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Corals and starfish waves on the great barrier reef: Analytical trophodynamics and 2-patch aggregation methods
. Mathl. Comput. Modelling Vol. 27, No. 4, pp. 121-135, 1998 Copyright@1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/98 sl9.00 + 0.00
Pergamon
PII: SOS95-7177(98)00012-O
Corals and Starfish Waves on the Great Barrier Reef: Analytical Trophodynamics and Z-Patch Aggregation Methods P. ANTONELLI Department of Mathematical Sciences University of Alberta Edmonton, Alberta, Canada T6G 2Gl P. AUGER Universitb Claude Bernard, Lyon 1, URA CNRS 243 43 Bd. du 11 Novembre 1918 69622 Villeurbanne Cedex, France R. BRADBURY National Resource Information Centre, GPO Box 858 Canberra ACT 2601, Australia Abstract-In this paper, singular perturbation theory is applied to the Antonelli/Kazarinoff limit cycle model of the Crown-of-Thorns starfish (COTS) predation of corals on the Great Barrier Reef. At the microscale of individual reefs, the previously published l-patch dynamics based on aggregation of individual behaviour is extended to include social interactions, spatial diffusion, and a&e&ion currents. As a consequence, the parameters which characterize the well-known wave solutions (i.e., the Reichelt starfish waves) are reinterpreted in terms of individual behaviour parameters. Likewise, the analytical trophodynamics of coral community production (without starfish), involving geometric invariants of second-order ODl% is defined at the microscale by direct manipulation of the reefal structure. The aggregation method leads to new insights into Liapunov stability of production of the coral community ss a whole. Keywords-Aggregation,
Geometrictheory of second-orderODEs, Diffusion, Reef ecology.
1. ANALYTICAL
TROPHODYNAMICS
That branch of ecology known as empirical trophodynamics is concerned with ‘stocks and flows’ models: the stocks of biomass and the flows of input (consumption) and output (production). These are often measured as caloric energy, forms of carbon, wet weight, and so on, but the idea is the same in each instance and is tied up with the so-called trophic pyramid of Lindeman. Although this approach is understandable, we do not follow it. Instead, we directly link ecological equations (competition, etc.) with production of biomass and consumption of energy as did Volterra in 1936 [l]. In analytical trophodynamics, production (consumption) is specified by Volterra’s equation,
xi(t)
=
t J
e,, *
iVi(s) ds + Xi(O),
0
Supported
in part by NSERC-A-7667.
121
(1.1)
122
P. ANTONELLIet al.
where Ni is size of ith population and t(i) is a positive constant. Such Xi are surrogates of biomass and include various useful examples such as the above, secondary compounds for defense, and allometric measures of biomass. This approach has been systematically explored since the early 1980s by our group [2,3]. Whereas activity in empirical trophodynamics has been concerned with refinement of measurements, as for example, the “Big Green Machine” of Birch, and various energy budgets, it has focused on the transformations between surrogate variables. The focus for us is different in that we explore the dynamical structures in an ecosystem which are invariant of the particular surrogate variables and their transformations. Our main interest is in species of colonial organisms, the ‘modular organisms’ of Harper [4], in which the biomass surrogate variables Xi and their production rates c = e,,)Ni are independent, reflecting the great plasticity in size of individual colonies (e.g., plants, corals, bryozoans, etc.). Given these two sets of quantities at time zero (i.e., the initial production levels and rates), we seek to predict their values at all future times. Accordingly, we postulate n Volterm production equations and n ecological interaction equations which combine to yield a second-order system of ODES. Bather than attempt numerical or analytic solutions, we plan to study the intrinsic (i.e., geometric) properties of this system of n second-order equations. Our approach does not require any optima&y assumptions, generally. Analytical trophodynamics studies properties of Volterra production (consumption) inherent in the second-order ODES and which are invariant under transformation between particular sets of surrogate variables. We claim, furthermore, that the efforts in empirical trophodynamics have been misdirected because of its focus on establishing the nature of the transformations, rather than on the underlying invariant properties of the dynamics (e.g., production stability, species interaction patterns and their efficiency, etc.).
If our stated goal is to understand the intrinsic properties of a postulated second-order system of ODES, just what is it that we gain by achieving this? These intrinsic properties are described by curvatures which give the complete spec@cation of the stability of production OTconsumption processes. For example, for each of two nearby similar islands with the same number and kinds of species, one can decide by computing curvature whether or not production levels and rates will remain close, given they are close initially. More generally, the curvature array completely determines the identity of the dynamical system according to the KCC-theory of Kosambi-CartanChern [5-71. Although computations are in general difficult, they can be made considerably simpler in the cases we need for ecology, namely, for a large class of ecological interactions with densitydependent effects which scale quadratically, according to experimental results of Wilbur [8,9] and Hutchinson’s theory of social interactions [lO,ll]. We have previously used this method on two problems and have had considerable success. The first is the homogeneous Hutchinson competition equations [12] and the second is higher-order predator-prey equations [13]. Both of these problems involve the experimental findings which reveal that social interactions first studied by Hutchinson in 1947 [lO,ll] and found in real data of ecological communities by Wilbur [8,9], Hairston et al. [14], and others (see [15]) to be of equal importance (from the analysis of variance method) as those of the usual quadratic terms in community interactions. We model this by requiring density-dependent terms added to, say, Gause-Witt competition equations, shall be (positively) homogeneous of degree two. Systems of second-order ODES with this property sre called sprays and are the topic of at least two recent books [2,3]. In the present work, we examine social models in the context of aggregation methods [16], both with and without spatial diffusion, which have been successful in modelling the Acanthaster plan& devastation of corals of the Great Barrier Beef. The two basic references for this problem are [17,18]. We shall generalize the previous aggregation model [16] to include social interactions among corals, which are relevant for the genus Acropom, the most abundant and most attacked coral on the Great Barrier Beef [19]. In Section 2, we go into the details of our model.
Corals and Starfish Waves
2.
SOCIAL INTERACTIONS
The literature
on ‘social’ interactions
123
IN COLONIAL
has had solid experimental
ORGANISMS
and theoretical
contributions
beginning with Hutchinson’s 1947 classic paper, in which cubic terms replace the classical quadratic terms in the 2-species competition equations of Gause and Witt [lO,ll]. Since then, the work of Hairston et al. [14] on bacterial and protozoan communities and that of Wilbur [8,9] have rigorously established their existence via analysis of variance techniques. There is a thorough discussion of Wilbur’s data on Ambystoma salamanders in Hutchinson’s ecology textbook [ll], where he explains the social interaction between A. latemle and its sexual parasite, the all-female species, A. tremblayi. Both Wilbur and Vandermeer
[15] discuss formal mathematical
models,
but state that systems
of equations with cubic, quartic, and higher Taylor series terms are not mathematically tractible. This may well be true, but addition of such terms do not conform to Wilbur’s all important statistical
evidence.
We quote directly
from Wilbur
(1972).
Frequently the higher-order interactions are as important as the main effects. This means that competition among salamander larvae is not a simple additive process that is a function of the total number of larvae in the community, but is a complex interaction between the proportions as well as the abundances and the identity of the species. (Our italics.) This quote will be easier to understand system may always be expressed by
if we recall that
a 2-species
competitive
dN1 = N1.F(N’,N2), dt dN2 = N2.G(N1,N2). dt
P-1)
Thus, the ‘additivity’ refers to the linearity of F and G in the usual Gause-Witt models. But, what of higher-order interactions being as statistically important as the usual quadratic terms? The way to incorporate this finding into a model is to require that social interaction terms scale the same way as quadratics. For example, terms like or
[N’ (N~)~]“~
are (positively) second-degree homogeneous in that either are replaced by pN1 and pN2 where p > 0 is an arbitrary
is multiplied by p2 when Ni and N2 real number. This property does not
hold for any higher-order term in a Taylor series, although it seems that Hutchinson thought it did [lo] in his discussion of the statistical meaning of the coefficients of his cubic terms. See [12] for a more complete discussion of Hutchinson’s error and for analysis of the homogenized version of Hutchinson’s equations, which use terms like the two above. It is noteworthy that Wilbur states that proportions are major influences in the dynamics of his salamander larvae populations. The first term above is obviously of this sort. Letting m 1 2 be an integer, we introduce social interactions, as in [13], for the starfish/coral dynamics as follows: m-2
dN1 -=X1N1-~I(N1)2-~2 dt
.N1N2+L
dN2 -=X2N2-~2(N2)2-~1 dt
.N’N2+L
. (N2)” - &FN’,
m-l m-2
. (N1)2 _ b2FN2,
(2.2)
m-l
All coefficients (~1, CQ, Xi, X2,&, 62, p, y,p are positive with N1, Iv2 being coral densities and F the starfish density. Xi, Xz, E are growth rates, Xi/cri, Xz/cus are single species carrying capacities,
