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Growth forms of hermatypic corals: stable states and noise-induced transitions
Ecological Modelling 141 (2001) 227– 239 www.elsevier.com/locate/ecolmodel
Growth forms of hermatypic corals: stable states and noise-induced transitions Z. Kizner a,*, R. Vago b , L. Vaky c a
Department of Physics and Faculty of Life Sciences, Bar-Ilan Uni6ersity, Ramat Gan 52900, Israel b The Institute for Applied Biosciences, Ben-Gurion Uni6ersity, Beer-She6a 8405, Israel c Faculty of Life Sciences, Bar-Ilan Uni6ersity, Ramat Gan 52900, Israel Received 14 July 2000; received in revised form 7 February 2001; accepted 26 February 2001
Abstract Field observations show that some coral species may assume different growth forms, or morphotypes, within apparently uniform habitats. The present paper examines the causes of this morphological diversity in hermatypic cnidarians and more particularly the following questions: Are there stable morphotypes among all the possible growth forms of a coral; if so, can a colony belonging to a certain stable morphotype change its growth form, and what can induce transition from type to type when the environment is relatively steady and only limited random (and short-term) variations of the external conditions occur? The approach adopted was to construct a mathematical model of the form dynamics of a coral colony and validate it by field data on the Red Sea hydrocoral Millepora dichotoma. The ratio between the volume and the surface area of a coral served as an index of its form and enabled treatment of the coral morphology in terms of dynamical systems. Such a dynamical system must have certain stable states corresponding to specific growth forms of a coral colony. Computer simulations showed that limited stochastic disturbances in the processes of biomass and skeletal growth caused by random fluctuations in environmental conditions may induce transitions of a coral from one stable growth form to another. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Coral; Dynamical system; Environmental fluctuations; Growth form; Modelling; Stability; Transition
1. Introduction
This study is dedicated to the memory of one of the authors, Lior Vaky, who was accidently killed while conducting field work on the biology of corals in the Red Sea. * Corresponding author. Tel.: +972-3-5318244; fax: + 9723-5351824. E-mail address: [email protected] (Z. Kizner).
Reef corals typically display high morphological diversity, which is apparent at both the interand the intra-specific levels. Growth of corals depends strongly on environmental conditions (Yonge, 1968; Mergner, 1971; Jokiel, 1978; Denny et al., 1985; Roberts et al., 1992), changes in the environment being reflected both in the calcifica-
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tion process (Weber, 1974; Graus and Macintyre, 1976; Barnes and Lough, 1989, 1997) and in the growth of living tissue (Vago et al., 1998). Therefore, coral colonies of the same species living under different external conditions may grow into different forms (Weerdt de, 1981, 1984; Lewis, 1989; Vago et al., 1998), while a long-term change in the environment may cause a long-term transformation in the form of a coral colony (Weerdt de, 1981, 1984). However, field observations show that even within apparently uniform habitats, the same coral species may appear in different growth forms. In cells, algae, plants and animals it is useful to describe body form in a reductionist manner in terms of the ratio of volume to surface area (Lewis, 1976; Foy, 1980; Reynolds, 1984; Thompson, 1992; Niklas, 1994). Such ratios generally have a physiological, an energetic, and also a biomechanical meaning (Foy, 1980; Reynolds, 1984; Thompson, 1992). This ratio is also convenient in the case of hermatypic cnidarians, in which skeleton deposition proceeds from a thin, superficial veneer of living tissue into a volume. Barnes (1973) was apparently the first to deal with volume and surface area in connection with growth of massive corals (see also Barnes and Taylor, 1993; Darke and Barnes, 1993). However, to date, the volume to surface ratio has not been proposed as a coral form index. In the present paper, we apply this index to examine whether the development of a coral into a particular form follows a random path representing one of a multitude of conceivable directions, or whether the number of stable growth forms possible is limited by coral physiology. We then go on to assess the possibility of modeling the dynamics of coral form using nonlinear dynamical systems, and finally we establish whether stochastic fluctuations or ‘external noise’ superimposed upon a steady environment would induce transitions from one stable state of the system to another. Data on the Red Sea hydrozoan Millepora dichotoma constitute the numerical basis for this study. In response to ‘external noise’, corals transit mainly from delicate growth forms to more robust ones. Thus relative occurrence of different morphotypes may be an important indicator of
the load of short-term environmental disturbances upon ecosystems of coral reefs, even though in the long-term perspective the environment can be considered steady.
2. Materials The growth forms of colonies of Red Sea M.dichotoma Forskal were examined at two sites exhibiting characteristic but distinct types of bottom topography in the northern part of the Gulf of Elat (Aqaba), 29°30% N, 34°55% S (for map see Vago et al., 1998). At Site 1 (Nature Reserve region), 369 colonies were examined within a belt 40 m long and 4 m deep (0–4 m). This site is rather sheltered and protected from waves due to the shape of the coastline (Dafni and Tobol, 1986). Site 2 (in front of the Inter-University Institute of Elat), where 502 colonies were examined along a belt 20 m long and 6 m deep (0–6 m) parallel to the coastline, is exposed to strong winter storms and southern waves. The coastal slope consists of rubble with scattered patches of coral (Dafni and Tobol, 1986). Since light changes rapidly with depth (Mergner, 1971), light availability must generally be taken into account when considering the distribution of coral forms over a large range of depths. However, in the shallow oligotrophic waters of the northern part of the Gulf of Elat, coral colonies maintain maximal photosynthetic rates even under the lower winter irradiance levels, to which they undergo photoacclimation (Falkowski and Dubinsky, 1981; Dubinsky et al., 1984; Falkowski et al., 1990). Therefore, light is not a consideration in the context of our analysis, and we do not distinguish between data obtained from different depths. Hydrodynamically, both sites are relatively homogeneous. During the surveys, the dimensions of each non-encrusting colony were measured with a vernier caliper to provide an estimate of colony volume V (cm3) and surface area S (cm2) covered with living tissue. Colony form was approximated in terms of combinations of standard geometrical figures (cylinders, prisms, pyramids, spherical segments etc.), and V and S were calculated using
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known formulae. Since only large-scale measurements were carried out, with characteristic colony size ranging from a few millimeters to 1 cm (Fig. 1), the values obtained for the area of the skeletal surface were not affected by the roughness or possible fractality of the latter. Based on these data, the ratio L = V/S was calculated for each colony. Experiments with a number of dead skeletons of M. dichotoma belonging to different morphotypes proved that the possible relative error in these estimates did not exceed 25%. For encrusting colonies, the most widespread form at both sites, L was determined as the range of thickness of the calcium carbonate layer (not including area of bumps; see below for definitions of the main morphotypes), that is 0.0– 0.06 cm.
3. The model
3.1. Equations of tissue growth and calcification The skeleton of a coral is composed of calcium carbonate deposited by the living tissue. The layer of living tissue that covers the surface of the skeleton is so thin that the mass and the volume of a coral colony are actually those of its skeleton. Let C be the mass of the skeleton and B the biomass of the living tissue of a coral colony. We may describe the dynamics of B and C by the following equations: dB =rBB, dt
(1)
dC =rCB, dt
(2)
where t is time, and rB and rC are the growth rate of the biomass (tissue) and the calcification rate per unit of biomass, respectively. Due to the thinness of the tissue layer, it may be assumed that the tissue biomass B is proportional to the surface area S of the skeleton: B = vS. Since we are looking at the long-term dynamics of corals, where the time scale considerably exceeds a season or a year, C can be assumed to be proportional to the colony volume V, i.e., C =wV, and the coefficients of proportionality, m and n, can be
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taken as constant. (In principle we could increase the universality and accuracy of the model by taking into consideration temporal variability in skeletal density due to, say, seasonality.) Hence it is legitimate to use area and volume units (say, cm2 and cm3) to measure tissue and skeletal mass, that is, to assume that v = w=1 and thus C= V and B =S. Eq. (1) is a conventional equation for the dynamics of the biomass of populations and colonies (see, e.g. Odum, 1971; Edelstein-Keshet, 1988), and it says that the living tissue reproduces itself at the rate rB per unit biomass. According to Eq. (2), every unit of the tissue mass participates in building the skeleton by depositing calcium carbonate at the same rate rC. Such a relation is clearly appropriate for massive corals (like Porites), which grow in all three dimensions without exhibiting any preferred growth in certain particular directions, as well as for actually two-dimensional encrusting colonies. However, we will assume that this equation also holds for other colonies, i.e. that it represents a general relationship between the tissue biomass and rate of skeletal mass accretion independently of the form of a coral. Such an assumption is in agreement with the results of Pearse and Muscatine (1971), Taylor (1977), Rinkevich and Loya (1983), and Rinkevich and Loya (1984) for Acropora cer6icornis, Montastrea annularis and Stylophora pistillata showing transport of fixed carbon from peripheral areas to growing tips of these branching corals, i.e. participation of remote tissue parts in the deposition of calcium carbonate at branch tips. The validity of Eq. (2) for branched colonies can also be supported by the following reasoning. Consider an idealized colony consisting of a single cylindrical branch that grows longitudinally without thickening. Let DV and DS be the volume and surface area accretions that take place within time interval Dt in such a brunch. The ratio DV: DS is equal to the ratio between the area of the cylinder’s base and its perimeter and is therefore independent of Dt (for example, in the case of a circular cylinder, DV: DS equals to half a radius of the cylinder; for details see the next subsection). In other words, for such a colony dV/dt and dS/dt are proportional to each other with a
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Fig. 1. Colonies of M. dichotoma ranked according to value of their form index: encrusting (a); lace-like (b – d); bladed (e, f), boxwork (g – i).
