A CONTINUOUS SIZE-STRUCTURED RED CORAL GROWTH MODEL

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1 Mathematica Modes and Methods in Appied Sciences c Word Scientific Pubishing Company A CONTINUOUS SIZE-STRUCTURED RED CORAL GROWTH MODEL P-E. JABIN UMR 1, J-A. Dieudonné Department, Parc Varose Nice Cedex, 1, France jabin@unice.fr V. LEMESLE UMR 5, COM, Campus de Luminy Case 91 Marseie Cedex 9, 13, France vaerie.emese@univmed.fr D. AURELLE UMR 5, COM, Campus de Luminy Case 91 Marseie Cedex 9, 13, France didier.auree@univmed.fr The aim of this paper is to derive and anayse the mathematica properties of a new continuous size-structured mode for red cora (Coraium rubrum, L.) growth. Since historica Lesie modes are often used to dea with some ecoogica probems, a new approach is here proposed and give some promising resuts. The main advantage of using continuous mode is that we hope to describe precisey the mass mortaity events, observed in Mediterranean sea, and its consequences on red cora dynamics. Simuations studies aow us to quaitativey discuss some questions about red cora popuations dynamics. The deveopment of this method shoud be usefu for the study of the conservation of red cora popuations. Keywords: Popuation dynamics; size-structured continuous mode; asymptotic anaysis; conservation bioogy. 1. Introduction The anaysis of the consequences of environmenta changes on the dynamics of natura popuations takes advantage of modeing studies. For exampe the deveopment of specific modes may aow to evauate the extinction risks of popuations under various hypotheses (see exampes in ). Such approaches may be particuary usefu for the study of ong ived marine invertebrates for which popuations dynamics may take pace at arge time scaes. Moreover the actua cimate change J-A. Dieudonné Department, University of Nice Sophia-Antipois Oceanoogica Center of Marseie, Mediterranean University Oceanoogica Center of Marseie, Mediterranean University 1

2 P-E. Jabin, V. Lemese & D.Auree and the increasing human impact raises important questions about the resiience and the conservation of the associated ecosystems, as in the case of cora reefs 7 or of Mediterranean hard-substrate communities 1. Some modes have aready been deveoped for some species in order to describe the growth and structure of individua coonies of tropica coras 1. These modes were based on growth rate, nutrient transport and the hydrodynamics of the fow with different structure (more or ess compact) for the cora. Other works deat with some branching mode using parameters and variabes reated to some specia moduar cora species 1, on the cacification phenomenon 1 or with the anaysis of tropica cora popuations dynamics 9,. Such modes are rare concerning Mediterranean sessie invertebrates such as the gorgonian red cora (Coraim rubrum, L.). This embematic and patrimonia species is heaviy expoited and has been affected by mass mortaity events in the summers 1999 and 3 aong the coasts of Provence. These events are probaby inked with cimate change 15 and their potentia repetitions might hinder the conservation of popuations of this ong-ived species with potentiay ow dispersa abiities 19. The main consequence of this mass mortaity was the variabe necrosis of some branches of the coony and their breaking. Moreover the expoitation of red cora for jeweery eads to the destruction or breaking of the argest coonies. The remaining popuation has thus an inferior size which may have a negative impact on popuation growth or persistence. A continuous mode of red cora growth is proposed in this paper taking into account possibe mass mortaity events and their consequences. The mathematica toos used for this mode are differentia systems for the whoe red cora popuation (i.e. the number of cora branches) n(t, ) depending on time t and on the size of branches. Continuous mode is here considered since historicay Lesie-Lewis matrix transition of popuation growth have been frequenty used 17 1 in ecoogy. This shoud enabe us to describe more precisey the mass mortaity and the impact on the red cora popuation dynamics. We start by modeing the probabiity of break-up of one cora branch by estimating the connection between the size of a branch and its diameter. This woud aow to take mass mortaities into account simpy by rendering the branches weaker (with a smaer diameter). This eads us to a precise but rather compicated mode which is simpified by assuming a ow probabiity of break-up. We then anayze the mode that we obtained both mathematicay and numericay.. The derivation of the mode.1. Modeing one cora branch Let us first consider ony one cora branch. It begins with a very short ength corresponding to one individua at time s. Its ength wi then increase unti the first break-up and its diameter reguary

