Modelling allelopathy among marine algae

Ecological Modelling 183 (2005) 373–384

Modelling allelopathy among marine algae

Jordi Solea,∗, Emilio Garcıa-Ladonaa, Piet Ruardijb, Marta Estradaa

a Institut de Ciencies del Mar (CSIC), 08039 Barcelona, Spainb Royal Netherlands Institute for Sea Research, PO Box 59, NL-1790 AB Den Burg, Texel, The Netherlands

Received 30 September 2003; received in revised form 29 June 2004; accepted 13 August 2004

Abstract

Allelopathy among microalgae has been modelled in order to fit the dynamics of mixed cultures ofChrysocromulina polylepisandHeterocapsa triquetra, carried out with three different initial concentrations of the allelopathic speciesC. polylepis. Theexperimental data have been analysed with a simple Lotka–Volterra type model including an allelopathic term. Based on thepopulation dynamics of the two species in unialgal and mixed cultures, the model has been simplified and solved analytically.The best fit of the allelopathy parameter (γ ≈ 10−5 to 10−6) provides a good agreement between the theoretical curve andthe experimental data for the two highest initial concentrations ofC. polylepisbut, interestingly, shows a less accurate fit forlow initial concentrations of the toxic alga. Following the same procedures, a modified model, in which the allelopathic effectis dependent on the square of the concentration of the toxic alga, has been adjusted to the experimental data. This modifiedmodel presents a good fit for all the range of initial concentrations of the toxic alga. These results allow a quantification of thestrength of the allelopathic interaction between two marine phytoplankton species. The growth curve of the non-toxic alga iss supportsfi ped,b©

K

1

drf

,assgan-andmfulde

ongnd-to

0d

ignificantly affected by the allelopathy term only after the toxic alga has reached relatively high concentrations. Thiseld observations suggesting that, at least in the case ofC. polylepis, allelopathy may be important once a bloom is well develout is not likely to be a key factor in initial phases of the proliferation.2004 Published by Elsevier B.V.

eywords:Marine algae; Toxicity; Allelopathy; Mixed cultures; Fitting

. Introduction

Algal proliferations are part of the normal planktonynamics in aquatic environments and tend to presentecurrent seasonal patterns linked with environmentalorcings such as mixing conditions of the water column

∗ Corresponding author.E-mail address:[email protected] (J. Sole).

and nutrient availability(Franks, 1997). Sometimesalgal populations may reach unusually high biomlevels, with deleterious consequences for other orisms in the aquatic ecosystem or for human healtheconomy. These events are broadly known as haralgal blooms (HABs) and their effects, which inclufish kills and poisoning of shellfish, may have a strimpact on human activity and well being. Understaing and prediction of HABs is of crucial importance

304-3800/$ – see front matter © 2004 Published by Elsevier B.V.oi:10.1016/j.ecolmodel.2004.08.021

374 J. Sole et al. / Ecological Modelling 183 (2005) 373–384

prevent their undesired consequences(Hallam and deLuna, 1984; Vila, 2001; Garces et al., 2002).

The study of fluctuations in phytoplankton abun-dance has been addressed by a variety of numericalmodels(Steele and Henderson, 1981; Fasham, 1995;Truscott, 1995; Baretta and Baretta-Bekker, 1997;Recknagel et al., 1997). In this context, the particu-lar problems posed by marine HABs may help to un-derstand more general aspects of plankton dynamics.Some approaches to HAB modelling focus on the en-vironmental changes that may trigger bloom devel-opment. For example,Franks (1997)assimilated thedevelopment of HABs to the dynamics of excitablemedia(Truscott, 1995)in which the growth rate pa-rameter is a complex and evolving function of time.Other works have focused on complex ecological mod-els devoted to simulate a particular aquatic ecosystem(Yamamoto et al., 2002)or on the development of ex-pert systems to predict HABs events(Recknagel et al.,1997). Apart of processes such as cell growth, graz-ing and dispersion, addressed in most phytoplanktonmodels, a relevant aspect of some marine HABs is thatthey are caused by species (described hereafter as “al-lelopathic” or “toxic”) which are capable of producing“allelopathic agents” directly affecting other microal-gal species(Hallam et al., 1983; Arzul et al., 1999).