P. ANTONELLI et al.
124
p, Sr, 6s are the interaction coefficients for starfish preying on corals, and y is the weficient of starfish aggregation. If we set m = 2, we obtain the (2 corals/l starfish)-model of Antonelli and Kazarinoff [20,21], in which every term of degree greater than one is quadratic. It is m > 3, which forces the social interaction terms (there are only two) to be nonquadratic, but to nevertheless scale the same as quadratics. The reader should take note that for m 1 3, neither N’ nor N2 is allowed to be zero, but that years of observation of Acanthaster plan& predation on the corals of the Great Barrier Reef indicate that although polyp densities may be drastically reduced on a reef, they are never actually vanishing. We also remind the reader that the predator F catches prey based only on food preference asymmetry in &N’F and 62N2F and that nutritionally for the predator, either species of coral is as good as the other. It is because of this assumption that we use the term ,LIF(N’+N2) instead of ,&NlF + P2N2F. Finally, when 61 # 62 stability of a small amplitude limit cycle bifurcating out of an equilibrium (Ni, Ni, Fo) and caused by increasing y beyond a certain critical (Hopf) value “yc,has been shown [13]. For m = 2, this cycle can be of large amplitude [21]. It is still an open question for m 2 3, but it is expected that the system will admit large amplitudes, as well. Also, any factor lowering the value of the coefficient of starfish aggregation 7 controls starfish; severe climatic disturbances and fish predation on starfish and their larvae, are examples. On the other hand, an influx of larvae of A. plan& would tend to increase y (see [17,18]). The effect of increasing the social parameter m has been determined; at first the asymptotical orbital stability of the limit-cycle solution increases until m = 6 is reached, whereupon it decreases steadily, eventually completely destabilizing the cycle [13]. Since it is known from observation that the cycle persists [17], it follows that m 5 8 in this model. Coral species of the genus Acropom are the most frequently attacked by A. planci [19]. For species of Acropom, Pandolfi [22,23] has documented the presence of dimorphic groups of individual coral polyps within single clones for which growth and development is morphologically distinct from other such groups. These so-called astogenetic groups are subjected to colony-tide developmental control. There is evidence that these groups compete interspecifically within a given clone of Acropom [23,24]. Of course, there is intraspecific competition between the polyps of an astogenetic group. Thus, two species of Acmpom, adjacent on a reef, do admit higher-order (i.e., social) interactions and it remains to find the social parameter m for the above model. Note, that our model is not for interactions between a single species of Acropora and another species of a different genus than this one. This is interesting and important, but now we concentrate on Acropom versus Acropora competition. For the estimation of m, note that if N2 = constant, F = 0, and N1 = C < 1, then G N C2-m >> 1 so that a strong recooperative effect results for m > 3 due to the density dependent term, ai/(m - 1)(N2/N1)“-2. This provides a realistic estimation procedure and shows the above model of coral social interactions is falsifiable in the Popperian sense. That is, if this estimation resulted in m 1 8, the model must be modified. We wish now to introduce aggregation methods based on individual behawiour of the starfish and corals which generalize our previous work in [16], in order to better understand the role of social interactions in the Great Barrier Reef ecosystem. For more details and information, we refer to [25]. 3. THE TWO
PATCH
MODEL
(SINGLE
REEF)
Now, the reef (say of diameter l/2 km) will be described as a set of two adjacent patches and two coral species. Patch #l could be raised up above patch #2 on the reef front, for example. Then, during the day, corals of patch #l compete with those of patch #2 for sunlight, while during the night, they compete for microscopic animals in the plankton which flow in changing
Corals and Starfish Waves
125
currents over the reef. Let Cj denote the density of corn1 of species i on patch j, and let Fj be the starfish density on patch j.
Our model includes two parts, a fast part and a slow part (e is a small parameter). The fast part describes the migration of the starfish adult population between the two patches at a rate kij, from patch #j to patch #i, and a manipulation of the coral distribution in space (and hence, the social interactions) by divers. We consider this as a Gedankin experiment, our sole purpose being to determine the effects on outbreaking starfish populations and on the production (stability) of the coral community as a whole. We therefore study the influence of manipulation of the social structure on the critical Hopf value “(c in the aggregated system (this is the triggering mechanism of an outbreak), and also the effect on the KCC-curuature (i.e., production stability), 8s well. The model is as follows:
(3.la)
+5 dC; edt=
($)m
-kl’“C;
($)m(C,a)2
-/- k21C’ 1 1 +&
-
-4F2C;
m --azc;c;
m-l
+X14, -
a1 (CT)’
(3.lc)
+~($)m(~)m(C:)2-6:F&~+X2C~), edc2” = -ki2C; dt
+ k;lC;
+5
E%=-hfi dF2 Yz-=
($)m
+ e
-“-QII~;~;
m-l
($)m
+ h&t + E (&FI
-IE12F2 -I-kzlfi
(C;)‘-6;F2C;
(3.lb)
_ a2 (~22)~
+X24,
(C; + Cl) + -/I(FI)2 _ pIFI) ,
+ &(P2F2 (C; -I-C;) + 72(F2)2 - pzF2) .
(3.ld) (3.14 @lf)
There are two time scales here, the fast and the slow. The latter is easily detected by the presence of a small positive parameter E. The fast part is the linear subsystem determined by ky and kgj and is identical with that in [16]. But, the slow part is quite difJewnt from our prewious model. Here, for example the manipulation ratios ki2/ki1 and ki2/kF1, for patch #2 and #l, respectively, play a role in modelling the social interaction tens. Although the individual ky enter on the fast-time scale in the linear part, these two ratios are determined by the fast equilibrium values above and play a definite role in social behaviour at the macroscale of the slow dynamics. We suppose for definiteness that species i of corn1 are on patch i and only there to start, and that all starfish are in patch #S to start.
Note the and (3.lf) all indices, equilibrium
strong symmetry of (3.1). Equations (3.la) and (3.ld), (3.lb) and (3.lc), and (3.le) are dual in the sense that say, (3.ld) can be obtained from (3.la) by interchanging i.e., replacing 1 with 2 and 2 with 1. Therefore, it is of interest to study the fast which satisfies v;* = v;“’ = P > 0, V;” = Vf’ = Q > 0, P+Q=l,
(3.2a)
126
P. ANTONELLI
et al.
where F=Fl+Fz, (3.2b)
c1=c,1+c& c, = c; + c;, are constants
of the motion for the fast dynamics for which (3.2~)
and ki2 = k;2 + kf’(
‘:*
Vf’=l-vi’, (3.2d)
k;l ‘:’ = k;” + k;’ 7
VI:* =1-v;*.
For the starfish population, we have Fl = V;F,
Fz = V.F
and
v; =
b2
k12 + kzl
and
VJ=l-VT.
(3.2e)
Note A, that the manipulation ratio for patch #2 is multiplied by the cross-patch density ratio for species #2, C,“/CF, in the first equation, while that for patch #l is multiplied by the species #l cross-patch density ratio, C,‘/C,‘, in the second equation. Both kinds of ratios are raised to power m > 2, in keeping with our approach to social interaction modelling, as explained above. Note B, that the dual of equation #l is #4 and that of equation #2 is equation #3. This means we can recover an equation from its dual (or conversely) by merely interchanging indices 1 and 2, throughout. Thus, the dual of equation #6 is #5 and conversely. Note C, that all four social interaction terms are beneficial to the four subpopulations; all four subpopulations are self-inhibiting in the classical logistic sense; species #l and #2 compete with each other on each of the two patches (in the Gause-Witt sense) with the effect on $$
and $$
being the same, and likewise for $$ and $$. Thus, competition is independent of the patch number and may involve individual behaviours, such as use of stinging nematocysts, regardless of which patch they are on. The ‘carrying capacity’ for species #l is Xl/al by analogy with single time-scale ecological models. This can be read off (3.la). But, this refers to patch #l and the value for species #l on patch #2 is Al/as, so these values are generally different. But, Note D it is assumed that their ‘intrinsic growth rates’ (the A’s) are the same on each patch, but not generally (X’ # X2) on different patches. The aggregated system
is seen to be
-dC2 = -al
--& (
dt
z
+a2
(A)
(3.3a)
C1C2-a2(C212
> (!$)m(C2)2
= @ICI + b2Cz)F + 7F2 - pF,
-62CzF+rzC2,
Corals and Starfish Waves
where al
= al
(P2+ Q") ,
127
a2=a2(P2+Q2),
51 = V; (6:P + b;Q) ,
62 = V; (6;P + S;Q)
r1 = XIP + X2Q,
rz = X2P + X’Q,
h
bz =
=
Y=
PlVl+,
71 (W2
+
72 (v;)2
1
, (3.3b)
P2V;,
P=Plv;
+p2v;.
This aggregated system (3.3a) is of the form considered by Antonelli, Bradbury and Lin [13]. The case m = 2 is the same as the aggregated system of Antonelli and Auger [16], but (3.3b) is not the same. SIX REMARKS ON THE AGGREGATED SYSTEM (3.3). (1) If bl = b2 in (3.3a),(3.3b), then (3.3a) is invariant under index interchange. In fact, the first and second equations are dual while the starfish equation is self-dual. This is consistent with the definition of P and Q. (2) We assume for definiteness that corals of species i are all on patch #i at time zero. This leads to a continuous spectrum of Q values from 0 to 1. In the earlier work [16], Q could only take values between 0 and l/2 because the aggregated system coefficients (P2 + Q2), and no such ai = ai(P2 - Q2) must be positive. In the present model, ai = CE~ restriction is necessary. (3) The concept of ‘carrying capacity’ can be formulated for the aggregated system in the usual way once we understand that P (and Q) must be a priori specified. Then we define the carrying capacity for patch #l with jlxed Q (and P) to be rl -= al
X1P + X2Q cq(P2 + Q”)’
Likewise for patch #2, and with the same Q (and P). The biological interpretation is merely that this ratio gives the standing crop value for Ci (with this P and Q), which is the largest possible for the micro-environment of patch #i. (4) Specifying a condition like ri = rs in the aggregated system is equivalent to X1(1 - Q) + X2Q = X2(1 - Q) + X’Q, for al2 Q in the range [O,l]. Therefore, X1 = X2, which in turn refers back to remark (D) after (3.2). Thus, there are separate notions of ‘carrying capacity’ for (3.1) and (3.3a)(3.3~). They do agree when Q = 0, however. Therefore, we have enlarged the concept in a logically consistent fashion. (5) Different coral interactions can lead to the same aggregated system, formally, but the definitions of the coefficients expressed in terms of the fast equilibrium ratios V:* and Vi* (P and Q) must be different, generally (6) In the first equation of (3.1), a same-patch competition term E(-_PrC:Ct) may be added to the right-hand side and also to the third equation. A similar term may be added to the second and fourth equations of (3.1), E(-_P~C~C~). Application of the aggregation method leads to (3.3a), but with the term -2piPQ(Cr)2 added to the right-hand side of first equation of (3.3a) and -2&PQ(C2)2 added to the second. These terms are added to -YZ~(C~)~and -a2(Cs)2, respectively. Note that although the coeficients are altered, the process of including same-patch competition terms at the level of individual behaviour (microscale) does not change the final form of (3.3a). Moreover, if these terms are included in (3.1), the choice of a special equilibrium, Q = 0, for the fast dynamics (i.e., manipulation of the two patches keeping species #l on patch #l and species #2 on patch #2) will cause the resulting macroscale terms to vanish.
128
P. ANTONELLI et al.
Let, us now describe the equilibrium for this aggregated system. Let al = a2 = a, bl = b2 = b, b1 = ?i2 = 6, and ~-1= 7-2 = r. Then there exist a unique positive equilibrium (COO,CZO,Fo):
Go
=
r-6Fo a(1 + km)
a1 := a2 F.