3 A continuous size-structured red cora growth mode 3 increase too ring by ring (more or ess one by year 11 ). The size of this ring typicay depends on the externa conditions (an increase in temperature or ow food) meaning that at a particuar point of the branch ength the diameter wi aso typicay increase Denote by L(t, s) the ength of the branch and δ(t,, s) the diameter at distance L(t) from the beginning. We wi assume a inear growth of the branch (which seems reasonnabe) such that 3 L(t, s) = σ(t s), for t s. (.1) There is no need to be so specific for the growth rate of the diameter since as we said before, it may depend on externa conditions. Consequenty we take δ(t,, s) = δ + t s+/σ I(r) dr. (.) The integra is taken from s + /σ which is the time r such that L(r) =. The function of time I represents the externa conditions and in the simpest case woud be taken constant with δ(t,, s) = δ + I (t s /σ). (.3) Our main assumption concerns the probabiity of break-up of the branch at the point at a distance. This break-up coud be due to expoitation, necrosis caused by bacteria or a temperature increase as we said before. Focus on necrosis phenomenon, we assume that the fow of water around the branch is a shear fow and hence that the veocity of the water at a point of the branch is proportiona to. If the fow has a sma Reynods number, it is reasonnabe to take a drag force at this point proportiona to the veocity times the diameter. Of course it woud be iusory to try to compute exacty the distribution of mechanica strain aong the cora branch. First of a its mechanica properties are mainy unknown and then our assumptions on the shape of the branch and the forces acting on it are such that the resut woud probaby sti be far from exact. Instead et us make a very rough assumption, namey that the strain at one point is proportiona to the sum of forces appied on the part of the branch above L(t,s) δ(t,, s) d. This makes the strain vanish at the end of the branch ( = L) and it is maxima at the bottom = so it is not a together unreasonnabe. This shoud be mutipied by the corresponding physica constants but as this computation is ony heuristic, we wi not mention them. The resistance of the branch at is proportiona to d and we simpy assume that the probabiity of break-up is proportiona to the ratio between the strain and

4 P-E. Jabin, V. Lemese & D.Auree the resistance 1 p(t,, s) = α (δ(t,, s)) L(t,s) δ(t,, s) d. (.) After a branch broke at a position, it wi start to grow again, both in ength and in diameter. The new part of the branch wi however have an initia diameter of δ and (.) is therefore noonger true aong a the branch. Therefore consider a branch which broke k times: At time s 1 and position 1, then at time s > s 1 and position and so on unti time s k and position k. The times are of course increasing s 1 < s <... < s k but in fact we may aso assume that the positions are increasing. Indeed consider the case of a branch with a first break-up at s 1 and 1 and a second at s > s 1 and < 1. Its ength and diameter after s is exacty the same as a branch with ony one break-up at time s and position. Now taking 1 < <... < k with have for t > s k and L(t, s, s 1, 1,..., s k, k ) = k + σ(t s k ), (.5) t δ k (t,, s, s 1, 1,..., s k, k ) = δ + I(r) dr, for i < i+1, (.) s i+( i)/σ with the convention = and k+1 = L(t). Finay and according to the remark, we define δ k when 1,... k is not increasing as foows. We extract Σ(1),..., Ssigma(k ) such that this sequence is the ongest increasing sequence that may be extracted from 1... k. Then we pose δ k (t,, s, s 1, 1,..., s k, k ) = δ k (t,, s, s Σ(1), Σ(1),..., s Σ(k ), Σ(k )). (.7).. A first mode with many branches The first difficuty is to describe statisticay a very arge number of branches. Let us remark that these branches can describe more than ony one coony. Here, the tota number of branches considered in a parce is modeed. Then et us denote by n k (t,, s, s 1, 1,..., s k, k ) the density of coonies with tota ength at time t, being new at time s < t and having had k break-up at times s 1 <... < s k and at corresponding positions 1,..., k. Let us now write down the equations satisfied by the n k, starting by the one for n (t,, s) t n + σ n = (d + S (t, )) n (t, s, ), (.) where d is the death rate of a coony (assumed to be constant and independent of the ength or age of the coony to be simpe) and S (t, ) is the break-up probabiity of a branch of size with no previous break-up namey S (t, ) = α p(t,, t /τ) d, (.9)