The term “allelopathy”, coined by Molisch (1937)(cited by Rice, 1984) was defined byRice (1984)as the effect of one plant on the growth of anotherthrough the release of chemical compounds (calledat andn e oftf icalc in ac ofc thy,cb1 ns h inv ul,11 99;S 002;V ,t and

their role in marine aquatic ecosystems remain poorlyunderstood.

From a theoretical point of view, one of the firstmathematical representations of allelopathic interac-tions was proposed byMaynard-Smith (1974). Thisauthor considered a two species Lotka–Volterra compe-tition model and introduced a term to take into accountthe effect of a toxic substance released at a constant rateby one species and being nocive to another. Improve-ments and refinements of this model have followedmany directions with applications, including reaction–diffusion equations, to analyse allelopathy in bacterial(Nakamaru and Iwasa, 2000)and plant communities(seeDubey and Hussain, 2000; An et al., 2003andreferences therein). Concerning allelopathy in aquaticphytoplankton several models have examined and gen-eralised the classical Lotka–Volterra dynamics pro-posed byMaynard-Smith (1974). An exhaustive anal-ysis of stability of a general two-species competitionsystem including allelopathic interactions was carriedout by Chattopadhyay (1996). The effects of a timedelay of the growth of two competing phytoplanktonspecies with allelopathic effects on each other was ad-dressed byMukhopadhyay et al. (1998). Further devel-opments of this basic model to include effects of envi-ronmental fluctuations together with spatial and tem-poral heterogeneity in plankton populations have beenrecently studied(Tapaswi and Mukhopadhyay, 1999;Mukhopadhyay et al., 2003).

In spite of such existing theoretical developments,m al-l at-u triale died,t llel-le aly-s itho thicm sys-te ati-c mi-c tudeo andu s.

ef-f im-

llelochemicals byWhittaker and Feeny, 1971) intohe environment. This includes both positiveegative effects, but many authors restrict the us

he term to negative effects (Inderjit and Foy, 1999). Inact, allelopathy is part of a whole range of chemommunication mechanisms between organismsommunity. A recent review of different aspectshemical ecology of microalgae, including allelopaan be found inCembella (2003). Allelopathy haseen observed in diverse ecological systems (seeRice,984; Inderjit and Einhelling, 1995) and has beehown to occur among marine algal species botitro and in situ(Chan et al., 1980; Gentien and Arz990; Nielsen et al., 1990; Maestrini and Graneli,991; Myklestad et al., 1995; Uchida et al., 19chmidt and Hansen, 2001; Tillmann and John, 2ardi et al., 2002; Fistarol et al., 2003, 2004). However

he chemical nature of allelopathic compounds

odelling studies attempting to quantify the role ofelopathic interactions in culture experiments or in nral phytoplankton dynamics are scarce. In terrescosystems, where allellopathy has been better stu

here have been efforts to apply models including aopathy to real systems (ECOTONE model, seeGosleet al., 2001). These authors present a sensitivity anis of the model, but without direct comparisons wbservational data. Direct applications of allelopaodels to phytoplankton populations in marine eco

ems are even more scarce (one can be found inUchidat al. (1999)). The adequacy of the classical mathemal representation of the allellopathic interaction inroalgal population models and the order of magnif the parameters involved, are still open questionsnknowns for most experimental or natural system

The aim of this article is to model allelopathicects among marine microalgae and to quantify the

J. Sole et al. / Ecological Modelling 183 (2005) 373–384 375

portance of this interaction using experimental data.This approach is intended as a first step towards imple-menting such kind of interaction into more complexand general ecological models and to analyze the im-pact of allellopathy as a factor determining the develop-ment of marine HABs. We will begin by applying thegeneral two-species competition model proposed byChattopadhyay (1996). The formulation of allelopathyused in this model is of a phenomenological Lotka–Volterra type. The allelopathic term depends on theproduct of the square of the concentration of the tar-get species by the concentration of the toxic species(Maynard-Smith, 1974). In other words, ifP1 is thepopulation density of a non-toxic alga andP2 that ofa toxic one, the mathematical form of the allelopathicinteraction term isγP2

1P2, whereγ is the allelopathicparameter.