=
and
C20 = kClo,
a1 = up(
a,
(3.3c)
1 + km) - br( 1 + k)
ay(1 + km) - bs(l + k)’ The equilibrium still exists when the above equalities of the coefficients are relaxed, but, numerical methods and computer programs are necessary [13,20,21]. If bs(l + k) - ay(1 + km) > 0 and br(1 + k) - ap(1 + k”) > 0, the above equilibrium (3.3~) is stable until y increases beyond 7~ where
a@(1 + k”) ‘Yc = bX(1 + k) + a(A - p)(l + km)*
(3.3d)
Moreover, a periodic solution of small amplitude appears as soon as y is larger than 7~. If m = 2, it has been shown that this amplitude increases with y [21]. It, is most certainly true for m 2 3, too, but it is still only a conjecture. As in the case of [16], we can show (3.4) where E 1s * a certain positive quantity. The proof is much the same and is omitted. able to obtain the meaningful results.
Thus, we are
(A) $$$
> Sibi implies s
> 0; so taking Q close to unity ensures 7~ is as large as possible.
(B) 5$$
< s:ai
< 0; so taking Q close to zero ensures 3% is as large as possible.
implies $$
Thus Q, the fast equilibrium from manipulation of the coral community, is a measure of protection. In the absence of knowledge of the actual values of the coefficients and parameters of our model, the best strategy would be to take Q = l/2, or what is the same, ensure that each of the patches is as diverse as possible. But, what effect will the social parameter m have on ^yc? To answer this question, it, is crucial to rank the corals. That is, we must specij$ a distinguishing feature for the two species, and use it to define which is to be #l and which #2. Since growth rates X1 = X2, a natural ranking is carrying capacity as defined in (3.1). We shall call #l, that coral species with the largest Xi/ai. Thus, Xl/crl > X2/a2 or 01 < (~2. From this, we conclude from (3.3~) that k > 1 for any m 2 2. Now, it is easy to see that
aye
-= dm
a6pr( 1 + k) k”&k [br(l + k) + a(~ - p)(l + k-)12’
(3.5)
treating m as a continuous (or alternately, allowing m to take on any real number 2 2). We conclude that -yc increases with increasing m. Therefore, increasing m is beneficial to the coral community. But, as already mentioned, increasing m decreases orbital stability and m 5 8 follows (see [13]). The above ranking procedure is necessary and can actually be carried out in ‘tank’ experiments to determine Xi and Xi/ai for each species, separately. This can be done in a large aquarium (as in Townsville, Queensland) or directly on a nearby reef by glass enclosure techniques. This method is compatible with system (3.1). For example, taking Ci = Cf = Cz = 6: = 6; = Sf = 6; = 0, we obtain logistic behaviour for C: and likewise for C.$.
Cords
and Starfish Waves
129
4. FAST TIME-SCALE MANIPULATION OF THE ANALYTICAL TROPHODYNAMICS OF CORALS In this section, we investigate how fast time-scale manipulation by divers influences production stability of the analytical trophodynamics for the aggregated system (3.3) when no starfish are present, i.e., we take 61 = &is= 6 = 0. This is our model of a healthy reefal community of two coral species. It is easy to obtain ‘lo = (1 + km)ol(lr- 2Q + 2Q2) ’
c-20 = IcClO.
(4.1)
Note, Cm is invariant under interchange of P and Q. Also, as always, species #l is assumed to have the laqest carrying capacity of the two. Thus, k > 1. We see from (4.1) that Czc > Cm and that as m increases, both standing crop values decrease. Fixing k,m,cr in (4.1), we see that as Q increases from zero, Cis, C&J increase, at first, but as Q exceeds l/2, they steadily decrease to values given by Q = 1 (or Q = 0), in (4.1)) all the while maintaining C&J > Crs. Thus, Cm, as a function of Q only, is concave down, positive, and symmetric about its maximum at Q = l/2. Now, we use the Volterra production equations, first at the two time-scale level of (3.1) and then at the aggregated level as follows. With Volterra (1936), let us write
:=ej
x;(t)
’
J
t
C; ds + X;(O),
for i,j = 1,2.
0
(4.2)
These four variables measure production. They are surrogate variables, generally, and may represent many aspects of coral production. Typically, Xj measures aragonite CaCOs of the skeleton of species i on patch j. Often we take it to be a positive multiple of the log biomass [l-3,17,18]. Th e constants f?j are per capita production rates and are patch dependent, but not species dependent. This simplification holds for closely related species, as when they are invertebrates of the same genus (say, Acmpom). In any case, for simplicity, we take ei = Ls = 1. Now define total production on patch #l and #2 as x’(t)
= x,‘(t)
+X,2(t),
X2(t)
= x;(t)
+X,2(t),
(4.3a)
and note that dX1
-=e1q dt
and
dX2 = e2c2, dt
(4.3b)
since Cl = C,l + CT, CZ = Ci + Ci. The variables X1 and X2 are the aggregated Volterra production variables. (We use the upper index to indicate patch number here.) In terms of X”, the aggregated trophodynamics for (3.3) decoupled from starfish becomes upon substitution of $$
= Ci, just the system d2X’ m -a2xT+al dt2 + m-l
dX1 dX2
d2X2 m -aldtdt+a2 dt2 + m-l
dX’ dX2
of course, m 2 3, here. Note, these equations are invariant under interchange of P and Q. This models the analytical trophodynamics &s follows.
130
P. ANTONELLI
et al.
a-Coral System with Social Interactions Now, we follow [2,3]. If the trajectories
of (4.4) are varied into nearby ones according to
2 = xi + d&T, one obtains for the first-order approximation in 67, the (KCC) equations convention on upper and lower repeated indices) d2ui -$ + g;$
ofvariation
(summation
(4.5a)
+ g;ru’ = 0,
where (4.4) is written as (4.5b) defining the two functions g’ and g2, while the semicolon denotes partial differentiation to T and a comma, that relative to X’. overator Dui -::=-
Kosambi,
relative
Cartan and Chern [5-71 introduced the
dui dt +9p,
dt
so that (4.5) can be written as a linear equation D2Ui
-
dt2
= Q’,
(4.7a)
where (4.7b) This is called the KCC-path deviation equation [3]. It contains the relevant curvature information in the eigenstructure of l$. We write the torsion tensor [3],
(4.8)
and then find the KCC-scalar
curvature K from the formula [2,3],
(4.9) where (4.10) is often called the ecological metric [2,3]. Antonelli and Shimada compute! the curvature K in [26]. We will spare the reader the difficult calculation
and merely record the answer here. Using the more convenient Ci for 5,
[(c2/c,)"-'
K=m(m-2).a2
4(m-1)2
*
-km-j2
~[(c2/cI>"+
l]
we have
(4.11)
(C2/G)2m-1
Clearly, K 2 0 for all values of Xi, Ci, i = 1,2. In fact, K > 0, except when &/Cl = k, which includes the unique positive equilibrium (Cal, C&2) of (4.1). Note, that K is invariant under interchanging P and Q.
Corals and Starfish Waves
131
Well-known results now guarantee the system (4.4) is stable in the sense of Liapunov [2,27]. This means that any trajectory is stable, or put another way, any two trajectories will have close (Xl(t), X2(t)) if initially (X1(0),X2(O)) and (s(O), g(O)) were close [27, p. 1751. Note, that for m = 2, K E 0, and we have a situation similar to straight lines in Euclidean space which are, indeed, unstable. We conclude that social interactions are, indeed, beneficial to the healthy coral community in the strong sense that production is more dependable (or predictable), robust and repeatible [2,27]. Let us further determine the influence of Q on the production stability K in (4.11). It is easy to compute $$ and @ to conclude that the graph of K as a function of Q only, is concave up, positive, and symmetric about its minimum at Q = l/2. That is, g
= 4a2 . He (P2 + Q”) (29 - 1))
and (4.12)
$$
= 16a2 . H . (3Q2 - 3Q + 1) ,
where H = K/a2 > 0, (i.e., G/C1 # k). Recalling that the graph of the standing crop value COO,as a function of Q only, is concave down, positive, and symmetric about its maximum at Q = l/2, we are able to correlate our relations. Thus, a smaller standing crop Cl0 corresponds with a larger stability of production K, and, conversely. The maximum Cl0 happens exactly when K is at its minimum (i.e., Q = l/2).
5. THE REACTION-DIFFUSION-TRANSPORT LARGE-SCALE WAVES
MODEL:
In the system (3.1), we now add diffusion in space and transport due to ocean currents in a way similar to the original model of Antonelli et al. [28], but at the microscale. Refer also to [29, Chapter 21. Letting A = & + wa’ denote the Euclidean Laplacian with z and y denoting latitude and longitude in a flat ellipse about 20 times long (2000 km) as wide and oriented with its major axis approximately in north-south directions, we replace E$$ in (3.1) with constant coefficient expression, 8C: act T_$?$ , --D;AC+T,“K& ( di! > ~9
in (3.1) with constant coefficient expression, - D;AC,z -T:‘=
ac:
-
,z$$ in (3.1) with constant coefficient expression,
and E!$
in (3.1) with constant coefficient expression,
Upon aggregation expression
of (3.1) to (3.3), the left-hand side 9 -a9 at
_ DfAcl
_ T;*z
is replaced by constant
_ T;‘!?$
coefficient
P. ANTONELLI et al.
132
and similarly, F
is replaced by constant coefficient expression,
ac, _2 at
D*AC
where D; = D;V;’
+ D2V2’ 1 1 7
D*2 = D;V,2* + D2V2’ 2 2 9 p* = TllVl’ + 7712v2* 1
1
1
119
+ 7712V2’ 2 1 9 + T22V2’ 1 2 7
T,2’ = T;lV;*
+ T;2V;*,
are the aggregated diflusion and advection coeficients. Also, F = Fl + FZ and E$$
and ~9
(5.la)
T,* = T;‘Vl’ 1 772’ = 7721V” 1 1 2
They are constants.
is replaced by
by
in the last two starfish equations of (3.1). Both of these are constant coefficient expressions. Therefore, aggregation leads to replacement of $ in (3.3) by the constant coefficient expression, dF --D;AF-T$!&T,;‘ ~, at
aY
where the advention coefficients for starfish are T3’ = ‘j-‘3lV* + ~32~8 1 1 1 1 27
(5.lb)
T3* = ,;lV* + T32V* 2 1 2 2.