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constant factor of proportionality, and Eq. (1) immediately yields Eq. (2). This conclusion can be extended to the case of colonies composed of a number of cylindrical branches. In general, the rates rB and rC are not constant: they depend on a number of variables, including the current tissue and skeletal masses, B and C, and environmental conditions. Thus, the living tissue and skeletal masses are interdependent in our model; their self- and inter-limitations are discussed in the next subsection.
3.2. The form index and the dynamic equation Every specific form of a coral colony depends on how the colony volume, V, relates to the surface area of the colony, S. Therefore, the ratio L= V/S can be regarded as the form index representing the colony form quantitatively. In a previous work (Vago et al., 1998) four main morphotypes of M. dichotoma were defined and described, namely encrusting, lace-like, bladed and boxwork. The nearly two-dimensional, encrusting colonies have very thin skeletons, closely follow the contours of the substratum, and often have a few bumps on their surfaces (Fig. 1a). Clearly, the form index of an encrusting colony equals to the average thickness of its skeleton (a few hundredth of cm). Lace-like colonies are built of one or several parallel delicate erect ‘pages’ composed of dichotomous cylindrical branches (Fig. 1 b– d). In order to estimate the form index of such a colony let us suppose it to consist of N cylinders of equal average radius r, the latter being substantially smaller than their heights. Then the surface area and the volume of j-th branch can be assessed as 2yrhj and yr 2hj, where hj is the branch’s height. Correspondingly, the total surface area and volume are S =2yrSN j= 1hj and V=yr 2SN j = 1hj, respectively, and the form index is L= V/S =r/2. The typical branch diameter in our surveys was 0.4 cm (Fig. 1 b– d), so the value of L characterizing the lace-like colonies was 0.1 cm. The bladed morphotype can be described as a relatively massive, upright, leaf-like growth form in which most of the gaps between branches are closed, or partly closed; the top margins of such colonies often bear short
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branches (Fig. 1e and f). For the purpose of estimating the form index, a bladed colony can be regarded as a plate of thickness d (obviously, d] 2r) and base surface area s (area of single side). Its volume and total area (both-sides) can be estimated as V= sd and S= 2s, which gives L= d/2. The characteristic L value for bladed colonies was 0.3 cm (Fig. 1e and f). Finally, colonies belonging to the boxwork morphotype (Fig. 1 g-i), i.e., robust, fist-like, massive ones, grow essentially in all three dimensions. If X is the characteristic size of such a truly three-dimensional colony, then S8 X 2, V8X 3 and, consequently, L8X. So, unlike the situation in the forms considered above, in a boxwork colony growth implies an increase in the form index, and not only in colony surface area and volume. For this morphotype, therefore, the form index is characterized by the lower bound of possible values of L. According to our data, for box-work colonies L] 0.93 cm. Thus we see that the form index defined as a volume-to-area ratio has a clear physiological interpretation as it relates the coral growth form to the amounts of the skeleton (calcium carbonate) deposited by the living tissue and the tissue itself. As both the tissue growth and the calcification process are responsive to the changes in the environment, parameter L acquires also an ecological meaning. Introducing the concept of the form index facilitates ranking of the various coral growth forms, as manifested in the small values of L for thin and delicate growth forms versus the large values for massive colonies (Fig. 1). However, another merit of using L is the possibility it offers of analyzing the form dynamics of a coral colony in terms of dynamical systems, a mathematical notion with numerous applications in physics, chemistry and biology (see, e.g., Horsthemke and Lefever, 1984; Ornstein, 1974; Stanly, 1971). Using the definition of the form index, L=V/ S= C/B, we obtain the following equation for the form dynamics from Eq. (1) and Eq. (2): dL = rC − rBL. dt (for mathematical details see Appendix).
(3)
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In general, as noted above, the rates rB and rC depend on a number of parameters, including environmental ones. So far we have been discussing the morphological plasticity of corals under uniform, relatively steady conditions. In reality, however, even when the environment can be considered steady in the long-term perspective, limited short-term random fluctuations do occur. These cause relatively small random variations in the physiological parameters rC and rB and, hence, variations in the rate of change of the form, dL/dt. So, we may regard both the calcification and the tissue growth rates as sums: rC = c+m, rB = b+i, where c and b, independent of time, are the mathematical expectations, or deterministic components, of rC and rB, respectively, while the fluctuations m and i (‘external noise’) are stationary stochastic processes (see, e.g., Horsthemke and Lefever, 1984). Tissue growth in a coral is limited by the tissue biomass itself, as well as by the ability of tissue to grow skeleton. Hence, the deterministic component of the tissue growth rate, b, is a function of both S and V. Similarly, the deterministic component of the calcification rate, c, must generally be regarded as a function of S and V. Undoubtedly, among the possible growth forms there are some that are better or worse adapted to harvesting light energy and nutrients and hence exhibit higher or lower rates of tissue growth and calcification. So it would be quite realistic to assume that the rates b and c depend only upon the form index L = V/S, i.e. b =b(L) and c= c(L). Eq. (3) then becomes: dL = F(L)+ (m−iL), dt
(4)
where F(L) = c(L)+ b(L)L. A rapidly fluctuating environment can be characterized by the property that the correlation time of the stochastic processes representing the external noise is much smaller than the typical time of the macroscopic (or deterministic) evolution of the dynamical system. Such a short-term environ-
mental effect is conventionally modeled by Gaussian white-noise processes (Horsthemke and Lefever, 1984). That is what will be assumed regarding m and i.
3.3. Steady states and stability In our model, the deterministic rates b and c remain constant provided the coral does not change its morphology while growing, that is, provided the dynamical system remains in a steady state (equilibrium) in which the tissue and skeletal growth have become adapted to each other and L is constant. As the dynamical system (4) is nonlinear it may possess a number of steady states. For example, when a colony grows as an encrusting, lace-like or bladed morphotypes, its form index L remains nearly constant and each of the three colony forms can be regarded as a steady state. Indeed, as they grow, encrusting colonies expand in area but retain almost the same thickness of the calcium carbonate layer (Vago et al., 1998). Similarly, the branches of lace-lake colonies are almost cylindrical and grow nearly longitudinally without changing radius, so L=r/2= const. A bladed colony increases in volume and surface area mainly due to apical growth of new shoots on its periphery and subsequent gap filling (Ben-Zion et al., 1991), and its skeleton thickness d actually does not change; as a result, L= d/2= const. On the other hand, the boxwork morphotype is not steady because, as showed above, the form index of the colony increases as it grows. In general, some of the steady states of a dynamical system may be stable and others not. To understand the meaning of stability in the context of coral growth, let us consider (just for the sake of simplicity) the particular case of absence of noise and constant rates c and b. In this case the function F(L) will be linear, and the dynamical system represented by Eq. (4) will have a single equilibrium state, i.e., the derivative dL/dt is zero only at L=c/b (Fig. 2). This state is stable, which means that the ratio between c and b forces the coral colony to maintain a stable proportion between its volume and surface area, that is, to assume a certain form. (As the relation between
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Fig. 2. Stability of the steady state, L =c/b, of the dynamical system (3) at constant rates b and c.
Fig. 3. Frequency distribution of the estimated values of the form index L for colonies of M. dichotoma at two sites (tail of distribution in inset): Site 1 — the Nature Reserve (a), Site 2 — in front of the Inter-University Institute of Elat (b).