The experimental data used in this work weretaken from a laboratory study in which 15 speciesof marine phytoplankton were exposed to suspensionsof Chrysocromulina polylepis(Schmidt and Hansen,2001). In 1988, this species was responsible for a toxicbloom in Scandinavian waters which caused extensivefish kills (Maestrini and Graneli, 1991; Edvarsen andPaasche, 1998). Although no experiments were carriedout at that time, observations of moribund accompany-ing phytoplankton species at the peak of the bloom andsome later experiments suggested thatC. polylepisre-leased allelochemicals that could affect algae and otherplanktonic organisms(Nielsen et al., 1990; Myklestade ityS tod eso ct ofct loh ciestc s re-d nglyd ut thec toe

w ra-tt

In Section 3, we describe, simplify and solve analyti-cally the mathematical model for representing the allel-lopathic interaction according to the monoculture ex-periments. InSections 4 and 5, we describe the proce-dure for fitting to experimental data and the estimationof the allelopathic parameter according to a classicaland a slightly modified model. Finally, we discuss andsummarise the main results.

2. Allelopathic effect betweenC. polylepisandH. triquetra

For completeness, the most relevant aspects of theexperiments reported bySchmidt and Hansen (2001)will be summarized here. In these experiments, a pop-ulation of the non toxic dinoflagellateH. triquetrawassubjected to the influence of different cell concentra-tions of a toxic flagellate,C. polylepis. In mixed cul-tures of both species, after an initial phase of expo-nential growth ofH. triquetraduring the first 5 days,the dinoflagellate cells started to lose motility and thepopulation decreased (Fig. 1). When the initial concen-tration ofC. polylepiswas increased, the behavior wasqualitatively similar but the population ofH. triquetradecreased earlier. Senescent cultures ofC.polylepisap-peared to lose toxicity, but this effect occurred at longerculture times than those considered here.

In order to work with equivalent units, the cell con-centration data used inSchmidt and Hansen (2001)w bicm1

C

w ella tF n( on-t2u

cul-t o-g thes on-c and

t al., 1995). In order to further explore this possibilchmidt and Hansen (2001)exposed several algaeense suspensions ofC. polylepisand designed a serif experiments using batch cultures to test the effeell concentration and growth phase ofC. polylepisonhe dinoflagellateHeterocapsa triquetra. The generautcome of these experiments was thatC. polylepisad a harmful effect on all except one of the 15 spe

ested. In mixed cultures withC. polylepis,H. triquetraells became non-motile and population growth wauced. Of course, allelopathic interactions are stroependent on the particular species considered, base ofC. polylepismay serve as a starting pointxtend and generalise results to other species.

The paper is then organised as follows. InSection 2e summarise briefly the main findings of the labo

ory studies ofSchmidt and Hansen (2001)concerninghe allelopathy betweenC. polylepisandH. triquetra.

ere transformed into milligrams of carbon per cueter (mg C/m3) using the relationship(Verity et al.,992)

= 0.433× V 0.863 (1)

hereC is the cell concentration given in pg of C/cndV is the cell volume given in�m3. Remark thaigs. 1 and 2are redrawn fromSchmidt and Hanse2001) in the new units. The calculated carbon cent was 312.3 pg C/cell forH. triquetra (cell volume:050�m3) and 30.8 pg C/cell forC. polylepis(cell vol-me: 140�m3).

As can be clearly appreciated in the monoure results (Fig. 2), both species exhibited a listic behavior that saturated 10–14 days aftertart of the cultures. Even at the maximal cell centrations reached, nutrients were in excess

376J.S

oleetal./E

cologicalModelling183(2005)373–384

Fig. 1. Growth ofC. polylepis(a) andH. triquetra (b) in mixed cultures for different initial concentrations ofC. polylepis: (�) an initial concentration of 2000 cells/ml ofC.polylepis, (�) 5000 cells/ml and (�) a 10 000 cells/ml (fromSchmidt and Hansen, 2001).

Fig. 2. Growth ofC. polylepis(a) andH. triquetra (b) in monocHansen, 2001).

ultures for initial concentrations of 2000 cells/ml ofC. polylepisand 10 cells/ml ofH. triquetra (from Schmidt and


Schmidt and Hansen (2001)concluded thatC.polylepiswas neither N nor P-limited. It was not clear what lim-ited the growth ofC. polylepis, although the authorssuggested that it could have been pH or CO2 availabil-ity. The toxicity ofC. polylepiscould be affected bypH and perhaps by CO2 limitation, but these factorsdid not influence the results of the experiments withH.triquetra in the time interval considered in our study.