According to [28], we make two general assumptions. For i = 1,2, 9 = $$ = $$ = 0 in dn x R+ (i.e., no Aux boundary conditions on fl for all positive time), where Z denotes the outward pointing normal vector to the boundary 80.
ASSUMPTION
A.
B. (Constant southward drift of the East Australian current.) in (x = constant) direction is nil. That is, ASSUMPTION
Average transport
T;’ = Trr2 = T,2l = Tf2 = e > 0, T,‘l = Ti2 = T;l
= Ti2 = 0.
One transforms the equilibrium state (Ccl, Cos, Fo) to the origin by setting
~~=c~-co~, i =
1,2,
fl=F-Fo.
Using Assumptions A and B, we can rewrite our reaction-diffusion
transport
model as (i = 1,2)
(5.2)
. Corals and Starfish Waves
Here, 2 = (f 1, f 2, f s ) are obtained
directly
133
from (3.3a)-(3.3c).
We refer the reader
more details on the precise form of s. In order to discuss the plane-wave solutions to (5.2), we now introduce
to [28] for
the phase-variable
2 = ct - a12 - azy, and set Vi(Z)
(i = 1,2)
= z;i,
and
Us(Z)
= F.
We substitute these into (5.2) and further require Vi to be 27r-periodic in 2. Such solutions, when they exist, are called plane wave trains (or plane waves). They do not satisfy Assumption A, however. The phase-velocity is the angular frequency C, divided by the wave number aI + ai E la’12. The vector a’ denotes the direction of the wave-front. Equations (5.2) become the vector equation (5.3) where 2; = C + aiC is the phase-shi,fed of diffusion constants. This transformation studied
by Kopell
and Howard
angular
frequency
reduces
and D is a 3 x 3 diagonal
the equation
matrix
(5.2) to the no-transport case
[30]. The phase-shiped velocity is C/]L?]~ with direction a’.
By known results [30], (E:I, (?z, @) is a plane wave train solution of (5.2) if and only if c(Z) satisfies (5.3); there is a family of wave trains parametrized by the wave length 27r/]a’] and that large values of this give good approximations to the socially dependent limit cycle of (3.3). These waves can have large amplitude and phase velocity, if m = 2 [28]. Note, that a’ depends on Q here, as does the phase-shifted
velocity
and angular
Recall, that in large-scale diffusion models, Great Barrier Reef, i.e., on R. This means that of all reefs of the Great Barrier Reef so that Q Q effects the wave length 27r/]a’] of the wave train
frequency.
What
is the meaning
of this?
we suppose that (3.3) holds everywhere on the our Gedankin experiment requires manipulation (and P) are defined. We see from the above that solutions. Obviously, ]a’] = (P2+Q2)161, so that
for Q increasing from zero to l/2, the wave length is steadily increased, while for Q increasing beyond l/2, towards unity, the wave length steadily decreases. The direction vector for the wave train a’does not change its direction as Q varies in [O,l]. The Conway-Hoff-Smoller Theory [31] was used in [28]. In the same way, we are able to conclude from CHS that any no-flux solution starting out at t = 0 in the stability torus set C of (3.3) converges, at rate B > 0, to its averaged value over the whole s2 and that for large times, this average is a good (uniform) approximation to the socially dependent limit cycle of (3.3). In particular, the large time behaviour of the wave solution is determined completely by the cycle and convergence time is relatively short. Therefore, for large times, the plane-wave train solution will well-approximate this wave away from the boundary of 0, as long as 27r/la’l is large enough. Prom the previous paragraph, we can ensure this by taking Q close to l/2. Now, because of the slow and more variable increase in coral abundance compared to the boom in starfish abundance [19], we expect COTS diffusivity 0: > Dz and 0: > 0: in (5.2). Letting d* = min{Dr, D;}, we have the formula for rate of convergence CT =
where b is some positive
constant,
$ -M
- ;(3b)‘i2
B is the length
> 0,
of the major
axis of R (i.e., N 2000 km), and
A4 is the Euclidean norm of the Jacobian matrix of 3 maximized over the stability torus C (attractor). The positivity of o means that d* must be large relative to R2 and relative to the rates of reaction defining M and to advection per unit length e/R. This must be assumed before any consequences of the CHS theory can be brought to bear on the model (5.2). These statements
134
P. ANTONELLI et al.
apply also to the wave train solution, which for us models the central portion of the Great Reef, at large times. Now, from (5.la), we have 0;
= D;P + DTQ,
Barrier
Df = D;Q + D,2P,
so that d* will depend on patch-difisiwities for each coral species, say of genus Acropora. Then, we can reasonably take 0: = Di 3 D1 and 0: = 02 G D2. If D2 # D’, then 0: = D’ + (D2 Dl)Q, D,* = D2 + (D1 - D2)Q and Q increasing will increase the minimum d* of D;, D;. Thus, we would take Q as small as possible. If D1 N D2, then there is no effect from Q in this Gedankin experiment. We refer the reader to [28] for more biological discussion. In particular, we refer to one of the main conclusions of [28], that large-scale synchronous outbreaks in the central Great Barrier Reef cannot occur if there is no net advection current (! = 0). Our intent here has been to explore
the role of Q in this previously
developed
theory.
In conclusion, our analysis shows that the best strategy of conservation of reefal communities is to optimize diversity between the patches, i.e., Q M l/2, when the model parameters cannot be completely known. This conclusion was reached in our previous paper [16], but here social effects among corals have been added to the model. In the case of lowest COTS densities, as happens every 12 to 15 years, but with opposite phase to COTS peaks, coral community production stability is at its lowest, but standing crop values are at their highest for Q = l/2.
REFERENCES 1. V. Volterra, Principles de Biologie Matht?matique, Acta Biotheo. III, pp. l-36, (1936). 2. P.L. Antonelli and R.H. Bradbury, Voltewa-Hamilton models in the ecology and evolution of colonial organisms, World Scientific Press Series in Math. Biology and Medicine, New York, (1995). 3. P.L. Antonelli, R.S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic, Dordrecht, (1993). 4. J.L. Harper, The Population Biology of Plants, Academic Press, New York, (1977). 5. D. Kosambi, Systems of differential equations of the second order, Quart. J. Math. (Oxford Series) 6, 1-12 (1935). 6. E. Cartan, Observations sur le memoire p&&dent, Math. Zeitschrift 37, 619-622 (1933). 7. S. Chern, Sur la gbmetrie d’un systbme d’equations differentialles du second ordre, Bull. Sci. Math. 63, 206-212 (1939). a. H. Wilbur, The ecological relationship of the salamander Ambystoma latemle to its all-female, Gynogenetic associate, Evolution 25, 168-179 (1971). 9. H. Wilbur, Competition, predation, and the structure of the Ambystoma-Rana Sylvatica community, Ecology 53, 3-21 (1972). 10. G.E. Hutchinson, A note on the theory of competition between two social species, Ecology 28, 319-321 (1947). 11. G.E. Hutchinson, An Introduction to Population Biology, Yale University Press, New Haven, CT, (1978). 12. P. Antonelli, R. Bradbury and X. Lin, On Hutchinson’s competition equations and their homogenization: A higher-order principle of competitive exclusion, Ecol. Mod. 60, 309-320 (1992). 13. P. Antonelli. R. Bradbury and X. Lin, A higher-order predator-prey interaction with application to observed starfish waves and cycles, Ecol. Mod. 58, 323-332 (1991). 14. N. Hairston, J. Allan, R. Colwell, D. Futuyama, J. Howell, M. Lubin, J. Mathias and J. Vandermeer, The relationship between species diversity and stability: An experimental approach with protozoa and bacteria, Ecology 49, 1091-1101 (1969). 15. J. Vandermeer, The competitive structure of communities: An experimental approach with protozoa, Ecology 49, 362-371 (1969). 16. P. Antonelli and P. Auger, Corals and starfish devastation of the Great Barrier ReeE Aggregation methods, Acta Biotheoretic 43, 481-493 (1995). 17. P.L. Antonelli (Editor), Proceedings International Workshop on Population Dynamics of Outbreaks, Mathl. Comput. Modelling 13 (6) (1990). 18. R. Bradbury (Editor), Acanthaster and the Corn1 Reef: A Theoretical Perspective, Springer-Verlag, Lecture Notes in Biomath, (1990). 19. P.J. Moran, The Acanthaster phenomenon, Ocennogr. Marine Biol. Ann. Rev. 24, 379-480 (1986). 20. P. Antonelli and N. Kazarinoff, Starfish predation of a growing coral reef community, J. Theor. Biol. 107, 667-684 (1984). 21. P. Antonelli, K. Fuller and N. Kazarinoff, A study of large amplitude periodic solutions in a model for starfish predation on coral, IMA J. Math. Appl. in Med. and Biol. 4, 207-214 (1987).
. Corals and Starfish Waves 22. 23. 24. 25. 26. 27. 28.
29. 30. 31.
135
J. Pandolfi, Developmental sequences in colonial corals: An overview, Mem. Ass. Austdias Palaeontols 8, 69-81 (1989). J. Pandolfi, Heterochrony in colonial marine animals, In Heterochrony in Evolution, (Edited by M.L. McKinney), Plenum Press, (1988). D.J. Barnes, Growth in colonial scleractinians, Bull. Mar. Sci. 23, 280-298 (1973). P.M. Auger and R. Floussarie, Complex ecological models with simple dynamics: From individuals to populations, Acta Biotheoretica 42, 111-136 (1994). P.L. Antonelli and H. Shimada, On l-form Finsler connections with constant coefficients, Tensor, N.S. 50, 263-275 (1991). D. Laugwitz, Diflenmtial and Riemannian Geometry, Academic Press, New York, (1965). P. Antonelli, N. Kazarinoff, Ft. Reichelt, R. Bradbury and P. Moran, A diffusion-reaction-transport model for large-scale waves in Crown-of-Thorns starfish outbreaks on the Great Barrier Reef, ZMA J. Math. Appl. in Med. and Biol. 6, 81-89 (1989). in Hiemrchically Organized Systems, Pergamon Press, (1989). P.M. Auger, Dynamics and Thermodynamics N. Kopell and L. Howard, Plane wave solutions to reaction-diffusion equations, Studies in Appl. Math. 4, 291-324 (1973). E. Conway, D. Hoff and J. Smoller, Large time behaviour of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math. 1, 1-16 (1978).