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form and ratio V/S is not a one-to-one correspondence, it is impossible, based only on the known stable value of L, to predict what exactly this form will be, it is basically an intrinsic property of the species. For example, the branched form of M. dichotoma is not a common ‘tree’ but just a dichotomous lace-like structure.) The stability of the steady state L= c/b in this example arises from the negative slope of the graph of the function F(L) (i.e., of dL/dt vs. L) in Fig. 2, and can be explained as follows. Let us assume that for some reason L has slightly deviated to the left. In this region, to the left of the steady state point L= c/b, the rate of change of L is positive. Therefore, the form index L will tend to increase and return to the point L= c/b. Similarly, L will recover if shifted to the right. Figuratively, to use a physical analogy, one might say that the stable state acts as a point of attraction for the dynamical system. Correspondingly, the dynamical system under consideration may possess more than one stable steady state only if the function F(L) is nonlinear and its graph crosses the L-axis a number of times with negative slope. The presence of weak fluctuations in the system does not essentially change this conclusion. What is the actual shape of the function F(L)? It could clearly be different for different species and sites. In order to give concrete expression to our model, frequency distributions of the form index L were evaluated based on our data for Red Sea M. dichotoma from each site (Fig. 3a and b). The histograms demonstrate the existence of three modes at L values of about 0.00–0.06, 0.190.005 (for both sites) and 0.390.03 cm (Site 1), and one additional, though small group of data with relatively high values of the form index, L] 1 9 0.25 cm (Site 1). The confidence intervals of the modal values of L (the figures carrying9sign) confirm the reliability of the above modes and were obtained by bootstrapping the data (Efron, 1982): 1000 replicates were made based on the assumption that the distribution of the possible error in L data is log-normal with 25% S.D. (see section 2). The three modes correspond to the three main morphotypes of M. dichotoma, that is, encrusting, lace-like and bladed.
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Fig. 4. Hypothetical shape of the nonlinear function F(L) corresponding to M. dichotoma at Site 1 and yielding existence of three stable states, encrusting, lace-like and bladed forms (Len, Lll and Lbl), and a phase of boxwork growth (L\ Lbw) (full range graph in inset).
The existence of three distinct modes and the grouping observed in the histograms signify an increase in the occurrence of colonies of the corresponding form as L approaches one of the modal values, L= Lm. In terms of the dynamical system (4), this will happen if the state L = Lm possesses the property of ‘attraction’, that is, in either of two cases: (i) the modal point L =Lm is a stable steady state, and hence numerous colonies gather in its immediate vicinity with time; (ii) L changes at a slow rate in the vicinity of the modal point and the colony lingers in that region for a long time before eventually proceeding beyond. In either of these cases, an ‘instantaneous’ survey would provide a local maximum of the form index frequency distribution. However the second scenario does not seem to be relevant to our data. Indeed, if, say, Len were to represent a slowdown point, and not a point of stable equilibrium, then each colony would have to pass it sooner or later, and the colony form index would stabilize at one of the subsequent modal points, Lll or Lbl, or even proceed to the region of boxwork growth (L \ Lbw). But in this case the probability for the occurrence of encrusting colonies would be less than for the next stable morphotype in the series (or, if neither L= Lll nor L = Lbw were points of stable equilibrium, less than the probability for the occurrence of the boxwork colonies). The same reasoning applies if we take L = Lll or L = Lbw as points of slowing down. However, our
data reflect a different situation: the number of encrusting colonies (0BLB 0.06 cm) significantly exceeds the total number of lace-like and intermediate, apical shooting colonies (0.06B LB 0.16 cm), and the latter in turn is much larger than the number of bladed and transitional lace-likebladed colonies (0.16B LB 0.36 cm). Finally quite a few boxwork colonies (L\0.93 cm) were found only at Site 1 (Fig. 3 a, b, see also Vago et al., 1998). These data dovetail nicely with the first scenario and testify to the stability of the three main morphotypes of M. dichotoma. Based on this we identify the stable states of the dynamical system under consideration with the three modes, Len, Lll and Lbl. As shown above, the graph of the function F(L) must cross the L axis in the points Len, Lll and Lbl with a negative slope to ensure the stability of the corresponding states. In Fig. 4 the hypothetical shape of the nonlinear function F(L) for M. dichotoma in the Gulf of Elat at Site 1 is presented. It affords three finite stable states (encrusting, lace-like and bladed growth) and monotonous increase of the form index in the boxwork region (L \Lbw) where dL/dt is positive (for details see Section 4). So, in fact, the only two hypotheses underlying our model are: (i) that of the stability of the three main states (Len, Lll, Lbl) and (ii) the assumption that c and b are functions of L.
4. Computer simulations and fitting the model For the purpose of computer simulations, Eq. (4) was replaced with its non-dimensional finitedifference analog (Korn and Korn, 1968): L(t+ dt)= L(t)+[F(L)+ m(t)− i(t)L]dt.
(5)
In Eq. (5) the time variable t is discrete, changing with a small finite step dt, and hence all the other variables, including m and i are also discrete. Eq. (5) may be regarded as a recurrent formula for the successive determination of the form index: once given an initial value L(0)= 0, every value of L(t + dt) can be calculated from the previous one, i.e., through L(t). Each of the variables m(t) and i(t) in Eq. (5) can be treated as
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a random series with independent elements normally distributed with zero mean and standard deviation |m or |i, respectively. In the case of stochastic differential equations this is a natural finite-difference approximation, and Eq. (4), which includes white-noise components of the tissue growth and calcification rates, m and i, can be obtained from Eq. (5) by means of going to the limit at dt 0 subject to the conditions |m dt = const and |i dt=const (Horsthemke and Lefever, 1984). As noted above, we regard the random variations in the rates rB and rC to be small in relation to the deterministic ones. This implies two more conditions on |m, si and dt. First, the random variations in L occurring within the time step dt must be small in relation to the macroscopic or deterministic changes in L, which can be estimated as F*dt [here F* is the characteristic value of the function F(L)]. Second, when the dynamical system approaches a stable steady state, the random deviations from it must be small compared with the distance between this stable state and the one nearest: otherwise the system would be able to leap to the region of attraction of another stable state even within one time step, i.e., the fluctuations would not be small. Therefore, the parameters |m, |i and dt should be chosen so that the products m(t)dt and i(t)Ldt will be much smaller
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than both the product F*dt and the differences between those pairs of zeros of the function F(L) that determine the stable steady states of the dynamical system (3), i.e., of Len, Lll, Lbl; and Lbw in the case of M. dichotoma. Finally, the time step dt must be much smaller than the time typical for the macroscopic (deterministic) temporal evolution of the system, which can be identified with the time of relaxation of the system towards a steady state. One may justifiably expect that both the stability of the main growth forms (encrusting, lace-like and bladed in the case of M. dichotoma) and relatively rare transitions from one stable growth form to another will be reproduced in simulation runs of the properly tuned model given by Eq. (5). We fitted the model to the description of the growth dynamics of M. dichotoma so that to render a frequency distribution of the form index (evaluated from numerous runs) similar to that presented in Fig. 3a. This was the method used to generate the details in the graph of the function F(L) shown on Fig. 4. The best fit parameters |m, and |i were estimated as |m = 2.2 and |i = 35 at dt=0.002. Goodness-of-fit for each of the checked combinations of parameters was verified by a Monte-Carlo computer experiment consisting of 100 runs followed by a 2-test. Each simulation lasted for T= 15 non-dimensional time units (T represented the life span of a coral colony in the model).
5. Results and discussion
Fig. 5. Monte-Carlo histogram of the magnitudes of the M. dichotoma form index L obtained based on 100 model runs [for function F(L) shown in Fig, 5, T= 15, dt= 0.002, |m, = 2.2, and |i =35] (tail of distribution in inset).
The histogram of the computed L values for the best fit combination of the model parameters is presented in Fig. 5. This is quite similar to the observed frequency distribution of the forms of M. dichotoma surveyed at Site 1 (Fig. 3a). The closeness of the two frequency distributions (Fig. 3a and 5) was tested using a 2-criterion for the partition into five classes: [0, 0.08], [0.08, 0.16], [0.16, 0.24], [0.24, 0.32], [0.32, 0.4]; the boxwork colonies were excluded from the analysis due to their small number in the field data. The result, 2 = 7.4B 2cr = 9.5, confirms the resemblance of these distributions at a 95% confidence level (h= 0.05, four degrees of freedom).
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Fig. 6. An example of simulated successive transitions of a colony of M. dichotoma: Life history of the colony (a); enlarged transition from the encrusting to the lace-like form (b); enlarged transition from the lace-like to the bladed form (c).
