ϫ ␥␺៮ ͒ ͑ Ϫ ␥␺៮ ͒2 1 K0 1 t constant pressure gradient. For small perturbations, solutions take the form: where ٌ is the horizontal gradient. For nonlinear h ␣ ϭ ␣ ϩ ⑀␣ 1͵H m Ϸ 0 1 (A21) terms, we have assumed that H 0 f (z)dz 1 (where m is any integer), which is generally only ␺ ϭ ␺ ϩ ⑀␺ (A22) true for m Յ 3; however, more accurate integration 0 1 only introduces factors that are O(1), which can be ϭ ͑ ͒ ϩ ⑀ P P0 x P1 (A23) absorbed into the system by slightly redeﬁning con- ␥ ⑀ϽϽ stants such as K0, k, and . Since the equations are where 1. Substituting these solutions into the made nondimensional anyway (see below), these two-dimensional equations gives rise to the three factors have no effect on the ﬁnal governing equa- steady-state equations O(⑀0): 72 S. Mistr and D. Bercovici

␣ץ Ϫ ␣ ϭ ␤␣ ␺2 ␣ s 0 0 (A24) 1 ϩ 2␣ ͑1 Ϫ ␺ ٌ͒P ⅐ ٌ␺ 1 0 t 0 0 0 1ץ ␭␺ ϭ ␤␣ ␺2 0 0 0 (A25) Ϫ ͑ Ϫ ␺ ͒2ٌ ⅐ ٌ␣ Ϫ ͑ Ϫ ␺ ͒2␣ ٌ2 1 0 P0 1 1 0 0 P1 2 d P0 ␣ϭ 0 (A26) ϭ ٌ2␣ Ϫ ␣ Ϫ ␤␣ ␺ ␺ Ϫ ␤␺2 dx2 1 1 2 0 0 1 0 1 (A29) ␺ץ 1 (the solutions of which are the trivial case: ϭ ␭ٌ2␺ Ϫ ␭␺ ϩ 2␤␣ ␺ ␺ ϩ ␤␺2␣ (A30 t 1 1 0 0 1 0 1ץ ␺0 ϭ ␣ ϭ ␣ 0 0, 0 s (A27) ␺ץ 2ٌP ⅐ ٌ␺ 1 ϭ 0 1 Ϫ 1 2ٌ (A31) ץ and the two conjugate roots: P1 ͑ Ϫ ␺ ͒ ͑ Ϫ ␺ ͒2 1 0 1 0 t ␣ ␣2 1 s s 4 We assume that solutions to the above equations ␺Ϯ ϭ ͩ Ϯ ͱ Ϫ ͪ (A28) 0 2 ␭ ␭2 ␤ take the form: ␣ ϭ␣ Ϫ␭␺ ͑ ␣ ␺ ͒ ϭ ͑ ␣ ␺ ͒ ik⅐rϩst where for all solutions 0 s 0 and the P1, 1, 1 Pk, k, k e (A32) ϭϪ␯ ␯ 0th-order pressure gradient is dP0/dx , where ϭ where s is the growth/decay rate, k (kx,ky)isthe is constant. ϭ With pressure specifications, we have introduced horizontal wavenumber vector, and r (x, y)isthe a fourth parameter, ␯, and thus fix two of the horizontal position vector. Assuming solutions of nondimensional parameters so as to maintain a this type implies that for each wavenumber pair (kx, manageable parameter space. We assume that the ky) there is an associated growth or decay rate. variation in the diffusion coefficients is minimal, Since wavenumber is related to wavelength, estab- such that the ratio of solid to nutrient diffusion, ␭, lishing which wavenumber supports the fastest remains constant. We fix ␭ϭ0.001, corresponding growth allows us to predict the spatial scale over to nutrient diffusion that is 1000 times faster than which the solid field exhibits preferred growth. Ϫ␺ Substituting perturbation solutions (A32) into solid diffusion. Since the 0th-order porosity 1 0 Յ ␺ Յ Eqs. (A29), (A30), and (A31), we find the disper- must be between 0 and 1, then 0 0 1, thus ␣ ␭Ͻ ␣2␤ sion relation: demanding that s/ 1, as well as that s / 2 (4␭ ) Ͼ 1, to ensure real ␺ . Therefore, we set ␣ ϭ 2 ϩ ͓͑ ϩ ␭͒͑ ϩ 2͒ ϩ ␤␺ ͑␺ Ϫ ␣ ϩ ␣ ␺ ͒ 0 s s s 1 1 k 0 0 2 0 0 0 0.0009, which, recall, is nondimensionalized by the ϩ ␯͑ Ϫ ␺ ͒2͔ ϩ ͑ ϩ 2͓͒␭͑ ϩ 2͒ ϩ ␭␤␺2 molar volume of the pure solid. Dimensionally, this ikx 1 0 1 k 1 k 0 is effectively a surface concentration of around 200 Ϫ ␤␣ ␺ ͔ ϩ ␯͑ Ϫ ␺ ͒2͓␭͑ ϩ 2͒ Ϫ ␤␣ ␺ ͔ mol/m3, assuming solid density for the calcium car- 2 0 0 ikx 1 0 1 k 2 0 0 3 bonate skeleton to be around 2.5 g/cm . Setting ϭ 0 (A33) ␣ ␭ constant values for two of the parameters, s and , 2 ϭ 2 ϩ 2 permits us to examine changes in the system as the where k kx ky. Eq. (A33) has two complex background pressure gradient, ␯, and the reaction roots, sϮ (where the “Ϯ” subscript indicates the sign parameter, ␤, are varied. of the radical term in the quadratic formula solu- The trivial case (A27) implies that there is no tion), only one of which, sϩ, allows growth. matrix present and that the nutrient distribution We explore the growth rates Re(sϩ) versus k ϭ matches the surface supply. The two quadratic (kx,ky) over a wide range of the nondimensional roots, (A28), represent steady states characterized parameters for the three steady states and find that ␺ϩ ␺Ϫ by either high ( 0 ) or low ( 0 ) solid density. The in all cases, the maximum growth rate max(Re(sϩ)) ϭ high solid-density steady state evolves when there occurs for ky 0. As an example of this result, we ϭ is not enough nutrient to react and support growth show the growth rate Re(sϩ) versus k (kx,ky) for beyond that which balances diffusive loss, whereas one reaction parameter ␤ over a range of pressure ␯ ␺Ϫ the low solid-density state evolves since there is not gradients, for the steady state 0 (A28) (Figure ␺Ϫ enough solid to react and sustain growth beyond A1). We select 0 because growth is sustained for that which is lost to diffusion. perturbations to this steady state, whereas growth is ␺0 The stability of each of the high- and low-density sustained for neither the trivial case 0 nor for the ␺ϩ matrix steady states, as well as the trivial case, is high solid-mass concentration case 0 over the examined using equations O(ε1) from substitutions wide range of parameters that were sampled. Here of solutions (A21), (A22), and (A23) into the two- we see that the fastest growth rate for low solid- ␺Ϫ dimensional Eqs. (3), (4), and (5): mass concentration case 0 occurs at kx 0 and ky Theoretical Model of Coral Reefs 73