Pergamon
PII: SOS95-7177(98)00012-O
Corals and Starfish Waves on the Great Barrier Reef: Analytical Trophodynamics and Z-Patch Aggregation Methods P. ANTONELLI Department of Mathematical Sciences University of Alberta Edmonton, Alberta, Canada T6G 2Gl P. AUGER Universitb Claude Bernard, Lyon 1, URA CNRS 243 43 Bd. du 11 Novembre 1918 69622 Villeurbanne Cedex, France R. BRADBURY National Resource Information Centre, GPO Box 858 Canberra ACT 2601, Australia Abstract-In this paper, singular perturbation theory is applied to the Antonelli/Kazarinoff limit cycle model of the Crown-of-Thorns starfish (COTS) predation of corals on the Great Barrier Reef. At the microscale of individual reefs, the previously published l-patch dynamics based on aggregation of individual behaviour is extended to include social interactions, spatial diffusion, and a&e&ion currents. As a consequence, the parameters which characterize the well-known wave solutions (i.e., the Reichelt starfish waves) are reinterpreted in terms of individual behaviour parameters. Likewise, the analytical trophodynamics of coral community production (without starfish), involving geometric invariants of second-order ODl% is defined at the microscale by direct manipulation of the reefal structure. The aggregation method leads to new insights into Liapunov stability of production of the coral community ss a whole. Keywords-Aggregation,
Geometrictheory of second-orderODEs, Diffusion, Reef ecology.
1. ANALYTICAL
TROPHODYNAMICS
That branch of ecology known as empirical trophodynamics is concerned with ‘stocks and flows’ models: the stocks of biomass and the flows of input (consumption) and output (production). These are often measured as caloric energy, forms of carbon, wet weight, and so on, but the idea is the same in each instance and is tied up with the so-called trophic pyramid of Lindeman. Although this approach is understandable, we do not follow it. Instead, we directly link ecological equations (competition, etc.) with production of biomass and consumption of energy as did Volterra in 1936 [l]. In analytical trophodynamics, production (consumption) is specified by Volterra’s equation,
xi(t)
=
t J
e,, *
iVi(s) ds + Xi(O),
0
Supported
in part by NSERC-A-7667.
121
(1.1)
122
P. ANTONELLIet al.
where Ni is size of ith population and t(i) is a positive constant. Such Xi are surrogates of biomass and include various useful examples such as the above, secondary compounds for defense, and allometric measures of biomass. This approach has been systematically explored since the early 1980s by our group [2,3]. Whereas activity in empirical trophodynamics has been concerned with refinement of measurements, as for example, the “Big Green Machine” of Birch, and various energy budgets, it has focused on the transformations between surrogate variables. The focus for us is different in that we explore the dynamical structures in an ecosystem which are invariant of the particular surrogate variables and their transformations. Our main interest is in species of colonial organisms, the ‘modular organisms’ of Harper [4], in which the biomass surrogate variables Xi and their production rates c = e,,)Ni are independent, reflecting the great plasticity in size of individual colonies (e.g., plants, corals, bryozoans, etc.). Given these two sets of quantities at time zero (i.e., the initial production levels and rates), we seek to predict their values at all future times. Accordingly, we postulate n Volterm production equations and n ecological interaction equations which combine to yield a second-order system of ODES. Bather than attempt numerical or analytic solutions, we plan to study the intrinsic (i.e., geometric) properties of this system of n second-order equations. Our approach does not require any optima&y assumptions, generally. Analytical trophodynamics studies properties of Volterra production (consumption) inherent in the second-order ODES and which are invariant under transformation between particular sets of surrogate variables. We claim, furthermore, that the efforts in empirical trophodynamics have been misdirected because of its focus on establishing the nature of the transformations, rather than on the underlying invariant properties of the dynamics (e.g., production stability, species interaction patterns and their efficiency, etc.).
If our stated goal is to understand the intrinsic properties of a postulated second-order system of ODES, just what is it that we gain by achieving this? These intrinsic properties are described by curvatures which give the complete spec@cation of the stability of production OTconsumption processes. For example, for each of two nearby similar islands with the same number and kinds of species, one can decide by computing curvature whether or not production levels and rates will remain close, given they are close initially. More generally, the curvature array completely determines the identity of the dynamical system according to the KCC-theory of Kosambi-CartanChern [5-71. Although computations are in general difficult, they can be made considerably simpler in the cases we need for ecology, namely, for a large class of ecological interactions with densitydependent effects which scale quadratically, according to experimental results of Wilbur [8,9] and Hutchinson’s theory of social interactions [lO,ll]. We have previously used this method on two problems and have had considerable success. The first is the homogeneous Hutchinson competition equations [12] and the second is higher-order predator-prey equations [13]. Both of these problems involve the experimental findings which reveal that social interactions first studied by Hutchinson in 1947 [lO,ll] and found in real data of ecological communities by Wilbur [8,9], Hairston et al. [14], and others (see [15]) to be of equal importance (from the analysis of variance method) as those of the usual quadratic terms in community interactions. We model this by requiring density-dependent terms added to, say, Gause-Witt competition equations, shall be (positively) homogeneous of degree two. Systems of second-order ODES with this property sre called sprays and are the topic of at least two recent books [2,3]. In the present work, we examine social models in the context of aggregation methods [16], both with and without spatial diffusion, which have been successful in modelling the Acanthaster plan& devastation of corals of the Great Barrier Beef. The two basic references for this problem are [17,18]. We shall generalize the previous aggregation model [16] to include social interactions among corals, which are relevant for the genus Acropom, the most abundant and most attacked coral on the Great Barrier Beef [19]. In Section 2, we go into the details of our model.
Corals and Starfish Waves
2.
SOCIAL INTERACTIONS
The literature
on ‘social’ interactions
123
IN COLONIAL
has had solid experimental
ORGANISMS
and theoretical
contributions
beginning with Hutchinson’s 1947 classic paper, in which cubic terms replace the classical quadratic terms in the 2-species competition equations of Gause and Witt [lO,ll]. Since then, the work of Hairston et al. [14] on bacterial and protozoan communities and that of Wilbur [8,9] have rigorously established their existence via analysis of variance techniques. There is a thorough discussion of Wilbur’s data on Ambystoma salamanders in Hutchinson’s ecology textbook [ll], where he explains the social interaction between A. latemle and its sexual parasite, the all-female species, A. tremblayi. Both Wilbur and Vandermeer
[15] discuss formal mathematical
models,
but state that systems
of equations with cubic, quartic, and higher Taylor series terms are not mathematically tractible. This may well be true, but addition of such terms do not conform to Wilbur’s all important statistical
evidence.
We quote directly
from Wilbur
(1972).
Frequently the higher-order interactions are as important as the main effects. This means that competition among salamander larvae is not a simple additive process that is a function of the total number of larvae in the community, but is a complex interaction between the proportions as well as the abundances and the identity of the species. (Our italics.) This quote will be easier to understand system may always be expressed by
if we recall that
a 2-species
competitive
dN1 = N1.F(N’,N2), dt dN2 = N2.G(N1,N2). dt
P-1)
Thus, the ‘additivity’ refers to the linearity of F and G in the usual Gause-Witt models. But, what of higher-order interactions being as statistically important as the usual quadratic terms? The way to incorporate this finding into a model is to require that social interaction terms scale the same way as quadratics. For example, terms like or
[N’ (N~)~]“~
are (positively) second-degree homogeneous in that either are replaced by pN1 and pN2 where p > 0 is an arbitrary
is multiplied by p2 when Ni and N2 real number. This property does not
hold for any higher-order term in a Taylor series, although it seems that Hutchinson thought it did [lo] in his discussion of the statistical meaning of the coefficients of his cubic terms. See [12] for a more complete discussion of Hutchinson’s error and for analysis of the homogenized version of Hutchinson’s equations, which use terms like the two above. It is noteworthy that Wilbur states that proportions are major influences in the dynamics of his salamander larvae populations. The first term above is obviously of this sort. Letting m 1 2 be an integer, we introduce social interactions, as in [13], for the starfish/coral dynamics as follows: m-2
dN1 -=X1N1-~I(N1)2-~2 dt
.N1N2+L
dN2 -=X2N2-~2(N2)2-~1 dt
.N’N2+L
. (N2)” - &FN’,
m-l m-2
. (N1)2 _ b2FN2,
(2.2)
m-l
All coefficients (~1, CQ, Xi, X2,&, 62, p, y,p are positive with N1, Iv2 being coral densities and F the starfish density. Xi, Xz, E are growth rates, Xi/cri, Xz/cus are single species carrying capacities,