An example of simulated growth form dynamics that exhibits successive transition of a coral colony from the encrusting state to the lace-like and then to the bladed form with rather long periods of stable growth in each is shown in Fig. 6a. It is evident from Eq. (5) that the effect of random fluctuations in the rate of calcification on the rate of change of the form index remains statistically constant during the life history of a coral colony, as well as among colonies (as |m dt =const). By contrast, the effect of stochastic variations in the rate of tissue growth increases with the increase of the form index (as it is coupled with the product |iLdt). In the vicinity of the first, encrusting stable state, while the form index is small (within the range 0.005L 50.06), the effect of random variations in rC is the major one for the chosen model parameters. Beyond this range, the two effects first become comparable in order of magnitude, and then, in the vicinity of the second, lace-like state, the random variations
in rB become dominant. Based on the analysis carried out, the following scenario of transitions in coral growth can be suggested. While a coral grows as the encrusting morphotype, its form is primarily affected by stochastic variations in the calcification rate, which can be apparent in formation of bumps. This effect is possibly accompanied by minor changes in form caused by random variations in the tissue growth rates. Suppose that random perturbations in the environment, such as small-scale spatial variations in carbonate availability, cause a brief succession of predominantly positive short-term random changes in the calcification rate m, say, supersaturation of carbonate in the boundary layer over the colony surface. The form index L could than increase in value and enter the region of attraction of the next stable steady state, L\0.06 (Fig. 6b). An increase in L means that growth of colony volume outstrips growth of its surface. Some bumps that receive additional carbonate will con-
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tinue to develop to accommodate surplus skeleton material, giving rise to the so called apical shooting described in our previous paper (Vago et al., 1998), subsequent dichotomous splitting, and formation of a lace-like structure. Clearly, the probability for such a series of chiefly positive values of o is much smaller than for a series in which positive and negative values eventually balance each other. That is why transitions from encrusting to lace-like growth are rare. A lace-like colony (L 0.1) is more sensitive to the stochastic variations in rB than to those in rC. The term iLdt in Eq. (5), which represents the effect of these variations, carries a minus sign. Therefore, a lace-like colony may rapidly transform to a bladed or even boxwork morphotype (Fig. 6c, see also Fig. 1g) if environmental fluctuations induce a series of short-term, random but predominantly negative changes in rate of tissue growth, delaying the increase in surface area necessary to balance the increase in volume. Such a situation may arise, for example, when tissue growth is somewhat restrained by small but repeated damage, which is in agreement with our observations (Vago et al., 1998). Conformably with the small probability for an event of transition of coral form, most of the runs demonstrated encrusting growth within the interval of integration of Eq. (5), 05 t 5T. When a transition to the lace-like form did occurred in the course of a run, the simulated colony usually remained lace-like. The number of runs in which transition occurred from encrusting to the lacelike and thence to a massive form was relatively small, as can be seen in Fig. 5. So far, the transition of a coral colony from one stable growth form to the next higher (in terms of L) stable state was discussed. Indeed, such a development in the direction of increasing L is the most probable, however, this is not the only possible path. Along with the transitions to the closest states with higher L values, the model permits transitions in the opposite direction, to states with smaller L, as well as immediate ‘longdistance’ transitions, say, from lace-like to boxwork form, although the one-step transitions to higher L prevailed over the reversal and ‘long-distance’ ones in our 100 runs. The point is that the
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distances between zeros of the function F(L) are large enough in relation to |iLdt, and a transition from a non-encrusting state may in fact occur only if the random changes in the rate of tissue growth (the leading factor in such transition) cause net increase or decrease of L within a few successive time intervals dt. Since the effect of random changes in rB becomes stronger with the increase of L, this means that the probability of transition from Lll to the region of attraction of the point Lbl is higher than that of returning from Lll to the region of attraction of the point Len. The same is valid for the transitions from the bladed state to the boxwork form and from the bladed to the lace-like state, as well as regarding the ‘long-distance’ transitions as compared with the one-step transitions. Transitions from encrusting to lace-like (apical shooting phenomenon) and from lace-like to the bladed form have been described in previous works (Ben-Zion et al., 1991; Vago et al., 1998). The colony of M. dichotoma shown in Fig. 7 provides, in our opinion, convincing evidence of a retrogressive transition from bladed to lace-like morphotype: a distinct lace-like configuration is developing on the bladed basement. If nothing happens to prevent such growth, the form index of this colony will tend to the value typical for the lace-like morphotype. The boxwork colony shown in Fig. 1g, in all likelihood, started thickening and overgrowing with irregular bumps immediately after it passed the stage of apical shooting and initial dichotomous splitting, and continued developing in such a way. Thus it provides evidence of a ‘long-distance’ transition from lace-like to boxwork morphotype without lingering at L= Lbl.
6. Conclusions Based on the data and the model presented above, it appears that the intra-specific morphological diversity of hermatypic corals is an effect of the interplay between ‘order’ (deterministic behavior) and ‘noise’ (random behavior). This diversity provides a striking example of a phenomenon belonging to the category of noise-induced transitions. Order is apparent in the existence of several
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permitted typical growth forms, resulting mainly from the dependence of the rates of tissue growth and calcification on form. External noise, i.e., random fluctuations superimposed upon a steady environment, even when of short duration and limited in amplitude, may induce transition of a coral colony from a given stable growth form to another.
Acknowledgements We thank Professor Yair Achituv, Professor Rem Khlebopros, Professor Ray Taylor, Professor Zvy Dubinsky, Dr Boris Sherman and Dr Ian White for stimulating discussions. We also thank Mr Oren Levy for taking the photographs of M. dichotoma in the Red Sea.
Appendix. Derivation of the equation of growth form dynamics According to the definition of the growth form index given in subsection 3.2, L= V/S= C/B. The first derivative of L with respect to time is therefore:
dL d C = = dt dt B
B
dC dB −C dt dt . B2
Eqs. 1 and 2 allow substitution of rBB for dB/dt and rCB for dC/dt, thus implying: dL rCB 2 − rBCB C = = rC − rB . 2 dt B B Finally, replacing C/B with L, we obtain dL = rC − rBL. dt This completes the derivation of Eq. (3), which describes the form dynamics of a hermatypic coral colony.
References
Fig. 7. Reversal transition: a lace-like configuration developing on the bladed basement.
Barnes, D.J., 1973. Growth in colonial scleractinians. Bull. Mar. SCI. 23, 281 – 298. Barnes, D.J., Lough, J.M., 1989. The nature of skeletal density banding in scleractinian corals: fine banding and seasonal patterns. J. Exp. Mar. Biol. Ecol. 126, 37 – 54. Barnes, D.J., Taylor, R., 1993. On coralites apparent in X-radiographs of skeletal slices of Porites. J. Exp. Mar. Biol. Ecol. 173, 123 – 131. Barnes, D.J., Lough, J.M., 1997. Coral skeletons: storage and recovery of environmental information. Global Change Biology 2, 569 – 582. Ben-Zion, M., Achituv, Y., Stambler, N., Dubinsky, Z., 1991. A photographic, computerized method for measurements of surface area in Millepora. Symbiosis 10, 115 – 121. Dafni, J., Tobol, R., 1986. Population structure patterns of a common Red Sea echinoid (Tripneustes gratilla elatensis). Isr. J. Zool. 34, 191 – 204. Darke, W., Barnes, D.J., 1993. Growth trajectories of coralites and ages of polyps in massive colonies of the reef-building corals of the genus Porites. Mar. Biol. 117, 321 – 326. Denny, M.W., Daniel, T.L., Koehl, M.A.R., 1985. Mechanical limits to size in wave-swept organisms. Ecol. Monogr. 55, 69 – 102. Dubinsky, Z., Falkowski, P.G., Porter, J., Muscatine, L., 1984. The absorption and utilization of radiant energy by light and shade-adapted colonies of the hermatypic coral,
Z. Kizner et al. / Ecological Modelling 141 (2001) 227–239 Stylophora pistillata. Proc. Roy. Soc. Lond. 222B, 203 – 214. Edelstein-Keshet, L., 1988. Mathematical Models in Biology. Random House, New York. Efron, B., 1982. The Jackknife, the Bootstrap and other Resampling Plans. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania. Falkowski, P.G., Dubinsky, Z., 1981. Light-shade adaptation of Stylophora pistillata a hermatypic coral from the Gulf of Eilat. Nature 289, 172 –174. Falkowski, P.G., Jokiel, P.L., Kinzie, R.A., 1990. Irradiance and corals. In: Dubinsky, Z. (Ed.), Coral Reef Ecosystems. Elsevier, Amsterdam, pp. 89 – 107. Foy, R.H., 1980. The influence of surface to volume ratio on the growth rates of planktonic blue-green algae. Br. Physiol. J. 15, 279 – 289. Graus, R.R., Macintyre, I.G., 1976. Light control of growth form in colonial reef corals: computer simulation. Science 193, 895 – 897. Horsthemke, W., Lefever, R., 1984. Noise-induced Transitions. Theory and Applications in Physics, Chemistry, and Biology. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo. Jokiel, P.L., 1978. Effects of water motion on reef corals. J. Exp. Mar. Biol. Ecol. 35, 87 –97. Korn, G.A., Korn, T.M., 1968. Mathematical Handbook for Scientists and Engineers. Definitions, Theorems and Formulas for Reference and Review, 2nd edn. McGraw-Hill Book Company, New-York, San-Francisco, Toronto, London, Sydney. Lewis, W.M., 1976. Surface/volume ratio: implications for phytoplankton morphology. Science 192, 885 –887. Lewis, J.B., 1989. The ecology of Millepor. Coral Reefs 8, 99 – 107. Mergner, H., 1971. Structure, ecology, and zonation of Red Sea reefs (in comparison with South Indian and Jamaican reefs). Symp. Zool. Soc. Lond. 28, 141 –161. Niklas, K.J., 1994. The scaling of plant and animal body mass, length, and diameter. Evolution 48 (1), 44 –54.