Figure A1. Two-dimensional dispersion relation for pertur- ϩ ϩ bations of the form ei(kxx kyy) st ␺ to the equilibrium state 0 ␣ ϭ given by Eq. (6) with s 0.0009, ␭ϭ0.001, ␤ϭ100. Surfaces show the growth rate of the perturbation, Re(sϩ) (where sϩ is the one of two possible solutions for (a) that allows growth) versus wave- ␯ϭ numbers kx and ky for (a) 5, (b) ␯ϭ10; (c) ␯ϭ50, (d) ␯ϭ100. Note the symmetry in

kx and that the least stable mode (location of largest ϭ Re(sϩ)) always occurs at ky 0.

Figure A2. Growth rate Re(sϩ) versus wavenumber k for perturbations to the low solid-density quadratic root ␺Ϫ 0 (solid curves), to the high solid-density quadratic root, ␺ϩ 0 (dash–dot curves), and to ␺0 the trivial steady state 0 (dashed curves). Plots are for increasing reaction parameter, ␤ from left to right (␤ϭ5, 10, 100), and for increasing imposed pressure gradient, ␯, from top to bottom (␯ϭ5, 10, 100).

ϭ ϭ 0. Nonzero kx implies that there is a finite wave- we set ky 0 and analyze the system for spatial length that characterizes the fastest growth in x, dependence in x. ϭ while ky 0 implies that growth is uniform in y, We show for the one-dimensional case that perpendicular to the imposed pressure gradient. growth is not sustained (max(Re(s)) Ͻ 0) for the Thus, for further analysis of the dispersion relation, trivial case, and growth is sustained only over very 74 S. Mistr and D. Bercovici restricted ranges for the nutrient-limited steady Jain MK. 1984. Numerical solution of differential equations. ␺ϩ New York: Wiley. state 0 (Figure A2). For the solid-limited steady ␺Ϫ Jupp DLB. 1985. Landsat based interpretation of the Cairns state, 0 , however, growth is sustained over a wide range of parameters. This result does not contradict section of the Great Barrier Reef Marine Park. CSIRO Div Water Nat Resou 4:1–51. our findings for sampled parameter ranges for the two-dimensional case. We thus examine only this Kaandorp JA, Lowe CP, Frenkel D, Sloot P. 1996. Effect of ␺Ϫ nutrient diffusion and flow on coral morphology. Phys Rev consistently unstable root, 0 , in the main text. Lett 77(11): 2328–2331. Kawaguti S, Sakumoto D. 1948. The effects of light on the REFERENCES calcium deposition of corals. Bull Oceanogr Inst Taiwan 4: Aharonov E, Whitehead JA, Kelemen, PB, Spiegelman M. 1995. 65–70. Channeling instability of upwelling melt in the mantle. J Geo- Lewis DH. 1973. The relevance of symbiosis to taxonomy and phys Res 100(10): 20433–20450. ecology, with particular reference to mutualistic symbioses Bailyn M. 1994. A survey of thermodynamics. College Park, and the exploitation of marginal habitats. In: Heywood VH, (MD): American Institute of Physics. editor. Taxonomy and ecology. New York: Academic Press. p Baker PA, Weber JN. 1975a. Coral growth rate: variation with 151–172. depth. Earth Planet Sci Lett 27: 57–61. Maxwell WGH. 1968. Atlas of the Great Barrier Reef. Amster- Baker PA, Weber JN. 1975b. Coral growth rate: variation with dam: Elsevier. depth. Phys Earth Planet Int 10: 135–9. Mistr S. 1999. Pattern formation in a reaction-advection-diffu- Bear J. 1988. Dynamics of fluids in porous media. Mineola (NY): sion