The carrying capacities for the different speciescan be estimated from the saturation concentrationsobserved in the monocultures. Thus, the initial andsaturation values forH. triquetra in the new unitsare 3.1 mg C/m3 (10 cells/ml) and 31227.0 mg C/m3

(105 cells/ml), respectively. ForC. polylepis, thereare three initial concentrations, 62.0 mg C/m3

(2000 cells/ml), 154.0 mg C/m3 (5000 cells/ml)and 308.0 mg C/m3 (10 000 cells/ml), and the sat-uration concentration is around 9240.0 mg C/m3

(3 × 105 cells/ml).

3. Modelling the experimental allelopathicsystem

As a starting point to model the behavior ofmixed cultures, we adopt the general two-species com-petition system with allelopathic terms studied byChattopadhyay (1996), based on the previously pro-posed formulation ofMaynard-Smith (1974). In thesema

wr ec-o cc sc pa-t hici -p f al-l es as er isp

In this model, the saturation level or “carrying ca-pacity” of a population growing in isolation can be de-fined as the quotient between the specific growth andthe intraspecific competition coefficient. Other modelsapply a mechanistic approach and consider explicitlythe consumption of the limiting resource or resources,assumed to cause the intra and inter-specific competi-tion effects among the coexisting populations. Often,growth rates are parameterised using a Monod’s typefunction of the amount of resource. Such models havebeen applied, for example, to develop resource compe-tition theory and to test the role of different types ofinteractions, including allelopathy, in determining sta-bility and biodiversity in multispecies systems(Grover,1997; de Freitas and Fredrickson, 1978). Ideally, amechanistic formulation of competitive interactions in-volving the relevant limiting resources should be supe-rior to models involving a “carrying capacity” throughthe use of Lotka–Volterra type competition coefficients.However, in practice, lack of the relevant informationoften precludes the use of a explicit formulation of thelimiting resources.

All the parameters are positive constants exceptγi, which can be positive (inhibitory) or negative(stimulatory). The stability analysis of such a system(Chattopadhyay, 1996)indicates that an inhibitory al-lelopathic term has some stabilizing effect on the two-species competitive system. In the case of an inhibitoryaction (γ1 andγ2 positive,Mukhopadhyay et al., 1998),which is the one interesting here, the introduction of at ededf oxics abil-i

d in( theo , oneo n ofr , rel-e .F r-i ssinga del( tivei oura

ents( as

odels, the temporal evolution of the populationsP1ndP2 follows the expressions

dP1

dt= P1[k1 − α1P1 − β12P2 − γ1P1P2],

dP2

dt= P2[k2 − α2P2 − β21P1 − γ2P1P2] (2)

herek1, k2 are the growth rate parameters,α1, α2 theate of intra-specific competition of the first and snd species, respectively,β12, β21 are the inter-specifiompetition parameters andγ1, γ2 are the coefficientharacterizing the allelopathic interaction (or allelohy coefficients). The formulation of the allelopatnteraction as (γP2

1P2), although non-linear, is the simlest mathematical way to satisfy the concept o

ellopathy interaction in which one species producubstance toxic to the other, but only when the othresent.

ime delay in the system, representing the time neor toxic cells to mature and be able to release the tubstance into the medium, does not modify the stty properties of the coexistence point.

As discussed above, the type of model describe2) indicates that one species influences itself andthers, but not how these effects occur. Presumablyf the interaction mechanisms will be consumptioesources needed by both populations. Howevervant data to implement a mechanistic model (e.gdereitas and Fredrickson, 1978) are lacking in the expe

ments considered here and in most of those addrellelopathic interactions in microalgae. As our mo2) is basically used as a tool to explore the relamportance of the allelopathic term, we think thatpproximation is warranted.