5 A continuous size-structured red cora growth mode 5 with p given by (.) and δ by (.). In this case the age of the coony is simpy of course /τ. Note that if I does not depend on time, then S is aso independent of time. The equation for n k with k 1 is of course more compicated δ t n k + σ n k = dn k α n k (t,,...) d d k (δ k (t,, s, s 1,..., k )) + α n k 1 (t,, s, s 1,..., k 1 ) k δ k 1 (t,, s, s 1,..., k 1 ) d (δ k 1 (t, k, s, s 1,..., k 1 )) d (.1) The term d n k is the mortaity, the next term counts how many branches of n k breaks (and therefore go to another popuation). The third term in the right hand side takes into account the branches of n k 1 (having had k 1 break-up) which spits once more after their ast break-up position. The δ k are given by formuas (.) and (.7). These equations have to be suppemented by conditions on the border =, which are for k 1 n k (t,,...) =, (.11) as no new part of the coony may aready have had a break-up. For n the condition is n (t,, s) = δ(t s) B n k (t,, s, s 1, 1,..., s k, k ) k= <s <s 1<...<s k <t < 1<...< k < d 1... d k d ds ds 1... ds k. (.1) This represents a the new branches appearing in a parce, with a birth rate B which coud depend on a the parameters of the coonies. Notice that the Dirac mass δ(t s) simpy ensures that a new branch born at time t is indeed registered as such..3. A simpified mode The set of equations (.),(.1) is quite compicated and in addition there is an infinite number of them. This woud make studying or numericay compute soutions to this system quite chaenging. Instead we propose to derive a simpified mode by assuming that the initia diameter δ is much smaer than the average diameter some time ater. Consider the case of constant environmenta conditions and et us precise the range of vaues of the parameters for which this is correct. Take one branch with no break-up yet (or the part of the branch which did not spit yet), with ength L at time t; It was created at time t L/σ and so δ(t, ) = δ + I (L ). σ

6 P-E. Jabin, V. Lemese & D.Auree The probabiity of break-up at is then Assume that p = α (δ + I(L )/σ) L (δ + I σ (L )) d. δ << IL σ, (.13) and consider ony those such that L >> (σδ )/I, we obtain that p ασ I(L ) L (L ) d = ασ ( + L/). (.1) 3I The tota probabiity of break-up of the branch (or the ast part of it) is P 3ασ I L, and so the typica ength of the branch before break-up is I L ασ. Inserting this into (.13), one finay finds the condition δ < I3 ασ 3. (.15) Now, notice that the probabiity of break-up given by (.1) is increasing in, unti is cose enough to L (L of the order of σδ /I). So a branch wi typicay rather break-up at the end. Moreover consider a branch which aready had a previous break-up and which has a second one. The typica time apse between the two is of the same order than the time it took for the first break-up to occur. Hence the typica diameter of the first (or odest) part of the branch is twice the one of the youngest part, making it even more ikey that break-up wi occur on the youngest part. Therefore in the seque we aso assume that break-up aways occur on the youngest part of the branch. This enabes us to repace δ k in the equation (.1) on n k by δ(t,, L) given by t δ(t,, L) = δ + I(s) ds. (.1) t (L )/σ Define the tota popuation n(t,, s) by t n(t,, s) = n k (t,, s, s 1,..., k ) ds 1... ds k d 1... d k. k= 1,..., k = s 1,...,s k =s This function now satisfies the reativey simpe equation δ(t,, ) d d t n(t,, s) + σ n = dn α n ( δ(t,, )) + α n(t,, s) δ(t,, ) d ( δ(t,, )) d. (.17)