P. ANTONELLI et al.
124
p, Sr, 6s are the interaction coefficients for starfish preying on corals, and y is the weficient of starfish aggregation. If we set m = 2, we obtain the (2 corals/l starfish)-model of Antonelli and Kazarinoff [20,21], in which every term of degree greater than one is quadratic. It is m > 3, which forces the social interaction terms (there are only two) to be nonquadratic, but to nevertheless scale the same as quadratics. The reader should take note that for m 1 3, neither N’ nor N2 is allowed to be zero, but that years of observation of Acanthaster plan& predation on the corals of the Great Barrier Reef indicate that although polyp densities may be drastically reduced on a reef, they are never actually vanishing. We also remind the reader that the predator F catches prey based only on food preference asymmetry in &N’F and 62N2F and that nutritionally for the predator, either species of coral is as good as the other. It is because of this assumption that we use the term ,LIF(N’+N2) instead of ,&NlF + P2N2F. Finally, when 61 # 62 stability of a small amplitude limit cycle bifurcating out of an equilibrium (Ni, Ni, Fo) and caused by increasing y beyond a certain critical (Hopf) value “yc,has been shown [13]. For m = 2, this cycle can be of large amplitude [21]. It is still an open question for m 2 3, but it is expected that the system will admit large amplitudes, as well. Also, any factor lowering the value of the coefficient of starfish aggregation 7 controls starfish; severe climatic disturbances and fish predation on starfish and their larvae, are examples. On the other hand, an influx of larvae of A. plan& would tend to increase y (see [17,18]). The effect of increasing the social parameter m has been determined; at first the asymptotical orbital stability of the limit-cycle solution increases until m = 6 is reached, whereupon it decreases steadily, eventually completely destabilizing the cycle [13]. Since it is known from observation that the cycle persists [17], it follows that m 5 8 in this model. Coral species of the genus Acropom are the most frequently attacked by A. planci [19]. For species of Acropom, Pandolfi [22,23] has documented the presence of dimorphic groups of individual coral polyps within single clones for which growth and development is morphologically distinct from other such groups. These so-called astogenetic groups are subjected to colony-tide developmental control. There is evidence that these groups compete interspecifically within a given clone of Acropom [23,24]. Of course, there is intraspecific competition between the polyps of an astogenetic group. Thus, two species of Acmpom, adjacent on a reef, do admit higher-order (i.e., social) interactions and it remains to find the social parameter m for the above model. Note, that our model is not for interactions between a single species of Acropora and another species of a different genus than this one. This is interesting and important, but now we concentrate on Acropom versus Acropora competition. For the estimation of m, note that if N2 = constant, F = 0, and N1 = C < 1, then G N C2-m >> 1 so that a strong recooperative effect results for m > 3 due to the density dependent term, ai/(m - 1)(N2/N1)“-2. This provides a realistic estimation procedure and shows the above model of coral social interactions is falsifiable in the Popperian sense. That is, if this estimation resulted in m 1 8, the model must be modified. We wish now to introduce aggregation methods based on individual behawiour of the starfish and corals which generalize our previous work in [16], in order to better understand the role of social interactions in the Great Barrier Reef ecosystem. For more details and information, we refer to [25]. 3. THE TWO
PATCH
MODEL
(SINGLE
REEF)
Now, the reef (say of diameter l/2 km) will be described as a set of two adjacent patches and two coral species. Patch #l could be raised up above patch #2 on the reef front, for example. Then, during the day, corals of patch #l compete with those of patch #2 for sunlight, while during the night, they compete for microscopic animals in the plankton which flow in changing
Corals and Starfish Waves
125
currents over the reef. Let Cj denote the density of corn1 of species i on patch j, and let Fj be the starfish density on patch j.
Our model includes two parts, a fast part and a slow part (e is a small parameter). The fast part describes the migration of the starfish adult population between the two patches at a rate kij, from patch #j to patch #i, and a manipulation of the coral distribution in space (and hence, the social interactions) by divers. We consider this as a Gedankin experiment, our sole purpose being to determine the effects on outbreaking starfish populations and on the production (stability) of the coral community as a whole. We therefore study the influence of manipulation of the social structure on the critical Hopf value “(c in the aggregated system (this is the triggering mechanism of an outbreak), and also the effect on the KCC-curuature (i.e., production stability), 8s well. The model is as follows:
(3.la)
+5 dC; edt=
($)m
-kl’“C;
($)m(C,a)2
-/- k21C’ 1 1 +&
-
-4F2C;
m --azc;c;
m-l
+X14, -
a1 (CT)’
(3.lc)
+~($)m(~)m(C:)2-6:F&~+X2C~), edc2” = -ki2C; dt
+ k;lC;
+5
E%=-hfi dF2 Yz-=
($)m
+ e
-“-QII~;~;
m-l
($)m
+ h&t + E (&FI
-IE12F2 -I-kzlfi
(C;)‘-6;F2C;
(3.lb)
_ a2 (~22)~
+X24,
(C; + Cl) + -/I(FI)2 _ pIFI) ,
+ &(P2F2 (C; -I-C;) + 72(F2)2 - pzF2) .
(3.ld) (3.14 @lf)
There are two time scales here, the fast and the slow. The latter is easily detected by the presence of a small positive parameter E. The fast part is the linear subsystem determined by ky and kgj and is identical with that in [16]. But, the slow part is quite difJewnt from our prewious model. Here, for example the manipulation ratios ki2/ki1 and ki2/kF1, for patch #2 and #l, respectively, play a role in modelling the social interaction tens. Although the individual ky enter on the fast-time scale in the linear part, these two ratios are determined by the fast equilibrium values above and play a definite role in social behaviour at the macroscale of the slow dynamics. We suppose for definiteness that species i of corn1 are on patch i and only there to start, and that all starfish are in patch #S to start.
Note the and (3.lf) all indices, equilibrium
strong symmetry of (3.1). Equations (3.la) and (3.ld), (3.lb) and (3.lc), and (3.le) are dual in the sense that say, (3.ld) can be obtained from (3.la) by interchanging i.e., replacing 1 with 2 and 2 with 1. Therefore, it is of interest to study the fast which satisfies v;* = v;“’ = P > 0, V;” = Vf’ = Q > 0, P+Q=l,
(3.2a)
126
P. ANTONELLI
et al.
where F=Fl+Fz, (3.2b)
c1=c,1+c& c, = c; + c;, are constants
of the motion for the fast dynamics for which (3.2~)
and ki2 = k;2 + kf’(
‘:*
Vf’=l-vi’, (3.2d)
k;l ‘:’ = k;” + k;’ 7
VI:* =1-v;*.
For the starfish population, we have Fl = V;F,
Fz = V.F
and
v; =
b2
k12 + kzl
and
VJ=l-VT.
(3.2e)
Note A, that the manipulation ratio for patch #2 is multiplied by the cross-patch density ratio for species #2, C,“/CF, in the first equation, while that for patch #l is multiplied by the species #l cross-patch density ratio, C,‘/C,‘, in the second equation. Both kinds of ratios are raised to power m > 2, in keeping with our approach to social interaction modelling, as explained above. Note B, that the dual of equation #l is #4 and that of equation #2 is equation #3. This means we can recover an equation from its dual (or conversely) by merely interchanging indices 1 and 2, throughout. Thus, the dual of equation #6 is #5 and conversely. Note C, that all four social interaction terms are beneficial to the four subpopulations; all four subpopulations are self-inhibiting in the classical logistic sense; species #l and #2 compete with each other on each of the two patches (in the Gause-Witt sense) with the effect on $$
and $$
being the same, and likewise for $$ and $$. Thus, competition is independent of the patch number and may involve individual behaviours, such as use of stinging nematocysts, regardless of which patch they are on. The ‘carrying capacity’ for species #l is Xl/al by analogy with single time-scale ecological models. This can be read off (3.la). But, this refers to patch #l and the value for species #l on patch #2 is Al/as, so these values are generally different. But, Note D it is assumed that their ‘intrinsic growth rates’ (the A’s) are the same on each patch, but not generally (X’ # X2) on different patches. The aggregated system
is seen to be
-dC2 = -al
--& (
dt
z
+a2
(A)
(3.3a)
C1C2-a2(C212
> (!$)m(C2)2
= @ICI + b2Cz)F + 7F2 - pF,
-62CzF+rzC2,
Corals and Starfish Waves
where al
= al
(P2+ Q") ,
127
a2=a2(P2+Q2),
51 = V; (6:P + b;Q) ,
62 = V; (6;P + S;Q)
r1 = XIP + X2Q,
rz = X2P + X’Q,
h
bz =
=
Y=
PlVl+,
71 (W2
+
72 (v;)2
1
, (3.3b)
P2V;,
P=Plv;
+p2v;.
This aggregated system (3.3a) is of the form considered by Antonelli, Bradbury and Lin [13]. The case m = 2 is the same as the aggregated system of Antonelli and Auger [16], but (3.3b) is not the same. SIX REMARKS ON THE AGGREGATED SYSTEM (3.3). (1) If bl = b2 in (3.3a),(3.3b), then (3.3a) is invariant under index interchange. In fact, the first and second equations are dual while the starfish equation is self-dual. This is consistent with the definition of P and Q. (2) We assume for definiteness that corals of species i are all on patch #i at time zero. This leads to a continuous spectrum of Q values from 0 to 1. In the earlier work [16], Q could only take values between 0 and l/2 because the aggregated system coefficients (P2 + Q2), and no such ai = ai(P2 - Q2) must be positive. In the present model, ai = CE~ restriction is necessary. (3) The concept of ‘carrying capacity’ can be formulated for the aggregated system in the usual way once we understand that P (and Q) must be a priori specified. Then we define the carrying capacity for patch #l with jlxed Q (and P) to be rl -= al
X1P + X2Q cq(P2 + Q”)’
Likewise for patch #2, and with the same Q (and P). The biological interpretation is merely that this ratio gives the standing crop value for Ci (with this P and Q), which is the largest possible for the micro-environment of patch #i. (4) Specifying a condition like ri = rs in the aggregated system is equivalent to X1(1 - Q) + X2Q = X2(1 - Q) + X’Q, for al2 Q in the range [O,l]. Therefore, X1 = X2, which in turn refers back to remark (D) after (3.2). Thus, there are separate notions of ‘carrying capacity’ for (3.1) and (3.3a)(3.3~). They do agree when Q = 0, however. Therefore, we have enlarged the concept in a logically consistent fashion. (5) Different coral interactions can lead to the same aggregated system, formally, but the definitions of the coefficients expressed in terms of the fast equilibrium ratios V:* and Vi* (P and Q) must be different, generally (6) In the first equation of (3.1), a same-patch competition term E(-_PrC:Ct) may be added to the right-hand side and also to the third equation. A similar term may be added to the second and fourth equations of (3.1), E(-_P~C~C~). Application of the aggregation method leads to (3.3a), but with the term -2piPQ(Cr)2 added to the right-hand side of first equation of (3.3a) and -2&PQ(C2)2 added to the second. These terms are added to -YZ~(C~)~and -a2(Cs)2, respectively. Note that although the coeficients are altered, the process of including same-patch competition terms at the level of individual behaviour (microscale) does not change the final form of (3.3a). Moreover, if these terms are included in (3.1), the choice of a special equilibrium, Q = 0, for the fast dynamics (i.e., manipulation of the two patches keeping species #l on patch #l and species #2 on patch #2) will cause the resulting macroscale terms to vanish.
128
P. ANTONELLI et al.
Let, us now describe the equilibrium for this aggregated system. Let al = a2 = a, bl = b2 = b, b1 = ?i2 = 6, and ~-1= 7-2 = r. Then there exist a unique positive equilibrium (COO,CZO,Fo):
Go
=
r-6Fo a(1 + km)
a1 := a2 F.