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Odum, H.T., 1971. Fundamentals of Ecologym, 3rd edn. W.B. Saunders Company, Philadelphia – London– Toron. Ornstein, D., 1974. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, New Haven. Pearse, V.B., Muscatine, L., 1971. Role of symbiotic algae (zooxantellae) in coral calcification. Biol. Bull. 141, 350 – 363. Reynolds, C.S., 1984. The Ecology of Freshwater Phytoplankton. Cambridge studies in Ecology. Cambridge University Press, Cambridge. Rinkevich, B., Loya, Y., 1983. Oriented translocation of energy in Grafted Reef Corals. Coral Reefs 1, 243 – 247. Rinkevich, B., Loya, Y., 1984. Coral illumination through an optic glass-fiber: incorporation of 14C photosynthates. Mar. Biol. 80, 7 – 15. Roberts, H.H., Wilson, P.A., Lugo-Fernandez, A., 1992. Biologic and geologic responses to physical processes: examples from modern reef systems of the Caribbean-Atlantic region. Cont. Shelf Res. 12, 809 – 834. Stanly, H.E., 1971. Introduction to Phase Transitions and Critical Phenomena. Claredon Press, Oxford. Taylor, D.L., 1977. Intra-colonial transport of organic compounds and calcium in some Atlantic reef corals. In: Proc. 3rd Int. Coral Reef Symp. Miami, pp. 431 – 436. Thompson, Darcy W., 1992. On Growth and Form. Cambridge University Press, Cambridge. Vago, R., Achituv, Y., Vaky, L., Dubinsky, Z., Kizner, Z., 1998. Colony architecture of Millepora dichotoma Forskal. J. Exp. Mar. Biol. Ecol. 224, 225 – 235. Weber, J.N., 1974. Basis for skeletal plasticity among reef building corals. Geology 2, 153 – 154. Weerdt de, W.H., 1981. Transplantation experiments with Caribbean Millepora species (Hydrozoa, Coelenterata) including some ecological observations on growth forms. Bijdr. Dierkd. 51, 1 – 19. Weerdt de, W.H., 1984. Taxonomic characters in Caribbean Millepora species (Hydrozoa, Coelenterata) including some ecological observations on growth forms. Bijdr. Dierkd. 54, 243 – 362. Yonge, C.M., 1968. Living corals. Proc. Roy. Soc. Lond. Ser. B 169, 329 – 344.
Growth forms of hermatypic corals: stable states and noise-induced transitions Z. Kizner a,*, R. Vago b , L. Vaky c a
Department of Physics and Faculty of Life Sciences, Bar-Ilan Uni6ersity, Ramat Gan 52900, Israel b The Institute for Applied Biosciences, Ben-Gurion Uni6ersity, Beer-She6a 8405, Israel c Faculty of Life Sciences, Bar-Ilan Uni6ersity, Ramat Gan 52900, Israel Received 14 July 2000; received in revised form 7 February 2001; accepted 26 February 2001
Abstract Field observations show that some coral species may assume different growth forms, or morphotypes, within apparently uniform habitats. The present paper examines the causes of this morphological diversity in hermatypic cnidarians and more particularly the following questions: Are there stable morphotypes among all the possible growth forms of a coral; if so, can a colony belonging to a certain stable morphotype change its growth form, and what can induce transition from type to type when the environment is relatively steady and only limited random (and short-term) variations of the external conditions occur? The approach adopted was to construct a mathematical model of the form dynamics of a coral colony and validate it by field data on the Red Sea hydrocoral Millepora dichotoma. The ratio between the volume and the surface area of a coral served as an index of its form and enabled treatment of the coral morphology in terms of dynamical systems. Such a dynamical system must have certain stable states corresponding to specific growth forms of a coral colony. Computer simulations showed that limited stochastic disturbances in the processes of biomass and skeletal growth caused by random fluctuations in environmental conditions may induce transitions of a coral from one stable growth form to another. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Coral; Dynamical system; Environmental fluctuations; Growth form; Modelling; Stability; Transition
1. Introduction
This study is dedicated to the memory of one of the authors, Lior Vaky, who was accidently killed while conducting field work on the biology of corals in the Red Sea. * Corresponding author. Tel.: +972-3-5318244; fax: + 9723-5351824. E-mail address: [email protected] (Z. Kizner).
Reef corals typically display high morphological diversity, which is apparent at both the interand the intra-specific levels. Growth of corals depends strongly on environmental conditions (Yonge, 1968; Mergner, 1971; Jokiel, 1978; Denny et al., 1985; Roberts et al., 1992), changes in the environment being reflected both in the calcifica-
0304-3800/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S0304-3800(01)00276-9
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tion process (Weber, 1974; Graus and Macintyre, 1976; Barnes and Lough, 1989, 1997) and in the growth of living tissue (Vago et al., 1998). Therefore, coral colonies of the same species living under different external conditions may grow into different forms (Weerdt de, 1981, 1984; Lewis, 1989; Vago et al., 1998), while a long-term change in the environment may cause a long-term transformation in the form of a coral colony (Weerdt de, 1981, 1984). However, field observations show that even within apparently uniform habitats, the same coral species may appear in different growth forms. In cells, algae, plants and animals it is useful to describe body form in a reductionist manner in terms of the ratio of volume to surface area (Lewis, 1976; Foy, 1980; Reynolds, 1984; Thompson, 1992; Niklas, 1994). Such ratios generally have a physiological, an energetic, and also a biomechanical meaning (Foy, 1980; Reynolds, 1984; Thompson, 1992). This ratio is also convenient in the case of hermatypic cnidarians, in which skeleton deposition proceeds from a thin, superficial veneer of living tissue into a volume. Barnes (1973) was apparently the first to deal with volume and surface area in connection with growth of massive corals (see also Barnes and Taylor, 1993; Darke and Barnes, 1993). However, to date, the volume to surface ratio has not been proposed as a coral form index. In the present paper, we apply this index to examine whether the development of a coral into a particular form follows a random path representing one of a multitude of conceivable directions, or whether the number of stable growth forms possible is limited by coral physiology. We then go on to assess the possibility of modeling the dynamics of coral form using nonlinear dynamical systems, and finally we establish whether stochastic fluctuations or ‘external noise’ superimposed upon a steady environment would induce transitions from one stable state of the system to another. Data on the Red Sea hydrozoan Millepora dichotoma constitute the numerical basis for this study. In response to ‘external noise’, corals transit mainly from delicate growth forms to more robust ones. Thus relative occurrence of different morphotypes may be an important indicator of
the load of short-term environmental disturbances upon ecosystems of coral reefs, even though in the long-term perspective the environment can be considered steady.
2. Materials The growth forms of colonies of Red Sea M.dichotoma Forskal were examined at two sites exhibiting characteristic but distinct types of bottom topography in the northern part of the Gulf of Elat (Aqaba), 29°30% N, 34°55% S (for map see Vago et al., 1998). At Site 1 (Nature Reserve region), 369 colonies were examined within a belt 40 m long and 4 m deep (0–4 m). This site is rather sheltered and protected from waves due to the shape of the coastline (Dafni and Tobol, 1986). Site 2 (in front of the Inter-University Institute of Elat), where 502 colonies were examined along a belt 20 m long and 6 m deep (0–6 m) parallel to the coastline, is exposed to strong winter storms and southern waves. The coastal slope consists of rubble with scattered patches of coral (Dafni and Tobol, 1986). Since light changes rapidly with depth (Mergner, 1971), light availability must generally be taken into account when considering the distribution of coral forms over a large range of depths. However, in the shallow oligotrophic waters of the northern part of the Gulf of Elat, coral colonies maintain maximal photosynthetic rates even under the lower winter irradiance levels, to which they undergo photoacclimation (Falkowski and Dubinsky, 1981; Dubinsky et al., 1984; Falkowski et al., 1990). Therefore, light is not a consideration in the context of our analysis, and we do not distinguish between data obtained from different depths. Hydrodynamically, both sites are relatively homogeneous. During the surveys, the dimensions of each non-encrusting colony were measured with a vernier caliper to provide an estimate of colony volume V (cm3) and surface area S (cm2) covered with living tissue. Colony form was approximated in terms of combinations of standard geometrical figures (cylinders, prisms, pyramids, spherical segments etc.), and V and S were calculated using
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known formulae. Since only large-scale measurements were carried out, with characteristic colony size ranging from a few millimeters to 1 cm (Fig. 1), the values obtained for the area of the skeletal surface were not affected by the roughness or possible fractality of the latter. Based on these data, the ratio L = V/S was calculated for each colony. Experiments with a number of dead skeletons of M. dichotoma belonging to different morphotypes proved that the possible relative error in these estimates did not exceed 25%. For encrusting colonies, the most widespread form at both sites, L was determined as the range of thickness of the calcium carbonate layer (not including area of bumps; see below for definitions of the main morphotypes), that is 0.0– 0.06 cm.