7 A continuous size-structured red cora growth mode 7 If the birth rate B depends ony on the size of the coony, the boundary condition becomes n(t,, s) = δ(t s) t B()n(t,, s ) d ds. (.1) In the case where the environmenta conditions are constant, (.17) is even fuy expicit and one does not need to take the difference between new part and od part of the coony (the age of the coony) into account, defining n(t, ) = t n(t,, s) ds, we have that (δ t n(t, ) + σ n = d n α n + I( )/σ) d d (δ + I( )/σ) + α n(t, (δ + I( )/σ) d ) (δ + I( )/σ) d. 3. Mathematica study (.19) We are ed to study the foowing type of mode, where K(, ) is the probabiity of a branch of size to break in a branch of size < and S() the break-up rate t n + n = (d + S())n(t, ) + + K(, )S( )n(t, )d n(t, ) = + b()n(t, )d n(, ) = N () (3.1) where b() is the recruitment rate for a coony of size and d > a constant environmenta death rate. Since K(, ) is a probabiity density, we have: + K(, )d = 1, and since the probabiity of a branch of size to give a onger branch is nu: K(, )d = 1. We need some technica bounds on the various coefficients, namey we assume b, K, S are C 1 functions δ, C, s.t. b() C e δ, im inf b() >, C, k s.t. S() C (1 + ) k, K(, ) C (1 + ) k. (3.) These assumptions are quite reasonnabe as they are satisfied by about every coefficients we woud ike to consider. Note that if the environmenta conditions depend on time, both S and K shoud depend aso on the time variabe. As we mainy study the permanent regime of (3.1), we keep things simpe and forgot for the moment about the extra dependency.

8 P-E. Jabin, V. Lemese & D.Auree 3.1. Permanent regime To study the system (3.1), we first ook for specia soutions corresponding to popuations with a constant shape and a constant rate of increase (or decrease) in time. They are of the form n(t, ) = e γt n γ (). The function n γ () satisfyies the corresponding stationary equation: n γ = (d + S() + γ)n γ + K(, )S( )n γ ( )d, (3.3) together with the normaized condition n γ () = 1 = > b()n γ () d normaized condition. (3.) The normaized condition shoud determine the vaue of the parameter γ (Mathus parameter). In fact one has Proposition 3.1. Assume (3.). There exists a unique γ and a unique n γ such that (1 + b()) n γ L 1 (R + ), for any, S() K(, )n γ () L 1 (R + ) and n γ is a soution to (3.3) and satisfies (3.). The mathematica anaysis of this probem is very simiar to the one deveopped in 13 for inear modes of popuations structured by age. Here the size pays the roe of the so-caed maturation veocity. There are of course some minor differences (our kerne K is not symmetric in and for exampe) but not enough to justify doing again and thoroughy the anaysis. We therefore ony sketch the severa steps and refer the reader to 13 for detais. Cassica methods 5 give a condition for the existence of bounded soution n γ to (3.3) (with sufficient decay at infinity in order to define > K(, ) S( )n γ ( ) d, which we wi precise ater). Namey we require γ > σ where σ is such that for a ( ) d + S() 1 K(, )d σ > < In our case as K(, ) d = 1, which is the conservation of the number of coonies, integrating (3.3) of the whoe R +, we find (d + γ) n γ () d = 1. Hence σ = d, and we have an estimate on the L 1 norm of n γ. Now define B(γ) = b()n γ () d. There exists a unique γ such B(γ) = 1 and thus (3.) is satisfied.