=
and
C20 = kClo,
a1 = up(
a,
(3.3c)
1 + km) - br( 1 + k)
ay(1 + km) - bs(l + k)’ The equilibrium still exists when the above equalities of the coefficients are relaxed, but, numerical methods and computer programs are necessary [13,20,21]. If bs(l + k) - ay(1 + km) > 0 and br(1 + k) - ap(1 + k”) > 0, the above equilibrium (3.3~) is stable until y increases beyond 7~ where
a@(1 + k”) ‘Yc = bX(1 + k) + a(A - p)(l + km)*
(3.3d)
Moreover, a periodic solution of small amplitude appears as soon as y is larger than 7~. If m = 2, it has been shown that this amplitude increases with y [21]. It, is most certainly true for m 2 3, too, but it is still only a conjecture. As in the case of [16], we can show (3.4) where E 1s * a certain positive quantity. The proof is much the same and is omitted. able to obtain the meaningful results.
Thus, we are
(A) $$$
> Sibi implies s
> 0; so taking Q close to unity ensures 7~ is as large as possible.
(B) 5$$
< s:ai
< 0; so taking Q close to zero ensures 3% is as large as possible.
implies $$
Thus Q, the fast equilibrium from manipulation of the coral community, is a measure of protection. In the absence of knowledge of the actual values of the coefficients and parameters of our model, the best strategy would be to take Q = l/2, or what is the same, ensure that each of the patches is as diverse as possible. But, what effect will the social parameter m have on ^yc? To answer this question, it, is crucial to rank the corals. That is, we must specij$ a distinguishing feature for the two species, and use it to define which is to be #l and which #2. Since growth rates X1 = X2, a natural ranking is carrying capacity as defined in (3.1). We shall call #l, that coral species with the largest Xi/ai. Thus, Xl/crl > X2/a2 or 01 < (~2. From this, we conclude from (3.3~) that k > 1 for any m 2 2. Now, it is easy to see that
aye
-= dm
a6pr( 1 + k) k”&k [br(l + k) + a(~ - p)(l + k-)12’
(3.5)
treating m as a continuous (or alternately, allowing m to take on any real number 2 2). We conclude that -yc increases with increasing m. Therefore, increasing m is beneficial to the coral community. But, as already mentioned, increasing m decreases orbital stability and m 5 8 follows (see [13]). The above ranking procedure is necessary and can actually be carried out in ‘tank’ experiments to determine Xi and Xi/ai for each species, separately. This can be done in a large aquarium (as in Townsville, Queensland) or directly on a nearby reef by glass enclosure techniques. This method is compatible with system (3.1). For example, taking Ci = Cf = Cz = 6: = 6; = Sf = 6; = 0, we obtain logistic behaviour for C: and likewise for C.$.
Cords
and Starfish Waves
129
4. FAST TIME-SCALE MANIPULATION OF THE ANALYTICAL TROPHODYNAMICS OF CORALS In this section, we investigate how fast time-scale manipulation by divers influences production stability of the analytical trophodynamics for the aggregated system (3.3) when no starfish are present, i.e., we take 61 = &is= 6 = 0. This is our model of a healthy reefal community of two coral species. It is easy to obtain ‘lo = (1 + km)ol(lr- 2Q + 2Q2) ’
c-20 = IcClO.
(4.1)
Note, Cm is invariant under interchange of P and Q. Also, as always, species #l is assumed to have the laqest carrying capacity of the two. Thus, k > 1. We see from (4.1) that Czc > Cm and that as m increases, both standing crop values decrease. Fixing k,m,cr in (4.1), we see that as Q increases from zero, Cis, C&J increase, at first, but as Q exceeds l/2, they steadily decrease to values given by Q = 1 (or Q = 0), in (4.1)) all the while maintaining C&J > Crs. Thus, Cm, as a function of Q only, is concave down, positive, and symmetric about its maximum at Q = l/2. Now, we use the Volterra production equations, first at the two time-scale level of (3.1) and then at the aggregated level as follows. With Volterra (1936), let us write
:=ej
x;(t)
’
J
t
C; ds + X;(O),
for i,j = 1,2.
0
(4.2)
These four variables measure production. They are surrogate variables, generally, and may represent many aspects of coral production. Typically, Xj measures aragonite CaCOs of the skeleton of species i on patch j. Often we take it to be a positive multiple of the log biomass [l-3,17,18]. Th e constants f?j are per capita production rates and are patch dependent, but not species dependent. This simplification holds for closely related species, as when they are invertebrates of the same genus (say, Acmpom). In any case, for simplicity, we take ei = Ls = 1. Now define total production on patch #l and #2 as x’(t)
= x,‘(t)
+X,2(t),
X2(t)
= x;(t)
+X,2(t),
(4.3a)
and note that dX1
-=e1q dt
and
dX2 = e2c2, dt
(4.3b)
since Cl = C,l + CT, CZ = Ci + Ci. The variables X1 and X2 are the aggregated Volterra production variables. (We use the upper index to indicate patch number here.) In terms of X”, the aggregated trophodynamics for (3.3) decoupled from starfish becomes upon substitution of $$
= Ci, just the system d2X’ m -a2xT+al dt2 + m-l
dX1 dX2
d2X2 m -aldtdt+a2 dt2 + m-l
dX’ dX2
of course, m 2 3, here. Note, these equations are invariant under interchange of P and Q. This models the analytical trophodynamics &s follows.
130
P. ANTONELLI
et al.
a-Coral System with Social Interactions Now, we follow [2,3]. If the trajectories
of (4.4) are varied into nearby ones according to
2 = xi + d&T, one obtains for the first-order approximation in 67, the (KCC) equations convention on upper and lower repeated indices) d2ui -$ + g;$
ofvariation
(summation
(4.5a)
+ g;ru’ = 0,
where (4.4) is written as (4.5b) defining the two functions g’ and g2, while the semicolon denotes partial differentiation to T and a comma, that relative to X’. overator Dui -::=-
Kosambi,
relative
Cartan and Chern [5-71 introduced the
dui dt +9p,
dt
so that (4.5) can be written as a linear equation D2Ui
-
dt2
= Q’,
(4.7a)
where (4.7b) This is called the KCC-path deviation equation [3]. It contains the relevant curvature information in the eigenstructure of l$. We write the torsion tensor [3],
(4.8)
and then find the KCC-scalar
curvature K from the formula [2,3],
(4.9) where (4.10) is often called the ecological metric [2,3]. Antonelli and Shimada compute! the curvature K in [26]. We will spare the reader the difficult calculation
and merely record the answer here. Using the more convenient Ci for 5,
[(c2/c,)"-'
K=m(m-2).a2
4(m-1)2
*
-km-j2
~[(c2/cI>"+
l]
we have
(4.11)
(C2/G)2m-1
Clearly, K 2 0 for all values of Xi, Ci, i = 1,2. In fact, K > 0, except when &/Cl = k, which includes the unique positive equilibrium (Cal, C&2) of (4.1). Note, that K is invariant under interchanging P and Q.
Corals and Starfish Waves
131
Well-known results now guarantee the system (4.4) is stable in the sense of Liapunov [2,27]. This means that any trajectory is stable, or put another way, any two trajectories will have close (Xl(t), X2(t)) if initially (X1(0),X2(O)) and (s(O), g(O)) were close [27, p. 1751. Note, that for m = 2, K E 0, and we have a situation similar to straight lines in Euclidean space which are, indeed, unstable. We conclude that social interactions are, indeed, beneficial to the healthy coral community in the strong sense that production is more dependable (or predictable), robust and repeatible [2,27]. Let us further determine the influence of Q on the production stability K in (4.11). It is easy to compute $$ and @ to conclude that the graph of K as a function of Q only, is concave up, positive, and symmetric about its minimum at Q = l/2. That is, g
= 4a2 . He (P2 + Q”) (29 - 1))
and (4.12)
$$
= 16a2 . H . (3Q2 - 3Q + 1) ,
where H = K/a2 > 0, (i.e., G/C1 # k). Recalling that the graph of the standing crop value COO,as a function of Q only, is concave down, positive, and symmetric about its maximum at Q = l/2, we are able to correlate our relations. Thus, a smaller standing crop Cl0 corresponds with a larger stability of production K, and, conversely. The maximum Cl0 happens exactly when K is at its minimum (i.e., Q = l/2).
5. THE REACTION-DIFFUSION-TRANSPORT LARGE-SCALE WAVES
MODEL:
In the system (3.1), we now add diffusion in space and transport due to ocean currents in a way similar to the original model of Antonelli et al. [28], but at the microscale. Refer also to [29, Chapter 21. Letting A = & + wa’ denote the Euclidean Laplacian with z and y denoting latitude and longitude in a flat ellipse about 20 times long (2000 km) as wide and oriented with its major axis approximately in north-south directions, we replace E$$ in (3.1) with constant coefficient expression, 8C: act T_$?$ , --D;AC+T,“K& ( di! > ~9
in (3.1) with constant coefficient expression, - D;AC,z -T:‘=
ac:
-
,z$$ in (3.1) with constant coefficient expression,
and E!$
in (3.1) with constant coefficient expression,
Upon aggregation expression
of (3.1) to (3.3), the left-hand side 9 -a9 at
_ DfAcl
_ T;*z
is replaced by constant
_ T;‘!?$
coefficient
P. ANTONELLI et al.
132
and similarly, F
is replaced by constant coefficient expression,
ac, _2 at
D*AC
where D; = D;V;’
+ D2V2’ 1 1 7
D*2 = D;V,2* + D2V2’ 2 2 9 p* = TllVl’ + 7712v2* 1
1
1
119
+ 7712V2’ 2 1 9 + T22V2’ 1 2 7
T,2’ = T;lV;*
+ T;2V;*,
are the aggregated diflusion and advection coeficients. Also, F = Fl + FZ and E$$
and ~9
(5.la)
T,* = T;‘Vl’ 1 772’ = 7721V” 1 1 2
They are constants.
is replaced by
by
in the last two starfish equations of (3.1). Both of these are constant coefficient expressions. Therefore, aggregation leads to replacement of $ in (3.3) by the constant coefficient expression, dF --D;AF-T$!&T,;‘ ~, at
aY
where the advention coefficients for starfish are T3’ = ‘j-‘3lV* + ~32~8 1 1 1 1 27
(5.lb)
T3* = ,;lV* + T32V* 2 1 2 2.