As can be seen from the outcome of the experimFigs. 1 and 2), the growth curve of the toxic alga w


similar for the monocultures and all the mixed culturesfor different initial concentrations. Therefore, we cansuppose that the competition term affecting the toxicalga (P2) can be neglected. In the case of the non-toxicalga (P1), a series of numerical simulations (data notshown) for values ofβ of a factor of 10 higher andlower than the intra-specific competition coefficients(i.e. between 10−3 and 10−6), indicated that the effectof the competition term was only significant after a timeperiod longer than the experimental time. Taking intoaccount these considerations we may putβ12 = β21 =0 and considering that there is only one toxic species,γ2 = 0. Thus Eq.(2) may be simplified to

dP1

dt= P1[k1 − α1P1 − γ1P1P2],

dP2

dt= P2[k2 − α2P2] (3)

This system may be solved analytically because theequation forP2 is decoupled fromP1. The equation forP2 is the classical logistic growth equation the solutionof which is

P2(t) = 1

(a − c) e−bt + c(4)

and wherea, b, andc are constants defined as

a ≡ 1, b ≡ k2, c ≡ 1 = α2

H -u city( ate(b owni

i ea

w ffi-c g,1 allyb

1/P1(t). Thus, Eq.(5) becomes

du(t)

dt= α + γ

(a − c) e−bt + c− ku(t). (6)

This equation is now a first-order linear one the gen-eral solution of which can be formally written as(Gradshteyn and Ryzhik, 1980),

u(t) = A0 e−kt + α

k+ γ e−kt

∫ t

0

e(k+b)ξ

(a − c) + c ebξdξ,

and by making the changez ≡ ebξ in the integral, wearrive at

u(t) = A0 e−kt + α

k+ γ

e−kt

b

×∫ z=ebt

1

zk/b

(a − c) + czdz (7)

where

A0 ≡[u(t = 0) − α

k

]

The integral term of the right-hand side of(7) canbe written in terms of the hypergeometric function2F1(a′, b′, c′; z) (Abramowitz and Stegun, 1972), as

u(t) = α

k+ e−kt

sen-t er am viora etricf tlyl

P0 Ps k2

ere,P0 is the initial concentration andPs is the satration (maximum) concentration or carrying capaPs ≡ ki/αi), which only depends on the growth rki) and the intraspecific competition coefficient (αi)ut not on the initial cell concentration values, as sh

n the monocultures ofC. polylepis(Fig. 1a).Now, from(4)we can substituteP2 in (3)and rewrit-

ng k1, α1 andγ1 ask, α andγ, respectively, we arrivt

dP1

dt= P1

[k − αP1 − γ

(a − c) e−bt + cP1

](5)

hich is a non-linear equation with variable coeients. This is a Bernoulli equation(Bender and Orsza978) that can be linearized and solved analyticy making the following change of variable,u(t) =

× A0 −γ2F1[(b + k)/b, 1, 1 + (b + k)/b,

− c/(a − c)]

(a − c)(b + k)

+ γ e(b+k)t

(a − c)(b + k)2F1

[b + k

b, 1, 1

+ b + k

b,−c ebt

a − c

] (8)

This is a rather complicated expression but, esially, has an initial decreasing behaviour and aftinimum value, has a dominant exponential behattenuated by the last term where the hypergeom

unction dominates, growing to infinity, at sufficienong times (t → ∞).


Fig. 3. Growth ofH. triquetra for different initial conditions ofC. polylepis. Solid, dashed and pointed lines for theoretical values obtainedwith the model given in Eq.(8). Experimental values for an initial concentration ofC. polylepisof 2000 cells/ml (�), 5000 cells/ml (�) and10 000 cells/ml (�).

4. Fitting the experimental observations

A direct fit of the experimental results of mixed cul-tures to the analytical expression given in(8) requiresadjustment of a number of parameters and is very sen-sitive to the numerical procedure used. Our approachwas to simplify the general model of(2) to first adjustthe values ofαi andki for each species from the results

Table 1Estimated values ofk andα from monocultures and mixed experi-ments ofC. polylepis

Initial concentrations (cell/ml)

Monoculture Mixed cultures

2000 2000 5000 10000

k 0.45 0.49 0.46 0.39α 8.02× 10−5 9.86× 10−5 9.32× 10−5 7.89× 10−5

Table 2Estimated values ofk andα from monocultures and mixed experiments ofH. triquetra

Initial concentration (×10 cell/ml)

Monoculture Mixed cultures (initialC. polylepisconcentrations (cell/ml))

2000 5000 10000

k 0.59 0.53 0.53 0.57α 1.89× 10−5

of monoculture experiments (Fig. 2). Afterwards, weused the final solution of(8) to fit γ. Because of thenon-linear character of(4), we guessed the values ofαi

andki during the initial exponential growth phase andsubsequently, we performed a non-linear regression bymeans of a least squares technique(Press et al., 1992).Iterations were continued until the chi-square changeswere less than 1.0 × 10−3. The parameter values ob-tained from the regression are listed inTables 1 and 2.Note that they differ from those proposed bySchmidtand Hansen (2001)because these authors made an esti-mate based on a pure exponential growth model, whichis different from the logistic one adopted here. In fact,the exponential behavior lasts only for a very short timeafter the beginning of the experiment.