3. The model
3.1. Equations of tissue growth and calcification The skeleton of a coral is composed of calcium carbonate deposited by the living tissue. The layer of living tissue that covers the surface of the skeleton is so thin that the mass and the volume of a coral colony are actually those of its skeleton. Let C be the mass of the skeleton and B the biomass of the living tissue of a coral colony. We may describe the dynamics of B and C by the following equations: dB =rBB, dt
(1)
dC =rCB, dt
(2)
where t is time, and rB and rC are the growth rate of the biomass (tissue) and the calcification rate per unit of biomass, respectively. Due to the thinness of the tissue layer, it may be assumed that the tissue biomass B is proportional to the surface area S of the skeleton: B = vS. Since we are looking at the long-term dynamics of corals, where the time scale considerably exceeds a season or a year, C can be assumed to be proportional to the colony volume V, i.e., C =wV, and the coefficients of proportionality, m and n, can be
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taken as constant. (In principle we could increase the universality and accuracy of the model by taking into consideration temporal variability in skeletal density due to, say, seasonality.) Hence it is legitimate to use area and volume units (say, cm2 and cm3) to measure tissue and skeletal mass, that is, to assume that v = w=1 and thus C= V and B =S. Eq. (1) is a conventional equation for the dynamics of the biomass of populations and colonies (see, e.g. Odum, 1971; Edelstein-Keshet, 1988), and it says that the living tissue reproduces itself at the rate rB per unit biomass. According to Eq. (2), every unit of the tissue mass participates in building the skeleton by depositing calcium carbonate at the same rate rC. Such a relation is clearly appropriate for massive corals (like Porites), which grow in all three dimensions without exhibiting any preferred growth in certain particular directions, as well as for actually two-dimensional encrusting colonies. However, we will assume that this equation also holds for other colonies, i.e. that it represents a general relationship between the tissue biomass and rate of skeletal mass accretion independently of the form of a coral. Such an assumption is in agreement with the results of Pearse and Muscatine (1971), Taylor (1977), Rinkevich and Loya (1983), and Rinkevich and Loya (1984) for Acropora cer6icornis, Montastrea annularis and Stylophora pistillata showing transport of fixed carbon from peripheral areas to growing tips of these branching corals, i.e. participation of remote tissue parts in the deposition of calcium carbonate at branch tips. The validity of Eq. (2) for branched colonies can also be supported by the following reasoning. Consider an idealized colony consisting of a single cylindrical branch that grows longitudinally without thickening. Let DV and DS be the volume and surface area accretions that take place within time interval Dt in such a brunch. The ratio DV: DS is equal to the ratio between the area of the cylinder’s base and its perimeter and is therefore independent of Dt (for example, in the case of a circular cylinder, DV: DS equals to half a radius of the cylinder; for details see the next subsection). In other words, for such a colony dV/dt and dS/dt are proportional to each other with a
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Fig. 1. Colonies of M. dichotoma ranked according to value of their form index: encrusting (a); lace-like (b – d); bladed (e, f), boxwork (g – i).
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constant factor of proportionality, and Eq. (1) immediately yields Eq. (2). This conclusion can be extended to the case of colonies composed of a number of cylindrical branches. In general, the rates rB and rC are not constant: they depend on a number of variables, including the current tissue and skeletal masses, B and C, and environmental conditions. Thus, the living tissue and skeletal masses are interdependent in our model; their self- and inter-limitations are discussed in the next subsection.
3.2. The form index and the dynamic equation Every specific form of a coral colony depends on how the colony volume, V, relates to the surface area of the colony, S. Therefore, the ratio L= V/S can be regarded as the form index representing the colony form quantitatively. In a previous work (Vago et al., 1998) four main morphotypes of M. dichotoma were defined and described, namely encrusting, lace-like, bladed and boxwork. The nearly two-dimensional, encrusting colonies have very thin skeletons, closely follow the contours of the substratum, and often have a few bumps on their surfaces (Fig. 1a). Clearly, the form index of an encrusting colony equals to the average thickness of its skeleton (a few hundredth of cm). Lace-like colonies are built of one or several parallel delicate erect ‘pages’ composed of dichotomous cylindrical branches (Fig. 1 b– d). In order to estimate the form index of such a colony let us suppose it to consist of N cylinders of equal average radius r, the latter being substantially smaller than their heights. Then the surface area and the volume of j-th branch can be assessed as 2yrhj and yr 2hj, where hj is the branch’s height. Correspondingly, the total surface area and volume are S =2yrSN j= 1hj and V=yr 2SN j = 1hj, respectively, and the form index is L= V/S =r/2. The typical branch diameter in our surveys was 0.4 cm (Fig. 1 b– d), so the value of L characterizing the lace-like colonies was 0.1 cm. The bladed morphotype can be described as a relatively massive, upright, leaf-like growth form in which most of the gaps between branches are closed, or partly closed; the top margins of such colonies often bear short
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branches (Fig. 1e and f). For the purpose of estimating the form index, a bladed colony can be regarded as a plate of thickness d (obviously, d] 2r) and base surface area s (area of single side). Its volume and total area (both-sides) can be estimated as V= sd and S= 2s, which gives L= d/2. The characteristic L value for bladed colonies was 0.3 cm (Fig. 1e and f). Finally, colonies belonging to the boxwork morphotype (Fig. 1 g-i), i.e., robust, fist-like, massive ones, grow essentially in all three dimensions. If X is the characteristic size of such a truly three-dimensional colony, then S8 X 2, V8X 3 and, consequently, L8X. So, unlike the situation in the forms considered above, in a boxwork colony growth implies an increase in the form index, and not only in colony surface area and volume. For this morphotype, therefore, the form index is characterized by the lower bound of possible values of L. According to our data, for box-work colonies L] 0.93 cm. Thus we see that the form index defined as a volume-to-area ratio has a clear physiological interpretation as it relates the coral growth form to the amounts of the skeleton (calcium carbonate) deposited by the living tissue and the tissue itself. As both the tissue growth and the calcification process are responsive to the changes in the environment, parameter L acquires also an ecological meaning. Introducing the concept of the form index facilitates ranking of the various coral growth forms, as manifested in the small values of L for thin and delicate growth forms versus the large values for massive colonies (Fig. 1). However, another merit of using L is the possibility it offers of analyzing the form dynamics of a coral colony in terms of dynamical systems, a mathematical notion with numerous applications in physics, chemistry and biology (see, e.g., Horsthemke and Lefever, 1984; Ornstein, 1974; Stanly, 1971). Using the definition of the form index, L=V/ S= C/B, we obtain the following equation for the form dynamics from Eq. (1) and Eq. (2): dL = rC − rBL. dt (for mathematical details see Appendix).
(3)
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In general, as noted above, the rates rB and rC depend on a number of parameters, including environmental ones. So far we have been discussing the morphological plasticity of corals under uniform, relatively steady conditions. In reality, however, even when the environment can be considered steady in the long-term perspective, limited short-term random fluctuations do occur. These cause relatively small random variations in the physiological parameters rC and rB and, hence, variations in the rate of change of the form, dL/dt. So, we may regard both the calcification and the tissue growth rates as sums: rC = c+m, rB = b+i, where c and b, independent of time, are the mathematical expectations, or deterministic components, of rC and rB, respectively, while the fluctuations m and i (‘external noise’) are stationary stochastic processes (see, e.g., Horsthemke and Lefever, 1984). Tissue growth in a coral is limited by the tissue biomass itself, as well as by the ability of tissue to grow skeleton. Hence, the deterministic component of the tissue growth rate, b, is a function of both S and V. Similarly, the deterministic component of the calcification rate, c, must generally be regarded as a function of S and V. Undoubtedly, among the possible growth forms there are some that are better or worse adapted to harvesting light energy and nutrients and hence exhibit higher or lower rates of tissue growth and calcification. So it would be quite realistic to assume that the rates b and c depend only upon the form index L = V/S, i.e. b =b(L) and c= c(L). Eq. (3) then becomes: dL = F(L)+ (m−iL), dt
(4)
where F(L) = c(L)+ b(L)L. A rapidly fluctuating environment can be characterized by the property that the correlation time of the stochastic processes representing the external noise is much smaller than the typical time of the macroscopic (or deterministic) evolution of the dynamical system. Such a short-term environ-
mental effect is conventionally modeled by Gaussian white-noise processes (Horsthemke and Lefever, 1984). That is what will be assumed regarding m and i.