According to [28], we make two general assumptions. For i = 1,2, 9 = $$ = $$ = 0 in dn x R+ (i.e., no Aux boundary conditions on fl for all positive time), where Z denotes the outward pointing normal vector to the boundary 80.
ASSUMPTION
A.
B. (Constant southward drift of the East Australian current.) in (x = constant) direction is nil. That is, ASSUMPTION
Average transport
T;’ = Trr2 = T,2l = Tf2 = e > 0, T,‘l = Ti2 = T;l
= Ti2 = 0.
One transforms the equilibrium state (Ccl, Cos, Fo) to the origin by setting
~~=c~-co~, i =
1,2,
fl=F-Fo.
Using Assumptions A and B, we can rewrite our reaction-diffusion
transport
model as (i = 1,2)
(5.2)
. Corals and Starfish Waves
Here, 2 = (f 1, f 2, f s ) are obtained
directly
133
from (3.3a)-(3.3c).
We refer the reader
more details on the precise form of s. In order to discuss the plane-wave solutions to (5.2), we now introduce
to [28] for
the phase-variable
2 = ct - a12 - azy, and set Vi(Z)
(i = 1,2)
= z;i,
and
Us(Z)
= F.
We substitute these into (5.2) and further require Vi to be 27r-periodic in 2. Such solutions, when they exist, are called plane wave trains (or plane waves). They do not satisfy Assumption A, however. The phase-velocity is the angular frequency C, divided by the wave number aI + ai E la’12. The vector a’ denotes the direction of the wave-front. Equations (5.2) become the vector equation (5.3) where 2; = C + aiC is the phase-shi,fed of diffusion constants. This transformation studied
by Kopell
and Howard
angular
frequency
reduces
and D is a 3 x 3 diagonal
the equation
matrix
(5.2) to the no-transport case
[30]. The phase-shiped velocity is C/]L?]~ with direction a’.
By known results [30], (E:I, (?z, @) is a plane wave train solution of (5.2) if and only if c(Z) satisfies (5.3); there is a family of wave trains parametrized by the wave length 27r/]a’] and that large values of this give good approximations to the socially dependent limit cycle of (3.3). These waves can have large amplitude and phase velocity, if m = 2 [28]. Note, that a’ depends on Q here, as does the phase-shifted
velocity
and angular
Recall, that in large-scale diffusion models, Great Barrier Reef, i.e., on R. This means that of all reefs of the Great Barrier Reef so that Q Q effects the wave length 27r/]a’] of the wave train
frequency.
What
is the meaning
of this?
we suppose that (3.3) holds everywhere on the our Gedankin experiment requires manipulation (and P) are defined. We see from the above that solutions. Obviously, ]a’] = (P2+Q2)161, so that
for Q increasing from zero to l/2, the wave length is steadily increased, while for Q increasing beyond l/2, towards unity, the wave length steadily decreases. The direction vector for the wave train a’does not change its direction as Q varies in [O,l]. The Conway-Hoff-Smoller Theory [31] was used in [28]. In the same way, we are able to conclude from CHS that any no-flux solution starting out at t = 0 in the stability torus set C of (3.3) converges, at rate B > 0, to its averaged value over the whole s2 and that for large times, this average is a good (uniform) approximation to the socially dependent limit cycle of (3.3). In particular, the large time behaviour of the wave solution is determined completely by the cycle and convergence time is relatively short. Therefore, for large times, the plane-wave train solution will well-approximate this wave away from the boundary of 0, as long as 27r/la’l is large enough. Prom the previous paragraph, we can ensure this by taking Q close to l/2. Now, because of the slow and more variable increase in coral abundance compared to the boom in starfish abundance [19], we expect COTS diffusivity 0: > Dz and 0: > 0: in (5.2). Letting d* = min{Dr, D;}, we have the formula for rate of convergence CT =
where b is some positive
constant,
$ -M
- ;(3b)‘i2
B is the length
> 0,
of the major
axis of R (i.e., N 2000 km), and
A4 is the Euclidean norm of the Jacobian matrix of 3 maximized over the stability torus C (attractor). The positivity of o means that d* must be large relative to R2 and relative to the rates of reaction defining M and to advection per unit length e/R. This must be assumed before any consequences of the CHS theory can be brought to bear on the model (5.2). These statements
134
P. ANTONELLI et al.
apply also to the wave train solution, which for us models the central portion of the Great Reef, at large times. Now, from (5.la), we have 0;
= D;P + DTQ,
Barrier
Df = D;Q + D,2P,
so that d* will depend on patch-difisiwities for each coral species, say of genus Acropora. Then, we can reasonably take 0: = Di 3 D1 and 0: = 02 G D2. If D2 # D’, then 0: = D’ + (D2 Dl)Q, D,* = D2 + (D1 - D2)Q and Q increasing will increase the minimum d* of D;, D;. Thus, we would take Q as small as possible. If D1 N D2, then there is no effect from Q in this Gedankin experiment. We refer the reader to [28] for more biological discussion. In particular, we refer to one of the main conclusions of [28], that large-scale synchronous outbreaks in the central Great Barrier Reef cannot occur if there is no net advection current (! = 0). Our intent here has been to explore
the role of Q in this previously
developed
theory.
In conclusion, our analysis shows that the best strategy of conservation of reefal communities is to optimize diversity between the patches, i.e., Q M l/2, when the model parameters cannot be completely known. This conclusion was reached in our previous paper [16], but here social effects among corals have been added to the model. In the case of lowest COTS densities, as happens every 12 to 15 years, but with opposite phase to COTS peaks, coral community production stability is at its lowest, but standing crop values are at their highest for Q = l/2.
REFERENCES 1. V. Volterra, Principles de Biologie Matht?matique, Acta Biotheo. III, pp. l-36, (1936). 2. P.L. Antonelli and R.H. Bradbury, Voltewa-Hamilton models in the ecology and evolution of colonial organisms, World Scientific Press Series in Math. Biology and Medicine, New York, (1995). 3. P.L. Antonelli, R.S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic, Dordrecht, (1993). 4. J.L. Harper, The Population Biology of Plants, Academic Press, New York, (1977). 5. D. Kosambi, Systems of differential equations of the second order, Quart. J. Math. (Oxford Series) 6, 1-12 (1935). 6. E. Cartan, Observations sur le memoire p&&dent, Math. Zeitschrift 37, 619-622 (1933). 7. S. Chern, Sur la gbmetrie d’un systbme d’equations differentialles du second ordre, Bull. Sci. Math. 63, 206-212 (1939). a. H. Wilbur, The ecological relationship of the salamander Ambystoma latemle to its all-female, Gynogenetic associate, Evolution 25, 168-179 (1971). 9. H. Wilbur, Competition, predation, and the structure of the Ambystoma-Rana Sylvatica community, Ecology 53, 3-21 (1972). 10. G.E. Hutchinson, A note on the theory of competition between two social species, Ecology 28, 319-321 (1947). 11. G.E. Hutchinson, An Introduction to Population Biology, Yale University Press, New Haven, CT, (1978). 12. P. Antonelli, R. Bradbury and X. Lin, On Hutchinson’s competition equations and their homogenization: A higher-order principle of competitive exclusion, Ecol. Mod. 60, 309-320 (1992). 13. P. Antonelli. R. Bradbury and X. Lin, A higher-order predator-prey interaction with application to observed starfish waves and cycles, Ecol. Mod. 58, 323-332 (1991). 14. N. Hairston, J. Allan, R. Colwell, D. Futuyama, J. Howell, M. Lubin, J. Mathias and J. Vandermeer, The relationship between species diversity and stability: An experimental approach with protozoa and bacteria, Ecology 49, 1091-1101 (1969). 15. J. Vandermeer, The competitive structure of communities: An experimental approach with protozoa, Ecology 49, 362-371 (1969). 16. P. Antonelli and P. Auger, Corals and starfish devastation of the Great Barrier ReeE Aggregation methods, Acta Biotheoretic 43, 481-493 (1995). 17. P.L. Antonelli (Editor), Proceedings International Workshop on Population Dynamics of Outbreaks, Mathl. Comput. Modelling 13 (6) (1990). 18. R. Bradbury (Editor), Acanthaster and the Corn1 Reef: A Theoretical Perspective, Springer-Verlag, Lecture Notes in Biomath, (1990). 19. P.J. Moran, The Acanthaster phenomenon, Ocennogr. Marine Biol. Ann. Rev. 24, 379-480 (1986). 20. P. Antonelli and N. Kazarinoff, Starfish predation of a growing coral reef community, J. Theor. Biol. 107, 667-684 (1984). 21. P. Antonelli, K. Fuller and N. Kazarinoff, A study of large amplitude periodic solutions in a model for starfish predation on coral, IMA J. Math. Appl. in Med. and Biol. 4, 207-214 (1987).
. Corals and Starfish Waves 22. 23. 24. 25. 26. 27. 28.
29. 30. 31.
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J. Pandolfi, Developmental sequences in colonial corals: An overview, Mem. Ass. Austdias Palaeontols 8, 69-81 (1989). J. Pandolfi, Heterochrony in colonial marine animals, In Heterochrony in Evolution, (Edited by M.L. McKinney), Plenum Press, (1988). D.J. Barnes, Growth in colonial scleractinians, Bull. Mar. Sci. 23, 280-298 (1973). P.M. Auger and R. Floussarie, Complex ecological models with simple dynamics: From individuals to populations, Acta Biotheoretica 42, 111-136 (1994). P.L. Antonelli and H. Shimada, On l-form Finsler connections with constant coefficients, Tensor, N.S. 50, 263-275 (1991). D. Laugwitz, Diflenmtial and Riemannian Geometry, Academic Press, New York, (1965). P. Antonelli, N. Kazarinoff, Ft. Reichelt, R. Bradbury and P. Moran, A diffusion-reaction-transport model for large-scale waves in Crown-of-Thorns starfish outbreaks on the Great Barrier Reef, ZMA J. Math. Appl. in Med. and Biol. 6, 81-89 (1989). in Hiemrchically Organized Systems, Pergamon Press, (1989). P.M. Auger, Dynamics and Thermodynamics N. Kopell and L. Howard, Plane wave solutions to reaction-diffusion equations, Studies in Appl. Math. 4, 291-324 (1973). E. Conway, D. Hoff and J. Smoller, Large time behaviour of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math. 1, 1-16 (1978).