In order to estimateγ and fit(8) to the experimen-tal data ofH. triquetra in mixed cultures (Fig. 1), weapplied the inverse transformation of the cell concen-


trations. In this way, only a linear fitting ofγ is neededfor each initial concentration ofC. polylepis. Thebest values found wereγ = 2.42× 10−6, γ = 6.97×10−6, γ = 2.19× 10−5 for initial concentrations of2000, 5000 and 10 000 cells/ml, respectively (standarddeviations: 6.79× 10−7, 9.50× 10−7, 4.99× 10−6).The solution(8)and the experimental results are shownin Fig. 3, where we represent the growth curves of thenon-toxic alga for the same initial concentration of thisalga in mixed cultures with different initial concentra-tions of the toxic alga. As can be seen in the figure,the analytical solution of the model reproduces qual-itatively the experimental data, although, for an ini-tial concentration of toxic algae of 2000 cells/ml, thetheoretical solution with fitted parameters fails to ad-equately reproduce the time at which the populationstarts to decrease (it occurs 1 day sooner in the model)and the corresponding rate of decrease.

5. Modified model

Although there is a rather good general qualitativeagreement between the model given by fitting(8) andthe experimental data, there are some discrepanciesfor low initial concentrations of the toxic species. Thisfinding is of more than theoretical interest here becauseof the often low cell concentrations in the real marineecosystems. The observed discrepancies suggest thatthe allelopathic effect should be a non-linear functiono , ino ayt tionc de-p ism thict

ialm

P tion,t

u(t) = α

k+ e−kt

×

A0 −

γ ′2F1[(2b + k)/b, 1, 1 + (2b + k)/b,

−c/(a − c)]

(a − c)2(2b + k)

+ γ ′ e(2b+k)t

(a − c)2(2b + k)2F1

[2b + k

b, 1, 1 + 2b + k

b,−c ebt

a − c

] (10)

Fitting analogously as in the previous case, the fol-lowing values are obtained for the allelopathic pa-rameter,γ ′ = 7.82× 10−10, γ ′ = 6.97× 10−6, γ ′ =2.23× 10−5 for 2000, 5000 and 10 000 cells/ml, re-spectively, with standard deviations of: 4.87× 10−10,9.50× 10−7 and 5.02× 10−6. As can be appreciatedin Fig. 4, there is a better agreement between the theo-retical solution and the experimental data for low con-centrations. For high and intermediate concentrations,the adjustment is similar to that found in the previouscase, but the new solution reproduces satisfactorily thetime at which the population of the non-toxic alga startsto decrease and the rate of decrease for the low con-centration case.

An interesting result is the dependence of the fitteda ′ -t esfr co-i st s. Ap xedc het indo ffecthb

6

thers ail-a e be-

f the amount of toxin present in the medium orther words, of the amount of toxic cells. A simple w

o make the model more sensitive to the concentrahanges of the toxic alga is to introduce a quadraticendency onP2. From a modelling point of view, thisathematically equivalent to consider the allelopa

erm asγ ′P21P2

2.Taking into account this modification in the init

odel, the equations to be solved become

dP1

dt= P1[k1 − α1P1 − γ ′P1P

22],

dP2

dt= P2[k2 − α2P2] (9)

roceeding in a similar way as in the previous seche analytical solution can be found as

llelopathic parameterγ on different initial concentraions ofC. polylepisand the finding that it decreasor lower initial concentrations ofC. polylepis. Thiseflects the fact that the concentration of toxic cellsnciding with the falling-off ofH. triquetradecreases ahe initial concentrations of the toxic alga decreaseossible explanation is that the non-toxic cells in miulture with a relatively low initial concentration of toxic cells have the possibility to develop some kf resistance to the allelopathic substance. This eas been observed and reported for the diatomDitylumrightwellii (Rijstenbil and Wijnholds, 1991).