3.3. Steady states and stability In our model, the deterministic rates b and c remain constant provided the coral does not change its morphology while growing, that is, provided the dynamical system remains in a steady state (equilibrium) in which the tissue and skeletal growth have become adapted to each other and L is constant. As the dynamical system (4) is nonlinear it may possess a number of steady states. For example, when a colony grows as an encrusting, lace-like or bladed morphotypes, its form index L remains nearly constant and each of the three colony forms can be regarded as a steady state. Indeed, as they grow, encrusting colonies expand in area but retain almost the same thickness of the calcium carbonate layer (Vago et al., 1998). Similarly, the branches of lace-lake colonies are almost cylindrical and grow nearly longitudinally without changing radius, so L=r/2= const. A bladed colony increases in volume and surface area mainly due to apical growth of new shoots on its periphery and subsequent gap filling (Ben-Zion et al., 1991), and its skeleton thickness d actually does not change; as a result, L= d/2= const. On the other hand, the boxwork morphotype is not steady because, as showed above, the form index of the colony increases as it grows. In general, some of the steady states of a dynamical system may be stable and others not. To understand the meaning of stability in the context of coral growth, let us consider (just for the sake of simplicity) the particular case of absence of noise and constant rates c and b. In this case the function F(L) will be linear, and the dynamical system represented by Eq. (4) will have a single equilibrium state, i.e., the derivative dL/dt is zero only at L=c/b (Fig. 2). This state is stable, which means that the ratio between c and b forces the coral colony to maintain a stable proportion between its volume and surface area, that is, to assume a certain form. (As the relation between
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Fig. 2. Stability of the steady state, L =c/b, of the dynamical system (3) at constant rates b and c.
Fig. 3. Frequency distribution of the estimated values of the form index L for colonies of M. dichotoma at two sites (tail of distribution in inset): Site 1 — the Nature Reserve (a), Site 2 — in front of the Inter-University Institute of Elat (b).
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form and ratio V/S is not a one-to-one correspondence, it is impossible, based only on the known stable value of L, to predict what exactly this form will be, it is basically an intrinsic property of the species. For example, the branched form of M. dichotoma is not a common ‘tree’ but just a dichotomous lace-like structure.) The stability of the steady state L= c/b in this example arises from the negative slope of the graph of the function F(L) (i.e., of dL/dt vs. L) in Fig. 2, and can be explained as follows. Let us assume that for some reason L has slightly deviated to the left. In this region, to the left of the steady state point L= c/b, the rate of change of L is positive. Therefore, the form index L will tend to increase and return to the point L= c/b. Similarly, L will recover if shifted to the right. Figuratively, to use a physical analogy, one might say that the stable state acts as a point of attraction for the dynamical system. Correspondingly, the dynamical system under consideration may possess more than one stable steady state only if the function F(L) is nonlinear and its graph crosses the L-axis a number of times with negative slope. The presence of weak fluctuations in the system does not essentially change this conclusion. What is the actual shape of the function F(L)? It could clearly be different for different species and sites. In order to give concrete expression to our model, frequency distributions of the form index L were evaluated based on our data for Red Sea M. dichotoma from each site (Fig. 3a and b). The histograms demonstrate the existence of three modes at L values of about 0.00–0.06, 0.190.005 (for both sites) and 0.390.03 cm (Site 1), and one additional, though small group of data with relatively high values of the form index, L] 1 9 0.25 cm (Site 1). The confidence intervals of the modal values of L (the figures carrying9sign) confirm the reliability of the above modes and were obtained by bootstrapping the data (Efron, 1982): 1000 replicates were made based on the assumption that the distribution of the possible error in L data is log-normal with 25% S.D. (see section 2). The three modes correspond to the three main morphotypes of M. dichotoma, that is, encrusting, lace-like and bladed.
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Fig. 4. Hypothetical shape of the nonlinear function F(L) corresponding to M. dichotoma at Site 1 and yielding existence of three stable states, encrusting, lace-like and bladed forms (Len, Lll and Lbl), and a phase of boxwork growth (L\ Lbw) (full range graph in inset).
The existence of three distinct modes and the grouping observed in the histograms signify an increase in the occurrence of colonies of the corresponding form as L approaches one of the modal values, L= Lm. In terms of the dynamical system (4), this will happen if the state L = Lm possesses the property of ‘attraction’, that is, in either of two cases: (i) the modal point L =Lm is a stable steady state, and hence numerous colonies gather in its immediate vicinity with time; (ii) L changes at a slow rate in the vicinity of the modal point and the colony lingers in that region for a long time before eventually proceeding beyond. In either of these cases, an ‘instantaneous’ survey would provide a local maximum of the form index frequency distribution. However the second scenario does not seem to be relevant to our data. Indeed, if, say, Len were to represent a slowdown point, and not a point of stable equilibrium, then each colony would have to pass it sooner or later, and the colony form index would stabilize at one of the subsequent modal points, Lll or Lbl, or even proceed to the region of boxwork growth (L \ Lbw). But in this case the probability for the occurrence of encrusting colonies would be less than for the next stable morphotype in the series (or, if neither L= Lll nor L = Lbw were points of stable equilibrium, less than the probability for the occurrence of the boxwork colonies). The same reasoning applies if we take L = Lll or L = Lbw as points of slowing down. However, our
data reflect a different situation: the number of encrusting colonies (0BLB 0.06 cm) significantly exceeds the total number of lace-like and intermediate, apical shooting colonies (0.06B LB 0.16 cm), and the latter in turn is much larger than the number of bladed and transitional lace-likebladed colonies (0.16B LB 0.36 cm). Finally quite a few boxwork colonies (L\0.93 cm) were found only at Site 1 (Fig. 3 a, b, see also Vago et al., 1998). These data dovetail nicely with the first scenario and testify to the stability of the three main morphotypes of M. dichotoma. Based on this we identify the stable states of the dynamical system under consideration with the three modes, Len, Lll and Lbl. As shown above, the graph of the function F(L) must cross the L axis in the points Len, Lll and Lbl with a negative slope to ensure the stability of the corresponding states. In Fig. 4 the hypothetical shape of the nonlinear function F(L) for M. dichotoma in the Gulf of Elat at Site 1 is presented. It affords three finite stable states (encrusting, lace-like and bladed growth) and monotonous increase of the form index in the boxwork region (L \Lbw) where dL/dt is positive (for details see Section 4). So, in fact, the only two hypotheses underlying our model are: (i) that of the stability of the three main states (Len, Lll, Lbl) and (ii) the assumption that c and b are functions of L.
4. Computer simulations and fitting the model For the purpose of computer simulations, Eq. (4) was replaced with its non-dimensional finitedifference analog (Korn and Korn, 1968): L(t+ dt)= L(t)+[F(L)+ m(t)− i(t)L]dt.