9 A continuous size-structured red cora growth mode 9 Notice first that B is decreasing in terms of γ. In fact n γ itsef is decreasing in γ; Indeed if γ 1 > γ, then denoting u = n γ1 n γ u = (d + S() + γ 1 )u + K(, )S( )u( )d + (γ 1 γ ) n γ, > and as (γ 1 γ ) n γ is non positive and u() =, by the maximum principe one has u. It is even possibe to be more precise. Define v = e (γ1 γ) n γ1, then v = (d + S() + γ 1 )v + K(, )S( ) e (γ1 γ)( ) v( )d > (d + S() + γ 1 )v + K(, )S( ) v( )d. > By the maximum principe on n γ u, we hence get for γ 1 > γ n γ1 () e (γ1 γ) n γ (). As b is dominated by an exponentia and n γ L 1 for every γ, this impies that B(γ) is finite for γ arge enough. It aso shows that B(γ), as γ. Note that this exponentia decay is aso enough to define K(, ) S( ) n γ ( ) d. It ony remains to prove that B(γ) as γ d. Of course n γ () d = 1 d + γ +, as γ d. On the other hand n γ aso satisfies ) n γ () = e (1 D() γ + e D( )+γ K(, )S( )n γ ( )d d > ( ) ) min(, = e (1 ) D() γ + e D( )+γ K(, )d S( )n γ ( )d. with D() such that D() = d + S(). Hence n γ () e D() γ (1 + S( )n γ ( ) d ). (3.5) Now fix L arge enough such that b() c > for a L. As γ d either L n γ or L n γ. In the second case as B(γ) c L n γ() d, we deduce immediatey that B(γ).

10 1 P-E. Jabin, V. Lemese & D.Auree In the first case we see that for any L n γ () e D() γ (1 + S() L n γ ( ) d ) and so n γ () + for any L as γ d which is enough to show that B(γ) = b()n γ () + as b is bounded from beow on L. Concuding B is decreasing and maps ] d, + [ onto ], + [. Therefore there is a unique γ such that B(γ ) = 1. Note that formua (3.5) may be used to obtain n γ through a fixed point argument, the operator ( ) min(, ) T n = e D() γ e D( )+γ K(, )d S( )n γ ( )d, satisfying (d + γ + S()) T n() d, S() n() d. 3.. Existence of soution for the time-dependent system (3.1) Obtaining soutions to (3.1) ony requires enough decay in on the initia data in order to define K(, )S( )n( ) d and the birth term. Therefore we assume that with φ non decreasing, continuous and satisfying φ() n () d <, (3.) We then have φ, φ C φ, φ() S() +, φ() b() +. (3.7) Proposition 3.. Existence : Assume (3.), (3.) and (3.7). There exists a unique n L ([, T ], L 1 (R + )) for any T with φ() n(t, ) d L ([, T ]), n(t =, ) = n () and n is a soution in the sense of distributions to (3.1). This is an easy consequence (stabiity through approximation) of the two a priori estimates d dt n(t, ) d = d n(t, ) d + b() n(t, ) d,

11 A continuous size-structured red cora growth mode 11 together with d dt φ()n(t, ) d = d φ()n(t, ) d + φ() b() n(t, ) d + φ () n(t, ) d + n(t, ) S() (φ( ) φ())k(, ) d d d φ()n(t, ) d + φ () n d + φ() b() n(t, ) d. As φ is asymptoticay arger than b, these two estimates provides a bound for any finite time for n d and φ n d. This is enough to hande every term in (3.1) Convergence of the time-dependent soution toward the permanent regime We foow again 13 to prove that n(t, ) converges towards βe γt n γ. For simpicity we assume that γ = (this can be ensured by rescaing every soution in time by e γt thus changing d in the equation). First define an auxiiary function H() that soves the dua probem to (3.3): Proposition 3.3. Assume (3.). There exists a positive H C 1 (R + ) soution to H = (d + S())H S() K(, ) H( )d b() H() = 1, H() n γ () d <. Proof. There is of course a unique soution H to H = (d + S())H S() K(, ) H( )d b(), (3.) with H() = 1. But a priori nothing garantees that H stays non negative for exampe. For any γ, note that H(L)n γ (L) =n γ () γ + L H() L L H()n γ ()d L b() n γ () d K(, )S( )n γ ( )d d. Taking γ = γ =, one first gets the positivity of H by H(L) n γ (L) >. On the other hand, for γ >, etting L, one finds Now γ n γ is a soution to γ n γ = (d + S() + γ) γ n γ + H() n γ () d = 1 b()n γ () d. γ > K(, )S( ) γ n γ ( )d n γ,