. Discussion

One of the objectives of this work was to test wheimple allelopathy models could be fitted to avble experimental data to represent adequately th


Fig. 4. Growth ofH. triquetra for different initial conditions ofC. polylepis. Solid, dashed and pointed lines for theoretical values obtainedwith the modified model (Eq.(10)). Experimental values for an initial concentration ofC. polylepisof 2000 cells/ml (�), 5000 cells/ml (�) and10 000 cells/ml (�).

haviour of real populations. In a simplified way, the dy-namics of a mixed microalgal culture was approachedusing a Lotka–Volterra type model(Maynard-Smith,1974). The results of laboratory experiments with atoxic and a non-toxic microalgal species suggestedthat inter-specific competitive effects could be ne-glected. In this case, without competition and withonly one alga with allelopathic activity, growth of thetoxic alga is uncoupled from that of the non toxicone and an analytical solution can be found for themodel.

The proposed model, which was the simplest onethat could reproduce the behaviour of the system, wasvalidated a posteriori using experimental data(Schmidtand Hansen, 2001)from mixed cultures ofC. polylepis(allelopathic) andH. triquetra (non-allelopathic). Thenecessary parameters were fitted in two steps. First,the parameters affecting the pure logistic part of thedynamics were estimated by using data obtained frommonocultures of both algae. Then, the allelopathic pa-rameter was fitted to the analytical solution using mixedculture data. Although qualitatively satisfactory, thisinitial model was not able to represent the experimen-tal behavior for the case of low initial concentrationsof the toxic cells, a situation which is characteristicof pre-proliferation states in marine ecosystems and isvery relevant in natural conditions.

To overcome this problem, we modified the for-mulation of the allelopathic interaction, introducing aquadratic term on the toxic alga concentration. Thischange was based on the observation that the allelo-pathic effect appeared to be a non-linear function ofthe amount of toxic cells in the medium. The resultingmodel gave a satisfactory fit to the data for the wholerange of experimental cell concentrations.

The good agreement between model prediction andexperimental data lead to conclude that, at least for theexperiments presented here, the allelopathic interac-tion can be adequately represented by the modified termproposed in this work.Uchida et al. (1999), used a sim-ilar formulation to represent intra-specific competitionbetweenHeterocapsa circularisquamaand Gymno-dinium mikimotoi, but they included only a term like(P1P2) to represent all the interactions among bothspecies. However, in their experiment, there were mu-tual inhibitory effects among the two algae (althoughof different intensity) which were strongly dependenton the initial cell densities and appeared to be causedmainly by direct cell contact. In the case presented here,the toxic effects ofC. polylepiswere attributed to therelease of toxins into the surrounding water(Schmidtand Hansen, 2001).

The mathematical expression of the allelopathic in-teraction included in the two equation model adopted


here can be introduced in more complex multispecificmodels as a additional term expressing allelopathic ef-fects between phytoplankton species. This approach isdifferent from that leading to mechanistic models ofallelopathy such as those proposed for living plants interrestrial environments(Inderjit and Foy, 1999; An etal., 1993, 2003).

As we noted inSection 1, our purpose was to eval-uate the importance of an experimentally observed al-lelopathic interaction by means of a model adjusted toa set of data. In this sense, our model was intendedbasically to reproduce the general features of the stud-ied experiments. In nature, the dynamics of interactingspecies may be affected by many factors not repre-sented in a simple model. For example, the toxicityof C. polylepisappears to be affected by pH, growthphase(Schmidt and Hansen, 2001)and cellular N:Pratio (Johansson and Graneli, 1999), although thesefactors did not affect the actual experiments consid-ered here(Schmidt and Hansen, 2001). In general, ex-pression of toxicity within the genusChrysocromulinaappears to be highly variable, even within the samespecies(Edvarsen and Paasche, 1998; John and Till-mann, 2002). Other complications could arise fromecophysiological variability related to the affected or-ganisms. Thus, in our case, the decrease of the fittedallelopathic parameter for decreased initial concentra-tion of the toxic cells suggests the possibility of someadaptation by the non-toxic alga. Another possibilityis that increasing concentrations of allelopathic toxini iaa 1;H db oxinc nt oft

lo-p lso w ofm -b ase,n bes n oft d ont iricald tools( ucha the

mechanisms leading to allelopathy, but by allowing amathematical formulation and a numerical estimate ofthe allelopathic interaction coefficients, they may beuseful to constrain such interactions in more complexecological models.