(5)
In Eq. (5) the time variable t is discrete, changing with a small finite step dt, and hence all the other variables, including m and i are also discrete. Eq. (5) may be regarded as a recurrent formula for the successive determination of the form index: once given an initial value L(0)= 0, every value of L(t + dt) can be calculated from the previous one, i.e., through L(t). Each of the variables m(t) and i(t) in Eq. (5) can be treated as
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a random series with independent elements normally distributed with zero mean and standard deviation |m or |i, respectively. In the case of stochastic differential equations this is a natural finite-difference approximation, and Eq. (4), which includes white-noise components of the tissue growth and calcification rates, m and i, can be obtained from Eq. (5) by means of going to the limit at dt 0 subject to the conditions |m dt = const and |i dt=const (Horsthemke and Lefever, 1984). As noted above, we regard the random variations in the rates rB and rC to be small in relation to the deterministic ones. This implies two more conditions on |m, si and dt. First, the random variations in L occurring within the time step dt must be small in relation to the macroscopic or deterministic changes in L, which can be estimated as F*dt [here F* is the characteristic value of the function F(L)]. Second, when the dynamical system approaches a stable steady state, the random deviations from it must be small compared with the distance between this stable state and the one nearest: otherwise the system would be able to leap to the region of attraction of another stable state even within one time step, i.e., the fluctuations would not be small. Therefore, the parameters |m, |i and dt should be chosen so that the products m(t)dt and i(t)Ldt will be much smaller
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than both the product F*dt and the differences between those pairs of zeros of the function F(L) that determine the stable steady states of the dynamical system (3), i.e., of Len, Lll, Lbl; and Lbw in the case of M. dichotoma. Finally, the time step dt must be much smaller than the time typical for the macroscopic (deterministic) temporal evolution of the system, which can be identified with the time of relaxation of the system towards a steady state. One may justifiably expect that both the stability of the main growth forms (encrusting, lace-like and bladed in the case of M. dichotoma) and relatively rare transitions from one stable growth form to another will be reproduced in simulation runs of the properly tuned model given by Eq. (5). We fitted the model to the description of the growth dynamics of M. dichotoma so that to render a frequency distribution of the form index (evaluated from numerous runs) similar to that presented in Fig. 3a. This was the method used to generate the details in the graph of the function F(L) shown on Fig. 4. The best fit parameters |m, and |i were estimated as |m = 2.2 and |i = 35 at dt=0.002. Goodness-of-fit for each of the checked combinations of parameters was verified by a Monte-Carlo computer experiment consisting of 100 runs followed by a 2-test. Each simulation lasted for T= 15 non-dimensional time units (T represented the life span of a coral colony in the model).
5. Results and discussion
Fig. 5. Monte-Carlo histogram of the magnitudes of the M. dichotoma form index L obtained based on 100 model runs [for function F(L) shown in Fig, 5, T= 15, dt= 0.002, |m, = 2.2, and |i =35] (tail of distribution in inset).
The histogram of the computed L values for the best fit combination of the model parameters is presented in Fig. 5. This is quite similar to the observed frequency distribution of the forms of M. dichotoma surveyed at Site 1 (Fig. 3a). The closeness of the two frequency distributions (Fig. 3a and 5) was tested using a 2-criterion for the partition into five classes: [0, 0.08], [0.08, 0.16], [0.16, 0.24], [0.24, 0.32], [0.32, 0.4]; the boxwork colonies were excluded from the analysis due to their small number in the field data. The result, 2 = 7.4B 2cr = 9.5, confirms the resemblance of these distributions at a 95% confidence level (h= 0.05, four degrees of freedom).
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Fig. 6. An example of simulated successive transitions of a colony of M. dichotoma: Life history of the colony (a); enlarged transition from the encrusting to the lace-like form (b); enlarged transition from the lace-like to the bladed form (c).
An example of simulated growth form dynamics that exhibits successive transition of a coral colony from the encrusting state to the lace-like and then to the bladed form with rather long periods of stable growth in each is shown in Fig. 6a. It is evident from Eq. (5) that the effect of random fluctuations in the rate of calcification on the rate of change of the form index remains statistically constant during the life history of a coral colony, as well as among colonies (as |m dt =const). By contrast, the effect of stochastic variations in the rate of tissue growth increases with the increase of the form index (as it is coupled with the product |iLdt). In the vicinity of the first, encrusting stable state, while the form index is small (within the range 0.005L 50.06), the effect of random variations in rC is the major one for the chosen model parameters. Beyond this range, the two effects first become comparable in order of magnitude, and then, in the vicinity of the second, lace-like state, the random variations
in rB become dominant. Based on the analysis carried out, the following scenario of transitions in coral growth can be suggested. While a coral grows as the encrusting morphotype, its form is primarily affected by stochastic variations in the calcification rate, which can be apparent in formation of bumps. This effect is possibly accompanied by minor changes in form caused by random variations in the tissue growth rates. Suppose that random perturbations in the environment, such as small-scale spatial variations in carbonate availability, cause a brief succession of predominantly positive short-term random changes in the calcification rate m, say, supersaturation of carbonate in the boundary layer over the colony surface. The form index L could than increase in value and enter the region of attraction of the next stable steady state, L\0.06 (Fig. 6b). An increase in L means that growth of colony volume outstrips growth of its surface. Some bumps that receive additional carbonate will con-
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tinue to develop to accommodate surplus skeleton material, giving rise to the so called apical shooting described in our previous paper (Vago et al., 1998), subsequent dichotomous splitting, and formation of a lace-like structure. Clearly, the probability for such a series of chiefly positive values of o is much smaller than for a series in which positive and negative values eventually balance each other. That is why transitions from encrusting to lace-like growth are rare. A lace-like colony (L 0.1) is more sensitive to the stochastic variations in rB than to those in rC. The term iLdt in Eq. (5), which represents the effect of these variations, carries a minus sign. Therefore, a lace-like colony may rapidly transform to a bladed or even boxwork morphotype (Fig. 6c, see also Fig. 1g) if environmental fluctuations induce a series of short-term, random but predominantly negative changes in rate of tissue growth, delaying the increase in surface area necessary to balance the increase in volume. Such a situation may arise, for example, when tissue growth is somewhat restrained by small but repeated damage, which is in agreement with our observations (Vago et al., 1998). Conformably with the small probability for an event of transition of coral form, most of the runs demonstrated encrusting growth within the interval of integration of Eq. (5), 05 t 5T. When a transition to the lace-like form did occurred in the course of a run, the simulated colony usually remained lace-like. The number of runs in which transition occurred from encrusting to the lacelike and thence to a massive form was relatively small, as can be seen in Fig. 5. So far, the transition of a coral colony from one stable growth form to the next higher (in terms of L) stable state was discussed. Indeed, such a development in the direction of increasing L is the most probable, however, this is not the only possible path. Along with the transitions to the closest states with higher L values, the model permits transitions in the opposite direction, to states with smaller L, as well as immediate ‘longdistance’ transitions, say, from lace-like to boxwork form, although the one-step transitions to higher L prevailed over the reversal and ‘long-distance’ ones in our 100 runs. The point is that the
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distances between zeros of the function F(L) are large enough in relation to |iLdt, and a transition from a non-encrusting state may in fact occur only if the random changes in the rate of tissue growth (the leading factor in such transition) cause net increase or decrease of L within a few successive time intervals dt. Since the effect of random changes in rB becomes stronger with the increase of L, this means that the probability of transition from Lll to the region of attraction of the point Lbl is higher than that of returning from Lll to the region of attraction of the point Len. The same is valid for the transitions from the bladed state to the boxwork form and from the bladed to the lace-like state, as well as regarding the ‘long-distance’ transitions as compared with the one-step transitions. Transitions from encrusting to lace-like (apical shooting phenomenon) and from lace-like to the bladed form have been described in previous works (Ben-Zion et al., 1991; Vago et al., 1998). The colony of M. dichotoma shown in Fig. 7 provides, in our opinion, convincing evidence of a retrogressive transition from bladed to lace-like morphotype: a distinct lace-like configuration is developing on the bladed basement. If nothing happens to prevent such growth, the form index of this colony will tend to the value typical for the lace-like morphotype. The boxwork colony shown in Fig. 1g, in all likelihood, started thickening and overgrowing with irregular bumps immediately after it passed the stage of apical shooting and initial dichotomous splitting, and continued developing in such a way. Thus it provides evidence of a ‘long-distance’ transition from lace-like to boxwork morphotype without lingering at L= Lbl.
6. Conclusions Based on the data and the model presented above, it appears that the intra-specific morphological diversity of hermatypic corals is an effect of the interplay between ‘order’ (deterministic behavior) and ‘noise’ (random behavior). This diversity provides a striking example of a phenomenon belonging to the category of noise-induced transitions. Order is apparent in the existence of several
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permitted typical growth forms, resulting mainly from the dependence of the rates of tissue growth and calcification on form. External noise, i.e., random fluctuations superimposed upon a steady environment, even when of short duration and limited in amplitude, may induce transition of a coral colony from a given stable growth form to another.
Acknowledgements We thank Professor Yair Achituv, Professor Rem Khlebopros, Professor Ray Taylor, Professor Zvy Dubinsky, Dr Boris Sherman and Dr Ian White for stimulating discussions. We also thank Mr Oren Levy for taking the photographs of M. dichotoma in the Red Sea.
Appendix. Derivation of the equation of growth form dynamics According to the definition of the growth form index given in subsection 3.2, L= V/S= C/B. The first derivative of L with respect to time is therefore:
dL d C = = dt dt B
B
dC dB −C dt dt . B2
Eqs. 1 and 2 allow substitution of rBB for dB/dt and rCB for dC/dt, thus implying: dL rCB 2 − rBCB C = = rC − rB . 2 dt B B Finally, replacing C/B with L, we obtain dL = rC − rBL. dt This completes the derivation of Eq. (3), which describes the form dynamics of a hermatypic coral colony.
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