12 1 P-E. Jabin, V. Lemese & D.Auree and hence for any γ > d, b() γ n γ () d is uniformy bounded. Consequenty the imit of H()n γ () d as γ is finite and H()n γ () d <. Now we may state Theorem 3.1. Assume that b() for amost a. Assume that (3.) hods and (3.), (3.7) are satisfied for the initia data n with H()/Φ() as. Assume finay that for a constant C, n () C n γ (). Let n(t, ) the soution to (3.1) given by Proposition 3.. Let β > be such that H() n () d = β H()n γ () d. Then as t, n(t, ) β n γ () converges to in L 1 oc (R +). Proof. Let us denote u(t, ) = n(t, ) βn γ (). Then of course by inearity of the system, u is a soution to (3.1). Now mutipying by a reguarization of sign(u) and passing to the imit, we get t u + u (d + S()) u + S( ) K(, ) u(t, ) d. ( ) u(t, ) b() u d min b() u + d, b() u d, with u + and u the positive and negative parts of u. Mutipying by H() and integrating one gets ( ) d H() u(t, ) d H (d + S())H + S() H( )K(, ) d u(t, ) d dt +H() u(t, ). As H satisfies (3.), we get ( d H() u(t, ) d min b() u + d, dt ) b() u d. This shows that H u d has a imit and that as t, either u + or u converges toward pointwise (thanks to b > a.e.). Moreover the same computation aso shows that d dt H() u(t, ) d =. Notice aso that the contro on the decay of n and H()n γ is good enough to make a the previous steps rigourous.

13 A continuous size-structured red cora growth mode 13 By the choice of β, we deduce that H() u(t, ) d =. Finay as n n γ then by the maximum principe n(t, ) Cn γ and therefore H() u(t, ) is uniformy integrabe in time. The imit of H() u d is equa to im H()u d. Therefore H() u(t, ) d, which aso impies the convergence of u toward.. Numerica simuations.1. The stationary soution Theorem 3.1 says that the behaviour of the soution is governed by the stationary one. The behaviour of this ast one in terms of the break-up rate S() (which is typicay modified by the environmenta conditions), is however not so easy to investigate theoriticay. So instead we performed some numerica simuations Popuation Popuation Size Size Fig. 1. Stationary soution of mode 3.1. The break-up rate S() = and S() = /1. We proposed to study the different cases taking the break-up rate S() = and S() = /1. The very cassica Euer scheme is used in oder to perform the simuations but since the integra term of equation (3.1) does not begin in zero we have to integrate this equation in reverse time. For the other parameters vaues, we take K(, ) = 1/ and d =.1 which seems reasonnabe compare with the avaiabe data. Since a ot of numerica simuations were performed with different initias conditions and since the same resut is obtained, we can assume that probaby the stationary soution is gobay asymptoticay stabe. Fig 1 shows the evoution of the stationary soution of equation (3.1). One can see that the more the break-up rate is important (i.e. S() = ) the more the death