To test the importance of allelopathy in a natu-ral situation, we have carried out some simulations toimplement an scenario that simulated the conditionsduring theC. polylepisbloom of 1988 in Skagerrak-Kattegat(Maestrini and Graneli, 1991). This has beendone with the ERSEM ecosystem model (EuropeanRegional Seas Ecosystem Model,Baretta and Baretta-Bekker, 1997; Ebenhoh et al., 1997; Ruardij et al.,1997) which solves many of the biological and chem-ical processes in a water column including a benthicsystem. We have added the allellopathic interactionfor C. polylepisusing expression (9) and have maderuns of 60 days, considering realistic temperature, lightconditions and nutrient concentrations(Maestrini andGraneli, 1991), to simulate the effect of different ini-tial concentrations of the toxic alga on a generic groupof non-toxic flagellates (with growth rate 1.4 day−1).The benthic model was switched off and initial con-centrations of the zooplankton functional groups wereset to zero so that the system did not have a graz-ing pressure. Results are not shown but in all casestested, that is, for several different initial concentra-tions of the toxic alga, the growth of the non-toxicalgal group was only affected when the concentra-tion of the toxic alga reached around 3000 cells/ml.T om-p int sup-pw p-u yC ent.O -s f-fi ef-f1

7

re-p ions

n the medium may inhibit further toxin production vfeed-back mechanism(Freedman and Shukla, 199allam and de Luna, 1984). This kind of problem coule approached by analyzing the dynamics of the toncentration itself as another separate componehe system(Dubey and Hussain, 2000).

The strongly non-linear formulation of the alleathic term in(9) is reminiscent of theoretical modef drug-receptor interactions which, based on the laass action, postulate thatn molecules of drug comine with each receptor (Riggs, 1963). In this c= 2 and the drug (or toxin) concentration may

upposed to be proportional to the concentratiohe toxic alga. However, although equations basehese assumptions appear to be useful for empescriptions, they cannot be taken as explanatoryRiggs, 1963). In a similar way, numerical models ss those applied here do not provide insight into

hus, allelopathic interactions between algal cetitors did not appear to play a significant role

he initial phases of the bloom. These resultsort the interpretation ofMaestrini and Graneli (1991),hich suggested that while still at relatively low polation density (104 cells/l), the toxin produced b. polylepismay have acted as a grazer repellnly later on, whenC. polylepisexceeded cell den

ities of 106 cells/l, its toxin (or toxins) became suciently concentrated to produce an inhibitoryect on its algal competitors(Maestrini and Graneli,991).

. Concluding remarks

In this work, we have used a simple model toroduce experimental data on allelopathic interact


between two phytoplankton species and have provideda two step procedure to quantify the parameter ofthe allelopathic term. The same methodology couldbe applied to other experimental data sets. Our re-sults indicate that the allelopathic effect betweenC.polylepisandH. triquetra is only important for highpopulation densities (>106 cells/l) of the toxic cells.In other words, the allelopathic term becomes of thesame order as the intra-specific competition term andthe allelopathic interaction starts to affect the popu-lation of the non toxic alga only when the bloom ofthe toxic alga is already well developed. However, asfound in the ERSEM simulations, allelopathic effectscould be important for inhibiting competition once asufficiently high cell concentration has been reached.It is difficult to generalise the results found here be-tweenC. polylepisandH. triquetra, to those whichhave been reported among other marine algae. How-ever, the general observation that detrimental effectstend to appear at concentrations of the toxic alga around106 cells/l suggests that, in general, allelopathy amongphytoplankton should not be a critical factor in bloominitiation.

An important step for the future should be tointegrate laboratory experiments, field observationsand modelling, in order to validate mathematical ex-pressions of allelopathy and to obtain numerical in-formation for application into more complex eco-logical models. Both, theoretical and experimentalapproximations are needed to ascertain the impor-t Bp

A

eanC lop-m t ofS n 3u i-o anc ul toP atau them celV att

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Modelling allelopathy among marine algae