system with applications to coral reefs. [thesis]. Honolulu: Dover. University of Hawaii. Bercovici D, Ricard R, Schubert G. 2001. A two-phase model for Muscatine L. 1973. Nutrition of corals. In: Biology and geology compaction and damage. I. General theory. J Geophys Res of coral reefs. Jones OA, Endean R, editors. New York: Aca- 106: 8887–8906. demic Press. p 77–116. Borowitzka MA. 1987. Calcification in algae: mechanisms and Nicolis G. 1995. Introduction to nonlinear science. Cambridge the role of metabolism. CRC Crit Rev Plant Sci 6 (1): 1–43. (England): Cambridge University Press. Chandrasekhar S. 1961. Hydrodynamic and hydromagnetic in- Porter JW. 1976. Autotrophy, heterotrophy, and resource parti- stability. New York: Oxford University Press. tioning in Caribbean reef-building corals. American Naturalist Drazin PG, Reid WH. 1989. Hydrodynamic stability. New York: 110: 731–742. Cambridge University Press. Preece AL, Johnson CR. 1993. Recovery of model coral commu- Gattuso JP, Allemand D, Frankignoulle M. 1999. Photosynthesis nities: complex behaviours from interaction of parameters and calcification at cellular, organismal and community levels operating at different spatial scales. In: Complex systems: from in coral reefs: a review of interactions and control by carbon- biology to computation. Green DG, Bossomaier T, editors. ate chemistry. Am Zool 39: 160–183. Amsterdam: IOS Press. p 69–81. Goreau T. 1963. Calcium carbonate depostion by coralline algae Press W, Teukolsky S, Vetterling W, Flannery B. 1992. Numer- and corals in relation to their roles as reef builders. Ann NY ical recipes in FORTRAN. New York: Cambridge University Acad Sci 109: 127–167. Press. Graus RR, Macintyre IG. 1989. The zonation patterns of Carib- Stumm W, Morgan JJ. 1981. Aquatic chemistry. 2nd ed. New bean coral reefs as controlled by wave and light energy input, York: Wiley. bathymetric setting and reef morphology: computer simula- tion experiments. Coral Reefs 8:9–18. Turing A. 1952. Chemical basis of morphogenesis. In: Saunders Helmuth BST, Sebens KP, Daniel TL. 1997. Morphological vari- PT, editor. Morphogenesis. Amsterdam: Elsevier. ation in coral aggregations: branch spacing and mass flux to Veron JEN, Pichon J. 1976. Scleractinia of Eastern Australia. Part coral tissues. J Exp Mar Biol Ecol 209: 233–259. 1. Canberra: Australian Institute of Marine Science Mono- HilleRisLambers R, Rietkerk M, Prins HHT, van den Bosch F, de graph Series; vol. 1. Australian Government Publishing Ser- Kroon H. 2001. Vegetation pattern formation in semi-arid vice. grazing systems. Ecology 82: 50–61. Ware J, Smith SV, Reaka-Kudla ML. 1991. Coral reefs: sources ϩϩ Isa Y, Okazaki M. 1987. Some observations on the Ca -binding or sinks of atmospheric CO2? Coral Reefs 11: 127–130. phospholipids from scleractinian coral skeletons. Comp Bio- Yonge CM. 1930. Studies on the physiology of corals. I. Feeding chem Physiol 87B: 507–512. mechanisms and food. Sci Rep Great Barrier Reef Expedition, Jackson J. 1991. Adaptation and diversity of reef corals. Bio- 1928–29; Scientific Reports of the Great Barrier Reef Expedi- Science 41(7): 475–482. tion, 1928–1929 1: 13–57.