14 1 P-E. Jabin, V. Lemese & D.Auree of the big size coony is important. The popuation is then mainy constituted of sma size coonies (ess than size with S() = or size 1 with S() = /1). This confirms the bioogica observation made in... Evoution of the compete soution Some simuations of the compete mode are then proposed in order to evauate quaitativey the demographic consequences of protection rues. To perform these simuations, the cassica Euer discretization in time and in size is used Size 1 Size Size. Popuation Popuation Size Size, Size Size Popuation Popuation Size Size Fig.. evoution of a the size cass. Break-up rate S() = and S() = /1. Fig shows the evoution of the initia mode without no new hypotheses. This confirm the simuation of the stationary soution since we obtain the same imit vaues for both simuations for a given size. Then Fig 3 iustrates the possibe protection of the coony. At time 5, we take a break-up rate S() equa to zero in order to simuate popuation without mortaity due to necrosis or expoitation. One can see that the behavior of the soution is aso governed by the stationary one. In this particuar case this stationary soution can be expicitey (and easiy) computed. One can compute this soution and find that it is proportionna to exp( d) which can be aso verified in Fig 3. More bioogicay speaking, Fig 3 shows the maintenance of the highest coonies after some times depending on the break-up rate. The more the break-up rate is important the more the recuperation of the arge coonies is ong. This quaitative resut seems to confirm the possibe conservation of the popuation

15 A continuous size-structured red cora growth mode 15 but it confirms aso the very ong time to obtain this preservation considering the bioogica vaues of the growth rate Size 1 Size Popuation Popuation Size Size Size 1,3.5 Size, Popuation Popuation Size Size Fig. 3. evoution of different size cass. Protection at time T = 5. Break-up rate S() = and S() = /1 In Fig we take S() = for t = 5 to t = 15 and S() = (or S() = /1) otherwise. This represents the protection of the coony during ten years. Once again, the behavior of the soution is governed by the stationary one. A discontinuity in the break-up rate S() is enabed to destabiize the behavior of the soution. This means that the mode is very robust to perturbations and it confirms in the probaby goba convergence of the system. More bioogicay speaking, this quaitative resut coud describe the expoitation phenomenon. Indeed during the period of protection the big size coonies reappear then a period of expoitation remove a this new big size coonies and ony the sma size remain aive. This resut can aso describe a mass mortaity event due to the temperature increase after ten years without any changes and one can see that the big size coonies are a affected. So this enightens the necessity of managing this species on ong time scaes.

16 1 P-E. Jabin, V. Lemese & D.Auree Size 1 Size Size Popuation Popuation Size Size, Size Size Popuation Popuation Size Size Fig.. evoution of different size cass. Protection at time T = 5 and Mortaity event at time T = 15. Break-up rate S() = and S() = /1 References 1. M. Abbiati et a, Harvesting, predation and competition effects on red cora popuation, Neth. Journa of Sea Research. 3 (199) 19.. Y. Artzy-Randrup et a, Size structured demographic modes of cora popuations, Journa of Theoretica Bioogy. 5 (7) L. Bramanti et a, Recruitment eary surviva and growth of the Mediterranean red cora Coraium rubrum (L 175), a -year study, Journa of Exp. Marine Bioogy and Ecoogy. 31 (5) H. Caswe, Matrix popuation modes, nd Edition, Sinauer Associates, (1). 5. R. Dautray and J-L. Lions, Anayse mathe matique et cacu nume rique pour es sciences et es techniques, Masson, (19).. J. Garrabou et a, Mass mortaity event in red cora Coraium rubrum popuations in the Provence region (France, NW Mediterranean), Marine Ecoogica Prog. Series. 17 (1) Hughes et a, Cimate Change, Human Impacts, and the Resiience of Cora Reefs, Science. 31 (3) D.B. Lindenmayer et a, How accurate are popuation modes? Lessons from andscapescae tests in a fragmented system, Ecoogy etters. (3) D. Lirman, A simuation mode of the popuation dynamics of the branching cora Acropora pamata Effects of storm intensity and frequency, Ecoogica modeing. 11 (3) Z. Kizner et a, Growth forms of hermatypic coras: stabe states and noise-induced transitions, Ecoogica Modeing 11 (1) C. Marsha et a, A new method for measuring growth and age in the precious red

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